protists , the grouping of eukaryotic microorganisms that encompasses such diverse entities as flagellated and ciliated protozoa ( e.g. , _ euglena _ , _ paramecium _ ) and motile green alga ( e.g. , _ chlamydomonas _ , _ volvox _ shown in fig . [ fig : protists ] ) , constitute an important class of organisms in the study of evolutionary biology , biological physics , and , recently , biological fluid dynamics . many flagellated protists display swimming behavior that is inherently three - dimensional ( 3d ) .a number of important questions in biology and physics are associated with how the motion of such organisms is related to their body plan and to external stimuli such as light , dissolved molecular species , gravity , temperature , boundaries , and electromagnetic fields .it is thus desirable to track their position and orientation in 3d with high spatio - temporal resolution and , unless desired , free from systematic bias introduced by external stimuli , background fluid motion , and hydrodynamic surface effects .the first apparatus able to track microorganisms in 3d was designed for bacteria and utilized an analogue feedback loop that moved the microscope stage to keep a bacterium centered in the field of view .larger microorganisms such as protists require larger sample chambers , reaching millimeters or even centimeters in depth to avoid boundary - induced hydrodynamic effects . in this regime , methods based on a moving microscope stage are not suitable , as they induce uncontrolled background fluid motion in the sample chamber .similar considerations enter the study of the millimetric nematode _ c. elegans _ crawling on the surface of agar , for which a moving substrate introduces unwanted mechanical stimuli . instead, the camera itself can by moved by motors controlled by an algorithm that dynamically centers the worm in the field of view .several apparati for tracking the 3d motion of microscopic particles without a moving stage have emerged in recent years , yet none is ideal for studying the swimming of protists .methods based on controlled defocussing of particles , placing a cylindrical lens in the imaging optics of the microscope , or measuring the deflections of a laser beam that is focused close to a particle all suffer from a small tracking range along the optical axis . observing a sample which is illuminated from the side with a continuous gradient of color in order to color - codethe third dimension suffers from low spatial resolution in that dimension , and may provide a photostimulus to protists .tracking objects with a confocal microscope is only possible when the objects move at very low speeds .digital in - line holography may also be used for 3d particle tracking , yet even vibrations with amplitudes  m of components along the optical path lead to a time - varying background in the hologram that can significantly degrade the signal of the moving object .two protists whose swimming motion is of interest in this work . ( a ) _ chlamydomonas reinhardtii _ ( scale bar  m ) and ( b ) _ volvox carteri _ ( scale bar  m ) . ]  the difficulties mentioned above can be overcome by using more than one camera to observe synchronously the sample chamber from different angles and then combining the images to yield 3d tracks of particles . yet, existing implementations of this technique can not be easily adapted to track microscopic objects ; they either have a controlled but undesirable shear flow , have not integrated any control of thermal convection , or have been implemented with a temperature gradient across the sample ( k / cm ) to eliminate thermal convection .the latter method is an important advance , but may introduce an unwanted behavioral stimulus to protists .  herewe present an apparatus that uses two identical imaging assemblies at right angles to each other that can be used in dark- and bright - field illumination , combined with systems to deliver photo - stimuli , to control temperature inside the sample , and to eliminate background fluid motion in large sample chambers ( up to cm ) filled with aqueous growth medium , by homogenizing the temperature inside the sample chamber to millikelvin precision .the apparatus uses almost entirely off - the - shelf components and requires minimal expertise in optics .online supporting material includes software to control the hardware and to perform 3d tracking .we discuss the resolution and limitations of the apparatus , present swimming trajectories of the protists _ chlamydomonas _ and _ volvox _ in 3d , and illustrate with dual - view particle imaging velocimetry ( piv ) the flow field that _ volvox _ generates near a surface .the 3d tracking system , shown schematically in fig . [ fig : apparatus ] , is based on a flexible but powerful imaging system , phototactic stimulus lights , and equipment to control and homogenize the temperature inside the sample chamber .these three elements are now explained in detail .the imaging system is comprised of two identical assemblies that are mounted at right angles on a vibration isolation table ( science desk , with mm breadboard , thorlabs , ely , uk ) .a monochrome firewire ccd camera ( pike f145b , allied vision technologies , stadtroda , germany ; pixels , each  m , maximal frame rate of fps , and support for external triggering ) was attached to each of the two microscopes ( infinivar cfm-2/s , infinity photo - optical , boulder , co ) .these are continuously focusable with a working distance between mm and , yielding a maximum magnification of at the smallest available working distance . to allow a variable working distance , the camera/ microscope assemblies were mounted on sliding rails ( prl-12 , newport corp ., irvine , ca ) via standard post / post support hardware .the horizontal rail was attached directly to the breadboard while the vertical one was attached to a movable and lockable rail carrier on a large optical rail ( x95 , newport corp . ) mounted vertically to the optical breadboard . the outer chamber ( see fig . [fig : apparatus ] , and details below ) limits the smallest working distance to mm , yielding a magnification of .while sufficient to image protists of size  m with dark - field illumination , such organisms are better visualized with an additional magnifier lens ( 2xdl , infinity photo - optical ) .this increases the working distance and the depth of field , as discussed below .schematic drawing of apparatus .the outer chamber ( light blue , lid and flange in grey ) contains a water bath , fed through inlets and outlets ( purple ) , and mixed with a magnetic stir bar ( white ) driven by a motor external to the tank ( not shown ) .the sample chamber ( white ) is suspended by stainless steel holders ( light green ) , and illuminated by annular led arrays ( red ) .microorganisms are visualized with two long - working - distance microscopes ( dark gray ) equipped with ccd cameras ( dark green ) .phototactic stimulus is provided by two led and lens assemblies ( yellow ) , and controlled by shutters ( orange ) . ]the imaging system can be used with dark- or bright - field illumination : bright - field can be advantageous for organisms larger than  m as it captures more details of the organism ( _ e.g. _ , the body axis ) ; dark - field can be desirable when the organism is so small that only center - tracking is possible , as under these conditions it yields a better signal - to - noise ratio .the flexibility to use both illumination techniques is obtained by using an annular led array ( lfr-100-r , ccs inc . ,kyoto , japan ) as the ( unpolarized ) light source for each microscope .dark - field illumination is achieved when the field of view of the microscope only includes the dark region in the center of the led annulus .bright - field illumination is achieved by inserting a diffuser plate ( bulk frosted acrylic cut to size , rs components , uk ) in between the led array and the outer chamber , as far from the led array as possible .we chose the color of the led array to be narrowband red ( nm , nm bandwidth ) as it has been shown that this color does not trigger a phototactic response in motile green algae .for phototaxis studies , two opposing light sources are required in order to observe reproducible light - induced u - turns .the photo - stimulus lights were two broad spectrum cool white luxeon leds ( mwled , thorlabs ) , collimated with a short focal length lens ( mm , la1027 , thorlabs ) , and mounted to a shutter with millisecond precision ( sh05 , thorlabs ) .the shutter / led assemblies are mounted via standard cage hardware ( thorlabs ) to posts on rotating stages ( rp01 , thorlabs ) , allowing the direction of the light to be controlled .the stimulus direction was typically chosen to be horizontal in order to avoid an additional gravitational stimulus .this choice forces the common axis of the two cameras to be horizontal , along the stimulus direction .the beam diameter was controlled with an iris ( sm1d12 , thorlabs ) to illuminate only those faces of the sample chamber perpendicular to the common axis of the two cameras , thereby avoiding reflections , and the resulting unclear stimuli , from the other four faces .the two ccd cameras and shutters for the photo - stimulus lights were controlled with labview including the add - on toolbox ni - imaq for 1394-ieee cameras ( national instruments , austin , tx ) , allowing precise synchronization of image acquisition .a labview program is part of the supporting online material .  in order to combine the images from both cameras to yield 3d swimming tracks, it is crucial that both microscopes operate at the same magnification ( i.e. the same working distance ) .this is easily achieved to sufficient precision by replacing the sample chamber with a tilted calibration ruler that is observable through both cameras , and adjusting the working distance until the field of view of both cameras has the same physical size .after this calibration step , the microscopes need to be aligned along their common axis such that the field of view of both cameras contains the same section of the common axis .this axis alignment greatly improves the ability to reconstruct 3d tracks , as explained in section [ sec : improcess ] .  to obtain swimming trajectories that are not influenced by hydrodynamic surface effects or background flows , it is desirable to have a sample chamber that is as large as possible , while maintaining the fluid within it perfectly still .a stationary fluid can only be obtained if the temperature in the chamber and of the chamber walls is very homogeneous , thereby eliminating thermal convection caused by heating from the two led arrays ( each led array consumes w ) .for a closed chamber , the critical rayleigh number above which thermal convection starts to occur is , where is the thermal expansion coefficient of the fluid , is the gravitational acceleration , is the thermal diffusivity , is the kinematic viscosity , and is the length scale across which there is a temperature difference  . while the precise value of depends on the geometry of forcing and on boundary conditions , the scale of temperature differences involved for water ( k , /s , /s ) is roughly mk for a one centimeter length .the largest chambers that have previously been temperature - homogenized below the thermal convection threshold , under comparable conditions to those presented here , have cm in sedimentation studies .the system presented here eliminates thermal convection in chambers as large as cm , implying temperature differences between faces of the chamber to be below mk .decay of convective motion in the inner sample chamber .curves show the root - mean - squared velocity within the chamber , obtained by piv , as a function of time for three different chamber dimensions : blue - cm , magenta - cm , green - cm .inset shows the time constant for each decay as a function of chamber size , consistent with the diffusive scaling . ]  recognizing that in stokes flow the effects of boundaries at a distance from compact objects acted upon by gravity fall off as , and given that swimming trajectories can easily sample a vertical scale that is times the organism diameter , the sample chamber should be times the organism diameter for surface effects to remain below the 5% level .a chamber of this size thus allows protists as large as  m to be studied with negligible hydrodynamic surface effects .  before explaining how the temperature in the sample chamber is controlled and homogenized , it is necessary to give details of the outer and sample chambers .the outer chamber has dimensions cm , two inlets and two outlets ( as shown in fig .[ fig : apparatus ] ) and is made from mm thick borosilicate glass ( custom made by fine glass finishers ltd ., great chesterford , uk ) .the flange and lid of the outer chamber were custom made out of pvc with a cnc machine .the three types of sample chamber used were ( i ) custom cuvettes made from standard microscope slides cut with a tungsten glass scriber ( lac-450-a , fisher scientific , uk ) and glued together with uv - curing optical glue ( noa68 , norland products , cranbury , nj ) , cured in a uv chamber ( elc-500 , electro - lite corporation , bethel , ct ) , ( ii ) non - standard commercial glass cuvettes ( vitrocom , mountain lakes , nj ) , and ( iii ) standard glass cuvettes . the sample chamber is held rigidly in the center of the outer chamber by four thin stainless steel rods ( mm diameter ) .each rod was inserted into a corresponding tight - fitting hole in the lid and fixed with a set screw , allowing easy modification of the holding arrangements for the different chambers .it is well - known that the flagella of protists such as _ chlamydomonas _ have a strong tendency to stick to glass surfaces .as cells swim around the chamber they inevitably collide with the chamber walls .to avoid any problem with sticking , the glass is coated with pdms ( sylgard , dow - corning , belgium ) and etched for minutes in a plasma cleaner ( femto , diener electronic , nagold , germany ) , following a published protocol .  the temperature in the sample chamber is controlled by cycling filtered water through the outer chamber from a large tank that contains a submersed thermostatic heater ( jger eheim , deizisau , germany ) and pump ( eheim , deizisau , germany ) . in order to homogenize the temperature in the sample chamber ,the pump is switched off , and strong rare earth magnets ( ep200 , e - magnets uk ) are spun at rpm by a sturdy motor ( 178 - 5112 , rs components , uk ) on the outside of the outer chamber , thereby moving a teflon - coated cm magnetic stir bar inside the outer chamber at the same speed .this stirring evens out the temperature within the outer chamber and , if the stir bar is spun at the appropriate speed and place in the outer chamber , moves the water past the faces of the sample chamber without setting up recirculating vortices on the faces . by injecting inexpensive tracer particles ( size  m , pliolitevtac - l , eliokem , villejust , france ) into the outer chamber in order to visualize the flow across the faces of the sample chamber , we found that the arrangement drawn in fig .[ fig : apparatus ] can homogenize the temperature inside the sample chamber below the threshold .the flows within the sample chamber were quantified with commercial piv software ( flowmanager , dantec dynamics , skovlunde , denmark ) . as a metric for the extent of flows throughout the chamber , we report the r.m.s . velocity obtained by uniform averaging over the field of view of one camera .figure [ fig : convection ] shows that decays exponentially as a function of time after the stirring of the water bath has begun , with a time constant that depends on the smaller chamber dimension .it is straightforward to see when the temperature difference falls below the threshold for convection , as tracer particles that are used to visualize the convective flow in the sample chamber will suddenly begin to fall out of the fluid at approximately their stokes sedimentation speed , forming a sedimentation front that propagates downward . for  m latex beads ( c37259 , invitrogen , carlsbad , ca ) , the stokes sedimentation speed is  m/s .this serves as a lower bound for in fig .[ fig : convection ] . we expect to arise from viscous dissipation , and thus to scale as . in waterthis yields times on the order of a few to ten minutes for in the range cm , consistent with the data in fig .[ fig : convection ] .each curve conforms well to a single exponential decay , and the fitted times obey well the expected quadratic scaling as shown in the inset to fig .[ fig : convection ] , with a value not far from that expected from the kinematic viscosity of water .the 3d tracking was done by analyzing the image sequences from each camera separately , giving a set of two two - dimensional ( 2d ) tracks , and combining suitable tracks from these two sets to yield 3d trajectories .modified matlab ( mathworks , natick , ma ) versions of freely available particle tracking routines written by j.c .crocker and d.g .grier were used for the 2d tracking , allowing many organisms to be tracked at the same time . to track organisms that are so small that they appear circular and without internal structure in dark - field images ( e.g. , _ chlamydomonas _ ) , no modifications to the original versions of the code need to be made . to track extended objects ( e.g. , _ volvox _ ) the routines that identify the center of the object need to be modified in a way that depends on the shape and structure of the object . for bright - field images of the spherical _ volvox _ , all internal structure of _ volvox _was removed by histogram equalization , followed by spatial band pass filtering .the resulting image was then convolved with a binary disk - shaped kernel , yielding an image in which the centroids of peaks correspond to the _ volvox _ centers in the original image .colony carries daughter colonies inside it ( see fig . [fig : protists ] ) , which are fixed in the posterior hemisphere and therefore act as convenient markers of the body axis .the axis of _ volvox _ can thus be determined by finding the vector between the geometric center of _ volvox _ and the center of brightness of the daughter colonies inside _ volvox _ for both directions and then combining these two vectors to obtain a 3d axis .the modified code also allows additional information for each _ volvox _ to be gathered , such as the orientation of the body axis .  to identify two 2d tracks that are suitable for synthesizing into a 3d track ,the two sets of 2d tracks were compared along the common axis of the field of view of the two cameras .consider one of the possible combinations of two 2d tracks .even if these two tracks are projections of positions of a single organism , the tracks usually do not completely overlap in time , because during the course of a long track the signal from the tracked object may fall below the tracking threshold so that the object ` drops out ' of the tracking data .this means that only the time - overlapping sections of each track can be compared . because of the precise alignment of the field of view of both cameras ( see sec .[ sec : experimentalapp ] ) , a decision upon whether the two 2d tracks are from the same organism can be made by finding the r.m.s .difference between the position - coordinate along the common axis .the two 2d tracks for which this value is minimal ( and below a certain threshold ) are then synthesized into a 3d trajectory .code that can perform all the operations described above is part of the supporting online material .the tracking precision of the apparatus has been determined by the standard method of observing fluctuations in the tracked position of particles that are fixed between two cover slips .the precision was tested at the minimum working distance the apparatus allows ( 60 mm , corresponding to magnification ) , for two different types of objects . for monodisperse 10 m latex beads imaged in dark - field illumination ,the uncertainty in the position was found to be  m , if the magnifier lens is mounted to the microscope . for objects in the size range 425 - 500 m ( _ volvox _ fixed with iodine ) , in bright - field illumination , the uncertainty was found to be  m without the magnifier lens .  the performance of the apparatus is determined not only by the uncertainty in spatial position , but also by the overall volume in which 3d tracking can be performed .this volume is set by the depth of field of the microscope . for the purpose of simply tracking the center of a spherical object, the object can be out of focus as long as the signal in the image intensity profile is sufficiently large. therefore the trackable depth " ( td ) , which we define as the depth in which the signal / noise , is a more suitable measure for the trackable volume than the depth of field .the td is strongly dependent on the object size and on the signal ( dark - field illumination gives a larger signal ) .we measured the td for 10 m beads in dark - field to be mm ( mm ) at a working distance of mm ( mm ) .imaged in bright - field , fixed _ volvox _ of size 425 - 500 m have td mm at the minimum working distance of mm .a limitation of the tracking software presented here is that the concentration of organisms in the sample chamber should not be so large that swimmers overlap frequently in the 2d images from each camera .as the tracking software can not distinguish overlapping objects , a high concentration of swimmers would result in very short 2d tracks , and therefore less accurate synthesizing of 2d tracks to a 3d track .an alternative method for obtaining 3d tracks from the images of more than one camera is to determine the 3d position of every particle at every time point .this approach is often taken in 3d lagrangian particle tracking , and can handle larger concentrations of trackable objects , but is usually implemented with at least three cameras in order to reduce frequent ambiguities in the 3d particle identification , and requires an elaborate calibration . this 3d tracking method could be implemented with the apparatus described here , if one of the stimulus lights is replaced by a third microscope - camera - assembly .  another limitation is the size of the outer chamber , which limits the magnification to values that are not sufficient for tracking small bacteria ( e.g. , _ e. coli _ ) , even though the microscope has a maximum magnification of ( including the magnifier lens ) .for tracking bacteria sample chambers of size mm may be used , for which the temperature - homogenizing outer chamber is not needed .reconstruction of a swimming trajectory of _ chlamydomonas reinhardtii_. gravity is along negative -direction .( a ) 3d trajectory , color - coded to indicate local speed .red dot indicates start of the trajectory .( b)-(d ) three components of position versus time . in ( b )are the two traces from the two cameras . ]the apparatus was tested on two low - reynolds number swimmers of very different size : _ chlamydomonas reinhardtii _( diameter of  m ) and _ volvox barberi _( diameter of  m ) .both species were grown axenically in standard _volvox _ medium ( svm ) with sterile air bubbling , in diurnal growth chambers ( binder kbw400 , tuttlingen , germany ) set to a daily cycle of h in cool white light ( lux ) at c and h in the dark at c. sample chambers were filled with svm and were of size cm for _ volvox _ , and cm ( standard cuvette ) for _ chlamydomonas_.  the biflagellated _ chlamydomonas _ beats its flagella at hz , primarily in the manner of the breast stroke .its most familiar swimming trajectory is helical , with a radius of  m and a speed on the order of  m/s .the cell has an eye - spot " that serves as a photosensor , and the changing illumination levels of the eye - spot lead to transient changes in flagellar beat dynamics in such a manner that the cell can turn toward the light .figure [ fig : reconstruction]a shows a s trajectory , obtained without phototactic stimulus , during which the cell explored a volume less than mm .the manner in which the two 2d trajectories from the cameras are synthesized into a 3d trajectory is indicated in panels ( b)-(d ) of the figure .the -axis is common to the two cameras , and the overlap between the two is clear in fig .[ fig : reconstruction]b .the very slight graded mismatch between the two -component curves reflects a slight misalignment of the cameras , and is easily removed by remapping the pixel coordinates .   a phototactic turn of _ volvox barberi_. ( a ) the positional and orientational measurements are illustrated by vectors indicating the body axis , and swimming - speed - dependent coloration of the track .to initiate the 180 change swimming direction , the light initially was from the right along the -axis and then changed to come from the left .gravity is directed along the negative -direction .sphere represents initial position along the track .( b ) evolution of the body axis during the phototactic turn is described in terms of two angles and . ]  _ volvox barberi _ typically has biflagellated somatic cells rigidly embedded at the surface of a transparent extracellular matrix .these beat at a typical frequency of hz , primarily from the anterior pole to the posterior pole , with a slight tilt of the beat plane which leads to the characteristic spinning motion as it swims at speeds up to  m/s .just as in _ chlamydomonas _ , each somatic cell has an eye - spot that modulates the beating of its two flagella , allowing the whole colony to perform phototaxis . a 3d track of the _ volvox _ center and body axis during a phototactic turn is shown in fig .[ fig : phototurns]a .determining the body axis of _ volvox _ by using the position of the daughter colonies , as explained in sec .[ sec : improcess ] , leads to the time series of the angles and in fig .[ fig : phototurns]b .this is slightly noisy because the daughter colonies are not distributed evenly .the track also shows an interesting balance between the bottom - heaviness of _ volvox _ ( due to the clustering of daughters in the posterior hemisphere ) , which tends to align the axis with the -direction , and the phototactic tendency to align the colonial axis with the direction of the light ( the -axis ) ._ volvox carteri _ swimming near a surface .flow fields from particle imaging velocimetry of a colony swimming upward against a horizontal cover slip that is glued into the sample chamber ( of size cm ) , as seen from the side ( a ) , and the top ( b ) .images were taken at magnification .scale bar is  m . ]  in addition to allowing a controlled systematic disturbance to the behaviour of microorganisms in the form of light , this apparatus is also suitable to study the interactions of microorganisms with a surface , as the sample chambers can be made so large and so temperature - homogenized that the effect from other surfaces and thermal convection can be neglected .one interesting effect , discussed in more detail elsewhere , occurs when colonies swim upward to a horizontal surface and rotate in place .as the colonies are denser than the water in which they swim , and thus have a net external force acting on them , the far - field flow around them is described by a stokeslet pointing downwards . in the neighborhood of a no - slip surface, the stokeslet induces a set of image singularities which together produce a characteristic lobed flow field .this is readily demonstrated by glueing a horizontal surface ( a microscope coverslip ) into the sample chamber and performing dual - view piv .the results of this are shown in fig .[ fig : piv]a , b , illustrating the fluid flow vector fields viewed from both the side and the top . from above , we see vectors oriented predominantly inward .this inward flow leads to complex behavior of nearby colonies .  when viewed from above as in fig .[ fig : piv]b this setup also provides a means to monitor the rotational dynamics of colonies in great detail , providing accurate measurements of the mean rotational frequency , the noise in rotational motion , and lateral drifts . as mentioned earlier , the bottom - heaviness of the colonies keeps the colonial axis oriented vertically , allowing the daughters to serve as convenient markers to track rotation .precise determination of time series of rotation can then be achieved by determining the correlation between successive images and a reference image , adjusted for centroid drift .the centroid dynamics itself serves as a sensitive measure of colony asymmetries , such as mismatch between the colonial axis and the axis defined by the center of buoyancy and the geometric center .figure [ fig : rotation ] shows the two components of the centroid position for a colony rotating against an upper surface , showing a clear periodic wobble at a frequency of hz .the systematic drifts in position , which may be due to the swimming dynamics themselves or residual convective currents , are in any event below  m/s .tracking of spinning _ v.carteri_. shown are the ( red ) and ( blue ) coordinates of a colony spinning near an upper surface with its axis vertical. recorded at fps in convection controlled chamber .the oscillations represent a small periodic wobble in the colony centroid . ]we presented an apparatus that can track swimming microorganisms in the size range  m in 3d , without the influence of systematic bias due to beavioral stimuli , hydrodynamic interactions with surfaces , and convective background flows .as the apparatus can eliminate these biases , it can also be used to study the influence of each of them .the simplicity of the apparatus compared to other 3d tracking systems , and the software that is part of the supporting online material , make this system easily reproducible .the heart of the apparatus can also be used as the basis for studies of more complex phenomena .for instance , the entire device can be mounted on a tiltable platform in order to examine the effects of varying direction of gravity with respect to the phototactic axis .likewise , a rotatable chamber can be substituted for the ones discussed here in order to examine the effects of fluid vorticity on phototactic swimming .we are grateful to d. page - croft , j. milton , and t. parkin for vital technical assistance , and to j.p .gollub , j.t .locsei , m. polin and i. tuval for important discussions .this work was supported by the epsrc , engineering and biological sciences program of the bbsrc , the schlumberger chair fund , and doe grant no .de - ac02 - 06ch11357 .

Abstract: we present an apparatus optimized for tracking swimming microorganisms in the size range  m , in three dimensions ( 3d ) , far from surfaces , and with negligible background convective fluid motion . ccd cameras attached to two long working distance microscopes synchronously image the sample from two perpendicular directions , with narrowband dark - field or bright - field illumination chosen to avoid triggering a phototactic response . the images from the two cameras can be combined to yield 3d tracks of the organism . using additional , highly directional broad - spectrum illumination with millisecond timing control the phototactic trajectories in 3d of organisms ranging from _ chlamydomonas _ to _ volvox _ can be studied in detail . surface - mediated hydrodynamic interactions can also be investigated without convective interference . minimal modifications to the apparatus allow for studies of chemotaxis and other taxes .


at the beginning of the 20th century `` mechanics had not yet lost its place as the central subject within physics , although it was on the verge of losing it '' .fifty years later classical mechanics was considered , by the majority of physicists , a closed field from which one could not expect new challenges , specially in fundamental issues .but from then on , fresh mathematical results , together with additional numerical data obtained by high - speed computers , revived a field of knowledge related to non - linear dynamics and irregular behaviour of deterministic systems . a field that had begun at the end of the 19th century mainly by the works of henri poincar in his attempt to develop a convergent perturbation theory of planetary orbits .the solar system , despite its complexity , shows an extremely regular behaviour which can be predicted very accurately for short periods of time .this is due , in part , to the weakness of the gravitational force , but also because kepler s two - body problem is completely integrable , despite the fact that any gravitational system of three or more bodies is not ( let us imagine for a moment the difficulties of guessing the law of gravitation had the earth been orbiting around a double star ) .the newtonian deduction of kepler s laws is based precisely on the properties of the integrable system of two bodies .nevertheless , the dynamical interactions of the many bodies that form the solar system necessarily lead us to consider deviations to the predictions based on kepler s laws .this makes us wonder : why does the solar system behave in such a regular way ? or , as in the inquiry of j. k. moser : is the solar system stable ? that is , will it follow in the future the order that we presently observe ?even today these questions do not have a complete and satisfactory answer .  for more than three hundred years , since the time of newton , the evolution and stability of mechanical systems , and specially that of the solar system , have been central to physics and mathematics . during the 18th century , euler , lagrange and laplace made substantial developments by predicting changes in the planetary orbits due to small perturbations , establishing a framework for the study of global stability .all of this culminated in the 19th century works of hamilton and jacobi who reformulated the lagrangian formalism of classical mechanics in terms of the phase space , a step which proved to be most fruitful for the developments of both statistical and quantum mechanics .these successes consolidated the idea of classical determinism .however , at the turn of the 19th century , the works of poincar not only closed the door to an age but generated the first serious fracture in the philosophical conception of determinism .these works showed the impossibility of proving the convergence of perturbation series and , therefore , central questions , as the stability of the solar system , remained unanswered .poincar ' e was thus the first to study what much later has been called ( apparently by j. yorke in 1975 ) _chaos_.  a plain , albeit not simplistic , definition of chaos is `` chaos is the complicated temporal evolution of simple systems '' .thus , and contrary to common thinking , chaos is not spatial and static disorder but a characteristic of certain motions .it is essentially a dynamical concept , in which two initially close trajectories may be divergent after a finite time .chaos may appear in a variety of fields : physics , chemical reactions , the spread of illnesses , internet , economics and , of course , in planetary motions .for contrary to intuition , chaotic behaviour is not an exclusive property of dissipative or random systems but of conservative and deterministic systems as well .focusing on conservative systems we will show that small denominators giving place to resonances are associated to the appearance of chaos , and that due to this irregular behaviour , it seems impossible to predict the long - term evolution of nonintegrable systems ( the vast majority of mechanical systems ) using perturbation techniques .since the end of the 19th century there was no remarkable advance and this fundamental question was practically forgotten until 1954 , when a. n. kolmogorov gave a sketch of the proof that most trajectories of nonintegrable conservative systems are quasiperiodic and that one can obtain them through a convergent perturbation development . in the 1960 s ,v. i. arnold and j. k. moser gave a more formal and rigorous proof of this result , known as the kolmogorov - arnold - moser ( kam ) theorem .in the coming sections we will explain and develop these ideas , stressing physical aspects , rather than mathematical concepts and technicalities , of which there are plenty in a field that was the starting point of modern mathematical developments such as topology . in this way the paper is intended as a comprehensive and intuitive introduction to deterministic chaos which, we hope , will be useful for either students and instructors .the quintessential mechanical system , to which most physical systems are reduced , consists in a collection of mass points interacting between them .experience shows that the state of the system is completely determined by the set of positions and velocities of all its particles .the reference system chosen to describe positions and velocities needs not be cartesian .thus , in the lagrangian formulation of mechanics , the representation of the state of the system is obtained through generalized coordinates and velocities .the minimum number of generalized coordinates needed to describe the state of the system is called the number of degrees of freedom . inwhat follows we will use bold letters to design vectors .accordingly , the set of all positions is represented by the vector and the set of all velocities by .the laws of mechanics are those that determine the motion of the system by providing the time evolution of the positions , , and the velocities , , of all its components . therefore , the main problem is the determination of how the state of the system evolves with time .there have been several approaches to achieve this goal .the first one was based on newton s laws which directly emanate from experience .this approach was subsequently refined and generalized by the lagrangian and hamiltonian formulations , being the latter the most appropriate for the present development .  in the hamiltonian formulation ,the description of the state the system is also given by generalized coordinates but instead of velocities by generalized momenta .the main problem is then finding the time evolution of these quantities , in other words , the knowledge of the functions and which give the temporal evolution of coordinates and momenta .this is achieved by solving the following set of first - order differential equations known as _ hamilton equations _ : where the function is the hamiltonian which , for inertial reference frames , coincides with the total energy of the system . in ( [ 1 - 4 ] )we have used the notation : to denote the gradient of with respect to and analogously for .  for those readers not familiar with the hamiltonian formulation, it may be useful to recall that for the three dimensional motion of a particle of mass inside a conservative field of potential energy , the energy ( i.e. , the hamiltonian ) is where and . in this casehamilton equations ( [ 1 - 4 ] ) read which coincide with newton s second law .  in the hamiltonian formulation ,the state of a system with degrees of freedom is described by a single point of a space of dimension called _phase space_. hence , starting at from an initial point , the state of the system in a later time will be given by the point , where and are the equations of motion , that is , the solutions of hamilton equations ( [ 1 - 4 ] ) with the initial conditions and . the trajectory described by the point as time progressesis called the _ phase - space trajectory _ of the system .one of the advantages of the hamiltonian formulation is that we can evaluate the time evolution of any property of the system without having to know the form of the equations of motion .that is to say , without solving hamilton equations which is usually quite involved .let us assume that a given property of the system ( for instance , the energy or the angular momentum ) can be represented by a function , , of the state of the system and the time .how does change when the state evolves with time ?the answer is simple and reads  , \label{1 - 5}\ ] ] where  ] in such a case is a _ constant of motion_. incidentally , for conservative systems the hamiltonian is time independent , hence and , since =0 ] is the poisson bracket defined as in _( [ poisson ] ) . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _  let us remark that the mere existence of independent constants of motions is not sufficient for integrability ; to ensure that any system is integrable it is also necessary that the constants of motion satisfy the property of involution ( [ 2 - 0 ] ) .property that guarantees the existence of a canonical transformation for which the transformed momenta are precisely the integrals of motion and , therefore , constants . for constantmomenta the integration of hamilton equations is immediate .this is essentially the content of a theorem attributed to liouville and what ultimately justifies the above definition of integrability ( see [ appena0 ] for a more complete discussion ) .unfortunately , the existence of integrals of motion in involution is the exception more than the rule .poincar proved , more than a century ago , that most mechanical systems only have the energy as a constant of motion .therefore , _ most systems with more than one degree of freedom are not integrable_.  a paradigmatic example is provided by a closed -body system , i.e. , a set of point particles interacting with each other without external forces acting on them .the body system has degrees of freedom ( for each particle ) and only integrals of motion .these are :  * the three components of the total momentum * the three components of the motion of the center of mass ( is the total mass of the system ) * the three components of the total angular momentum where is the relative angular momentum . *the energy where is the relative kinetic energy and is the potential energy .from these ten integrals _ only six of them are independent and in involution _ : the three components of the total momentum , the square of the relative angular momentum , the third component of the relative angular momentum , and the total relative energy  .  when , _ the two - body problem _ , there are degrees of freedom and the problem is integrable . however , when , _ the three - body problem _ , there are degrees of freedom but only independent constants of motion in involution ; whence , the three - body problem ( for example , the system sun - earth - moon ) is not integrable.   let us first focus on integrable systems and postpone nonintegrable systems for later sections .a first and most important consequence of the existence of involution constants is that the trajectories of the system can not pass through all points of the phase space .this is due to the fact that all trajectories are confined to move in a dimensional manifold determined by the constants of motion and embedded in the phase space of dimension .indeed , the manifold is defined by the set of algebraic equations : where are arbitrary constants .consequently , _ the phase - space trajectories never leave _ because otherwise the conservation laws would be violated .note that the manifold depends on the particular value of the constants of motion and , since the latter depend on the choice of the initial state of the system , we see that is fixed by the initial conditions .  in what follows we will assume that the motion of the system is bounded ( recall , for instance , the planetary motion ) .as a consequence of this , the manifold must be bounded as well .in fact , it can be shown , that is a compact ( closed and bounded ) manifold .we will now see that in this case _ has the topology of a dimensional torus_.  in effect , let us define the following velocity fields : ( ) .note that if , for example , then ( cf ( [ 1 - 4 ] ) ) .accordingly , in ( [ 2 - 1 ] ) is included the hamiltonian flux of the phase - space trajectories which , due to the existence of independent integrals in involution , must be completely contained in the manifold .  in order to visualize that has the topology of an dimensional torus ,let us first show that the fields ( ) are independent and tangent to .the property of independence immediately follows from the independence of the integrals .we next observe that the dimensional gradients : ( ) , are orthogonal to every surface .on the other hand , the scalar product of the fields with these gradients is and , by virtue of the involution property , we get =0\ ] ] .hence , the fields are orthogonal to and , since the gradients are orthogonal to , we conclude that _ the fields are tangent to the manifold .   dimensional torus .( b ) singular point of a vector field on a dimensional sphere . ]there is a theorem in topology the poincar - hopf theorem , jokingly known as the `` hairy ball theorem'' which states that if in a compact manifold of dimension , one can define independent vector fields tangent to , then has necessarily the topology of an dimensional torus .  without getting into technical details, we can intuitively imagine this result as follows . among the several possibilities for a manifold to be compact , andhence bounded , the simplest ones are : either is a dimensional sphere ( or a surface with the same topology , like an apple ) or it has the topology of an dimensional torus .however , it is impossible to define a vector field on a sphere in all its points while it is perfectly possible to define a vector field on the surface of a torus .this is easily visualized if we compare the act of combing down hair on a torus and on a sphere for the latter case , a hair always sticks up at the pole and , hence , it is not tangent to the sphere see fig .we have just seen that , for the bounded motion of integrable systems with degrees of freedom , the phase - space trajectories lie on dimensional tori embedded in the dimensional phase space .we will now see that the existence of these invariant tori ) are often called `` invariant tori '' because they are constructed by giving different values to the integral invariants . ]provides a way of defining a new set of canonical variables , the action - angle variables , which are crucial for the further development of the subject .  indeed, any dimensional torus is a periodic object which can be considered as the product of independent periodicities . in other words, we can define topologically independent closed curves , , on a given torus , where none of the can be deformed continuously into each other or shrunk to zero . in fig .[ fig2 ] we represent a dimensional torus for which turns around through the longest path while does it through the shortest path .note that , neither nor can be converted into each other by continuous transformations .having this set of closed curves at our disposal , we define the _ action variables _ by means of the line integrals : where are the momenta written in terms of the coordinates and the constants of motion .notice that ( [ momenta0 ] ) is obtained by solving for the s the set of algebraic equations ( [ integrals ] ) provided by the existence of independent integrals .  since in ( [ 2 - 2 ] )we integrate with respect to the generalized coordinates , the action variables only depend on the integral invariants :  , and are themselves constants of motion .   and on a dimensional torus . ]we now define a canonical transformation from the original variables to a new set of variables where the actions are the transformed momenta . a generating function for this transformationis given by the indefinite integral where is given by ( [ momenta0 ] ) in which the s have been written in terms of the s after solving ( [ j_alpha ] ) .we shall see below that the new coordinates , defined by play the role of angles , increasing by when the phase - space trajectory undergoes a complete turn around , while momenta ( i.e. the actions ) are the radii of the invariant tori ( see fig .[ fig3 ] ) .in action - angle variables , hamilton equations read where is the transformed hamiltonian which coincides with the original one since we are dealing with a time independent canonical transformation .  however , , because the s are constants of motion .hence , , and the transformed hamiltonian only depends on the actions : whence the solution of the hamilton equations ( [ he_aa ] ) thus reads where  .  in action - angle variables ,the equations of motion are , therefore , very simple : the actions ( transformed momenta ) are constants and the angle variables ( transformed coordinates ) are linear functions of time . in the original variables , ,the equations of motion are obtained by introducing ( [ 2 - 3 ] ) into the equations of the canonical transformation which gives the original variables in terms of action - angle variables .  before proceedingfurther let us note that the transformation to action - angle variables is global .that is , the entire phase space is filled with tori ( see fig . [ fig3 ] ) and a given trajectory will forever reside on the surface of the particular torus fixed by the initial conditions .indeed , the choice of any invariant torus depends on the value of the constants .the set of constants determines the torus where the trajectory stays and the value of the angle variable , at a given time , determines the position of the trajectory on that torus .  , and label the different tori and correspond to different initial conditions . are the angle coordinates on a given torus . ]we will see next that _the s are the angular frequencies of the multiply periodic motion of the system_. to this end we first show that when the state of the system performs a complete oscillation along a closed curve , the corresponding angle variable varies in . in effect, let us denote by the change of the angle variable when the coordinate realizes a complete oscillation ( also known by the astronomical name of `` libration '' ) . in the [ appena ] we show that in other words , the angle variable changes in only when the system executes a complete oscillation along the curve and no change otherwise .on the other hand , let ( ) be the different periods of each libration motion .we see from ( [ 2 - 3 ] ) that the change in after a period is and equating it with ( [ delta_ij ] ) we get therefore , the quantities defined in ( [ omega ] ) as derivatives of the hamiltonian with respect to the actions are the angular frequencies of the ( multiperiodic ) motion of the system .this is quite remarkable since , for integrable hamiltonians described by action - angle variables , to obtain the frequencies of the motion it is not necessary to integrate hamilton equations , it suffices to take the derivative of the hamiltonian with respect to the actions , a much easier task indeed .recall that the original variables , , and the action - angle variables , , are related by a canonical transformation with the generating function ( [ s ] ) .this obviously implies that the relation between these two sets of canonical variables is known .let us denote this relation by owing to the periodic nature of the angle variables , as expressed in equation ( [ delta_ij ] ) , we see from ( [ relation ] ) that the s and s are multiply periodic functions of the s with periods , and the same is true for any dynamical function .as is well known from analysis , any multiply periodic function can be represented by a multiple fourier series .thus , for instance , would appear as where the s are integer indices running from to . by treating the set of s also as an dimensional vector , the expansion can be written in a more compact form : where we use the abbreviated notation : the fourier coefficients , which depend on the actions , are given by   if we also write the time dependence of the angle variables given in ( [ 2 - 3]) as a vector equation , then the time dependence of appears in the form where .we easily convince ourselves that a similar expression holds for any momentum : notice that the right hand side of equations ( [ q(t ) ] ) and ( [ p(t ) ] ) correspond to a superposition of the equations of motion of linear oscillators . therefore ,  _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ the coordinates and momenta of integrable systems can be represented as a sum of n dimensional harmonic oscillators with angular frequencies given by the fundamental ones , and their harmonics . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _  the coordinates and momenta of integrable systems are multiply periodic functions of time with independent ( angular ) frequencies .however , this property _ does not imply that , in general , the s and s are ( simply ) periodic functions of time _ ; for it would be necessary that there exist _ a single period _ for which all s and s are periodic :   in the [ appenb ] we show that this is the case if , and only if , the frequencies are integer multiples of a single frequency :  , where are integer numbers , and equation ( [ fundamental ] ) means that in order to have periodic motion , the frequencies must be _ commensurable_. from ( [ fundamental ] ) we immediately see that this is equivalent to assuming that all frequencies are rational multiples of each other if the frequencies are _ incommensurable _ , in other words , if they are not rationally related , then the motion is termed multiply periodic or quasiperiodic or conditionally periodic , according to different terminologies in use , and never repeats itself .dimensional tori .the torus in ( a ) is a rational surface ( resonant torus ) where , that is , the trajectory closes over itself after three turns around and one turn around .the torus in ( b ) is an irrational surface ( nonresonant torus ) where the frequencies are not commensurable . in ( b )the phase - space trajectory will eventually cover the surface of the torus densely . ]a formal condition for the frequencies to be commensurable is the existence of relations of the form where are integers . indeed ,if ( [ commen_3 ] ) holds , then we have an homogeneous system of algebraic equations ( with integer coefficients ) for unknowns , the frequencies , which means that we can solve ( [ commen_3 ] ) by writing all frequencies in terms of one of them , say .in other words , we obtain , where and are integers . therefore , is a rational number , and all frequencies are commensurable .let us remember that the phase - space trajectories lie on the invariant tori defined by equation ( [ 2 - 3 ] ) , and since these curves are mathematically described by functions of coordinates and momenta , the above considerations on single or multiple periodicity apply to phase - space trajectories as well .we can , therefore , conclude that _ on a given torus , the phase - space trajectory will be a closed curve ( i.e. the motion of the system will be periodic ) if and only if the frequencies of the motion are commensurable .when frequencies are incommensurable the trajectory will densely cover the torus , never closing on itself_. in the first case , we call it _ rational , or resonant , torus _ , while in the latter _ irrational , or nonresonant , torus _ ( fig . [ fig4 ] ) .a classic example of an integrable hamiltonian is provided by the kepler problem .consider the relative motion of a planet of mass orbiting around a star of mass .the hamiltonian of the system in polar coordinates , , is where is the relative momentum of the two bodies , is the reduced mass and ( is the gravitational constant ) .  both the energy , , and the angular momentum , , of the two - body system are constants of the motion .the latter implies that the motion takes place on the plane perpendicular to the direction of .  after a canonical transformation from the original variables to the action - angle variables , the hamiltonian is only a function of the actions :   the equations of motion in the original variables are quite complicated ( elliptic orbits ) , but in terms of action - angle variables they are very simple :  where and are constants determined by the initial conditions .  the motion of the system in the phase space ( i.e. , the phase - space trajectory ) lies on a two dimensional torus embedded in the three dimensional manifold of constant energy .the torus has two angular coordinates and and two constant radii and which are fixed by the initial conditions .there are two frequencies , and , associated to the motion ; if they are commensurable , that is if , where and are integers , the trajectory is closed and the motion is periodic ( the trajectory closes on itself after turns around and turns around ) . if the frequencies are incommensurable the trajectory will never repeat itself and will eventually cover the entire surface of the torus .  for the kepler problemthe frequencies are ( recall that ) which is kepler s third law .the frequencies are commensurable ( in fact , equal ) and hence the phase - space trajectory is always closed . in other words ,the motion is periodic .we know that the existence of as many constants of motion in involution as degrees of freedom determines integrability and regular motion . on the other hand , conservative systems with one degree of freedom always have one integral of motion , the energy , and they are , therefore , integrablethe following question then suggests itself : can we know whether or not a conservative system with degrees of freedom is integrable ?unfortunately there is no general answer to this problem . in most situations ,integrability is guessed , but not proved , by constructing a _ poincar surface of section _ which we can define as any bidimensional surface , contained in the phase space , where one can conveniently analyze the successive intersections of the phase - space trajectory ( fig .[ fig5 ] ) .    to see how this concept works ,let us suppose a conservative system with two degrees of freedom .the hamiltonian is a constant of motion and we write : this restricts phase - space trajectories to evolve inside a dimensional manifold the surface of constant energy or `` energy shell '' represented by ( [ 4 - 1]) embedded in the phase space which has dimension .suppose the system has a second integral ( for instance , the angular momentum ) : where is an arbitrary constant ; then this integral also defines another dimensional manifold in the dimensional phase space and any trajectory must also evolve in it .the coordinates and momenta of the system can not take arbitrary values because they must satisfy the conservation laws ( [ 4 - 1 ] ) and ( [ 4 - 2 ] ) . from a geometrical point of view, this means that any phase - space trajectory is bounded to move on a dimensional surface which is the intersection of two dimensional manifolds and . indeed , from the conservation of energy ( [ 4 - 1 ] ) ,we can write , for instance , as a function of and the energy , that is , .substituting this expression into the second conservation law ( [ 4 - 2 ] ) we can obtain in terms of and and the constants and : this is the equation of a dimensional surface within the dimensional phase space .such a surface is determined by the specific values of the constants and , which , in turn , depend on the initial conditions .  ,i.e. , each time the trajectory passes through the plane .( b ) if there are two integrals of motion in involution , the trajectory will remain along one dimensional curves , , in the plane .( c ) if only one constant of motion exists ( the energy ) , the trajectory will disperse over a region of the plane . ]if we now consider the section of the surface ( [ 4 - 3 ] ) and this is an example of a poincar surface of section we see that the successive intersections of the phase - space trajectory with this poincar section are along the curve on the plane .  in general , for a given conservative hamiltonian , we do not know whether or not a second integral of motion exists .we can guess , nonetheless , its existence by solving hamilton equations numerically and then drawing and each time ( fig .[ fig6 ] ( a ) ) .if the system is integrable , the intersections of the trajectory in the poincar section will appear as a collection of points regularly distributed on the curve ( fig .[ fig6 ] ( b ) ) . on the other hand ,if the system is not integrable the intersections of the phase - space trajectory will appear randomly that is , chaotically scattered over a region only limited by the conservation of energy ( fig .[ fig6 ] ( c ) ) .this was the method used by henon and heiles to determine the possible existence of a third integral which , as the observations showed , constrained the motion of a star in a galaxy with an axis of symmetry .this system has three degrees of freedom and two known constants of motion , the energy and the component of the angular momentum along the direction of the symmetry axis . for a long timeit was believed that there was no third integral of motion because it had not been found analytically .  .]  . ]henon and heiles proposed the following hamiltonian without any known symmetries leading to a third integral that could model the basic traits of the problem : and numerically studied the behaviour of the equations of motion .the hamilton equations are  .observe the occurrence of non - linear quadratic terms in these equations of motion which are consequence of the cubic terms in the potential energy .note also that the magnitude of these non - linear terms increases with the energy .the representation of the intersections of the phase - space trajectory on the poincar section is given in figs .[ fig7]-[fig8 ] . for low energies ( fig .[ fig7 ] ) , the distribution of points is apparently regular , which seems to suggest the existence of a third integral , at least to the accuracy of the plotting . as the energy increases ( which increases the effect of the nonlinear terms ) the third integral appears to be destroyed , at least partially , but still remain some `` islands '' where the motion is regular ( fig . [ fig8 ] ) . for even higher energies the hypothetical third integralis completely destroyed .the scattered points appearing in fig .[ fig8 ] correspond to a single phase - space trajectory that , for obvious reasons , we will call chaotic , and which implies a highly sensitive dependence on initial conditions that makes the long - time evolution of the system totally unpredictable .as stated in previous sections , most mechanical systems are not integrable , having only the energy as a constant of motion .however , in many situations , integrable and nonintegrable hamiltonians differ very little . in such caseswe can obtain approximate solutions of the equations of motion by means of the canonical perturbation theory .we will outline this theory because it provides the clue to the appearance of chaos in deterministic systems , for it shows that chaotic behaviour is a direct consequence of the occurrence of resonances , i.e. , of small denominators leading to infinities .  in order to fix ideas ,let us return to the kepler problem discussed in section [ sec4 - 2 ] and suppose that , in addition to the planet orbiting the sun , there is another celestial body which perturbs the motion of the two - body system .we assume that the effect of this perturbation is small ( because , for instance , the third body is small or more distant ) , so that the total hamiltonian can be written in the form where is a small parameter and is the unperturbed hamiltonian of the kepler problem as given in ( [ 2 - 5 ] ) .according to our previous discussions , we know that the unperturbed motion , represented by , takes place on a bidimensional torus of radii , .we want now to find the corrections to the unperturbed motion due to the presence of a third body .as we generally can not obtain an exact solution to the equations of motion , we search for approximations via perturbation expansions in powers of .observe that the motion generated by the hamiltonian ( [ 5 - 1 ] ) is bounded .this allows us to expand the perturbation as a fourier series in the angular variables .therefore , following the notation of section [ subsec4 - 1 ] , we see that ( [ 5 - 1 ] ) can be written as where  and ( ) .the fourier coefficients , , are defined by let us note that we have generalized ( [ 5 - 1 ] ) to include multiply periodic systems with degrees of freedom .the main idea behind canonical perturbation theory is simple and consists in finding , by means of a power series in , a canonical transformation , to a new set of action - angle variables such that _ the transformed hamiltonian is only a function of the new actions . if we could achieve this , then the hamiltonian ( [ 5 - 1 ] ) would become completely integrable and its equations of motion would be easily obtained ; that is , the s would be constants and the s linear functions of time .unfortunately , this procedure can not be done exactly .however , we may proceed by perturbation expansions since is small .we take this path and introduce the following generating function for the canonical transformation ( [ ct ] ) : which is defined by a perturbation series up to first order in with the unknown fourier coefficients to be determined later . in terms of the generating function, the canonical transformation is obtained by ( cf ( [ canonical_3 ] ) ) that is   since we are dealing with a time - independent canonical transformation , the transformed hamiltonian , , coincides with the original one .therefore , the new hamiltonian is found from ( [ 5 - 2 ] ) by solving the equations ( [ j ] ) and ( [ theta ] ) for as a function of .we detail this procedure in the [ appenc ] where we show that \cos\left(\mathbf{k}\cdot\mbox{\boldmath}\right)+{\rm o}(\epsilon^2 ) .\label{h'}\end{aligned}\ ] ] in this expressions are the ( known ) frequencies of the unperturbed motion : whereas is the fourier coefficient of the perturbation corresponding to and reads ( see ( [ fc ] ) ) in other words , _ is the average of the perturbation with respect to the original angle variables_.  recall that the fourier coefficients of the generating function ( [ 5 - 3 ] ) are still unknown and we are , therefore , free to choose them in the most convenient way .let us take ( ) , so that the term inside the sum in equation ( [ h ] ) vanishes ; we get in such a case , and to the first order in , _ the transformed hamiltonian is only a function of the new action variables and is , therefore , integrable ( up to first order)_.  moreover , the perturbed frequencies ( given by the derivatives of the transformed hamiltonian with respect to the actions ) read where is the average of the perturbation ( cf ( [ 5 - 7 ] ) ) and are the unperturbed frequencies given by ( [ unperturb_freq ] ) .finally , the relation between the original actions and the new variables , is given by ( see ( [ j ] ) and ( [ theta ] ) ) equation ( [ 5 - 9a ] ) is , to the lowest order in , the solution to our problem .we have obtained the new action variables which contain the first - order correction due to the perturbation . to first order in  are constants and the new angles linear functions of time .unfortunately , the expressions ( [ 5 - 7 ] ) and ( [ 5 - 9a ] ) suffer a serious illness , for both and diverge if in other words , _ the perturbation method breaks down when internal resonances occur inducing perturbation expansions to diverge_.  let us observe that if the unperturbed motion takes place on a rational ( resonant ) torus , i.e. , if the frequencies of the unperturbed hamiltonian are commensurable , there always exists a set of integers such that condition ( [ resonance ] ) holds ( see ( [ commen_3 ] ) ) . at first sight , this would imply the restriction of canonical perturbations to the motion on irrational ( nonresonant ) tori .however , the problem is worse , because , even if the frequencies are incommensurable ( and recalling that any irrational number can be approximated , within any degree of accuracy , by a sequence of rational numbers ) there will always be integers that generate denominators as small as desired , with the result of convergence failure . ) , the term which appears in the denominator of the perturbation series , albeit never zero , can be arbitrarily small for and covering all integer numbers except .thus , for instance , jupiter and saturn , in their motion around the sun cover and seconds of arc in a day and the denominator is very small compared with each frequency . ]hence we get two series whose convergence is uncertain . for one hand , we have fourier series that can not converge due to internal resonances which lead to the famous problem of `` small denominators '' . on the other hand , far from resonant frequencies and also because of small divisors , it is quite unclear that the perturbation series in converge .in fact , poincar proved that all these series do not lead to a convergent representation of the perturbed problem .consequently , fundamental questions , as the long - time stability of planetary orbits in the solar system , could not be answered in a satisfactory way .let us remind moser s query and ask again : is the solar system stable ?at the end of the 19th century , the general consensus was that the addition of even the smallest perturbation would render any conservative system nonintegrable , and that any phase - space trajectory would densely explore the energy shell in a chaotic manner .this was the first crack in classical certainty and a severe blow on classical thinking .all efforts and attempts by some of the finest mathematicians and physicists of the time failed to solve the small divisor problem .poincar called it the `` fundamental problem of classical mechanics '' and seemed to be an unsurmountable obstacle .thus , around the beginning of the 20th century , this field of research dropped off and questions concerning long - time planetary stability were almost forgotten in mainstream research .  the breakthrough came in 1954 , when a. n. kolmogorov devised a way of obtaining a theory of canonical perturbations which is rapidly convergent and appropriate to nonresonant tori .kolmogorov s new idea was rigorously proved by v. i. arnold in 1961 and , independently , by j. k. moser in 1962 .the complete result is known as the kolmogorov - arnold - moser ( kam ) theorem .following arnold ( , see also ) suppose that an integrable hamiltonian is perturbed by a function , so that the total hamiltonian is  where the perturbation is assumed to be periodic in the angular variables : hamilton equations read  where are the frequencies of the unperturbed motion .kolmogorov s idea was that , far from resonant frequencies , it is possible to construct a perturbation series _ in even powers of _ .this leads to a fast convergent ( `` superconvergent '' ) perturbation series with the overall result that , for most initial conditions , the motion generated by the hamiltonian ( [ 6 - 1 ] ) remains quasiperiodic , that is , confined to reside on certain surfaces which resemble slightly deformed tori ; while the complementary of this regular motion that is , the chaotic motion has a small lebesgue measure if is small . in other words , the set of chaotic trajectories is negligible as .  in order to be more precise in the formulation of the theorem, one has to assume that is an analytic function and that the unperturbed motion generated by is nondegenerate .the last statement implies that the hessian of is different from zero : the next step consists in identifying , for the unperturbed system , a particular torus , , defined by a set of frequencies , , which are incommensurable , that is for all the integers .the kam theorem can then be stated as :  _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ if is small enough , then for almost all nonresonant frequencies there exists an invariant surface of the perturbed system such that is close to . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _  in other words , the trajectories of the perturbed hamiltonian will evolve on surfaces which are `` close to '' tori of the integrable system .this means that , for almost all ( but _ not _ all ) nonresonant frequencies , the phase - space trajectories are regular and not chaotic as far as is small enough ( the theorem does not tell how small has to be ) .moreover , the theorem also proves that the measure of the set of points in phase space which do not lie on any surface tends to zero when .that is to say , the probability of randomly choosing initial conditions not leading to motion on stable surfaces is vanishingly small as .this has momentous consequences for the stability of planetary orbits because it is extremely unlikely ( though not impossible ) that a given celestial body is located in some unstable ( i.e. , resonant ) region ( see , nonetheless , the next section ) .therefore , we see that , under the conditions of the theorem , small perturbations do not destroy invariant tori but only deform them . on these deformed tori , called _kam surfaces _ , the motion is regular and not chaotic .the rigorous proof of the theorem is highly technical and we will not give it here .there are , however , more accessible derivations such , for instance , that of r. barrar to which we refer the interested reader ( see also ) .as mentioned above the clue of the proof , discovered by kolmogorov , is a superconvergent perturbation method similar to the old newton - raphson technique for solving algebraic equations .kam tori are determined by nonresonant frequencies and on these deformed tori the motion is not chaotic .but , what about resonant tori ? what happens to them ?in other words , what is the fate , when a perturbation is added , of those tori defined by a rational relation of frequencies ? it can be shown that resonances completely destroy resonant tori .consequently the trajectories on these tori which , remember , in the absence of perturbation follow well defined closed and periodic curves on them lose their bounds and wander through the phase space ( in fact , only through the subspace given by the surface of constant energy ) .therefore , motion on resonant tori becomes completely irregular and unstable .this is an example of the so - called _ hard chaos _, as opposed to _ soft - chaos _ which relates to the motion near kam surfaces ( see also next section ) .fortunately , as kam theorem proves , the set of resonant tori , on which regular motion is destroyed , has a vanishingly small measure as long as the perturbation is small .equivalently , the remaining slightly deformed tori where the motion is still regular , have a measure close to if is small .this means that most tori are not destroyed for small perturbations and kam surfaces almost densely fill the energy shell and between them there appear `` islands '' with chaotic motion .this results in poincar sections which are extraordinarily complex and that some authors picture as in fig .in this closing section we briefly outline some additional developments and applications of the theory .   as kam theorem proves ,the energy shell of a conservative and nonintegrable hamiltonian of the form where is integrable and is small , is densely filled with kam surfaces on which phase - space trajectories are fairly regular. however , outside kam tori trajectories are random . in order to analyze these chaotic trajectorieswe must distinguish among the cases of two and more than two degrees of freedom .let us recall that , for integrable conservative systems with degrees of freedom , the phase space is dimensional , the energy shell is dimensional and invariant tori are dimensional .hence , for , the two dimensional tori are embedded in the three - dimensional energy surface .this means that any closed surface , such as a torus , divide the energy shell into an inner region and an outer region .the same applies to kam tori of conservative nonintegrable system with degrees of freedom .accordingly , any chaotic trajectory , located in the interspace between two tori , can not escape from it .no matter how complicated its motion is , the trajectory never leaves the region and the action variables remain close to their initial values .this is another aspect of the soft - chaos mentioned above , for there exists certain stability of motion because kam surfaces densely populate the energy shell and are , therefore , very close to each other .however , for chaotic trajectories located in regions between tori can escape to other regions of the energy shell ( see fig .[ fig10 ] ) .thus , for instance , if the energy shell is dimensional , kam surfaces dimensional , and chaotic trajectories have two additional dimensions to wander through .hence , _ for the existence of kam surfaces does not guarantee the stability of motion _ , as two trajectories which are initially very close to each other can be very distant after a finite period of time .this phenomenon , that exists for arbitrarily small perturbations , is known as _ arnold diffusion _ and only appears in systems with more than degrees of freedom .the set of wandering trajectories resulting from it constitute the so - called _arnold web_. there have been several attempts to determine the average speed of the spreading of trajectories defined as , where .several theoretical models and numerical studies seem to indicate that this velocity is very small .some models predict that  which means , that for small , the action at time , , remains close to its initial value , , for a very long time . in other words ,the average diffusion velocity is of the order which is exponentially small and practically undetectable if .let us return to the problem of the long - time stability of the solar system , one of the major challenges of mechanics for more than three centuries . as we know , the solar system is a nonintegrable many - body problem which , to first approximation , can be considered conservative . the key point is whether planetary motions are quasiperiodic or random .the solar system is too complicated and a direct application of the kam theorem does not bring practical results . on the other hand , planet orbits exhibit great regularity and until recently it was thought that the solar system evolves quasiperiodically . in spite of this, there seems to be some evidence that this may not be the case .  in 1857the american astronomer d. kirkwood discovered a series of discontinuities in the distribution of main - belt asteroids which circle the sun and lie between mars and jupiter .there are regions where there is a depletion in the number of asteroids .these regions are called _kirkwood gaps_. one of the most plausible explanation for them has been given by s. f. dermott and c. d. murray , who in 1983 proposed that the gaps are due to resonances in the three - body problem of the sun , jupiter , and the individual asteroids .that is , jupiter perturbs the asteroid s almost keplerian orbit around the sun and creates resonances in the asteroid s phase space .the location of the observed gaps are found to be very close to the resonant surfaces of jupiter and the asteroids , which would agree with kam theory about the destruction of resonant tori .it has also been argued that the mechanism of arnold diffusion might be responsible for removing asteroids from the gap regions and force some of them to collide with mars .let us finally mention that there seems to be some numerical evidence that the long - term motion of the former planet pluto is chaotic and also that of some of the main planets of the solar system .all these results seem to indicate that the long - time motion of the solar system may be chaotic and , to some extent , unpredictable .let us finish this exposition by giving a very brief view on the possible manifestation of chaotic behaviour in quantum systems .if one naively tries to extend , in a direct manner , the concepts of regular and chaotic trajectories to quantum systems , one is immediately faced with an unsurmountable difficulty : due to heisenberg s uncertainty principle , between canonically conjugated variables , it makes no sense to define trajectories in the quantum phase space .in fact , the phase space of any system of degrees of freedom is divided in cells of volume ( is planck s constant ) and one can not determine at which point inside a given cell the system is .this renders the construction of phase - space trajectories meaningless .therefore , the notion of invariant tori , over which the trajectories of integrable systems stay , also loses its meaning in the quantum world .the problem of the quantization of nonintegrable hamiltonians can be traced back to the early days of the old quantum theory ( see for a thorough review ) .however , strictly speaking , there is no such think as `` quantum chaos '' , although there have been detected some peculiar effects in quantum systems whose classical counterparts correspond to chaotic systems . quoting m. gutzwiller : `` the term ` quantum chaos ' serves more to describe a conundrum than to define a well - posed problem '' .what is understood by quantum chaos is essentially the relation between classical chaotic systems and their quantum counterparts .let us recall that in quantum mechanics , the state of the system is not specified by a point of the phase space but by a wave function giving the probability of finding the system at position in time .however , this wave - like character of quantum systems implies a smooth nature . how can the irregular character of classical chaos be reconciled with the smooth and wave - like nature of phenomena on the atomic scale governed by quantum mechanics ? . what are , therefore , the quantum manifestations of chaos ? .there are several places and situations where one can look for the `` quantum signatures '' of chaos , and substantial research has been done along this direction in the last three decades .we can single out one aspect where quantum chaos is quite significant .the first place to look is in the energy levels , not a particular level but the distribution of them .thus , and somewhat paradoxically , in nonchaotic quantum systems ( i.e. , those quantum systems whose classical counterparts are integrable ) the energy levels are completely random without any correlation , while in chaotic systems the energy levels show strong correlations between them , which implies a lower degree of randomness .there are other situations in which quantum manifestations of chaos appear as , for example , in the distribution of stationary states and in electron scattering , among others .we refer the interested reader to more specialized literature for further information ( see , for instance , ) .there is a line of thought , among some philosophers and scientists , affirming that science is a construction , being scientists designers who fit as many data as possible within consistent frames .opposite to that , others believe , and this is the traditional thinking , that science is discovery and the scientist is like a detective struggling to unveil the hidden truth , `` the secret of the old one '' , as einstein put it .obviously , mainstream scientists lie somewhere in between these two extremes .  whatever the interpretation one may support , it is beyond any doubt that classical mechanics is one of the finest and most creative works that scientific endeavour can yield . in its most sophisticated form , the hamiltonian approach , classical mechanics even arrives to forecast its own limitation : the long - time unpredictable behaviour of nonintegrable systems which , as mechanics also proves , are ubiquitous in nature . the solar system , galaxies , galaxy clusters , perhaps atoms and beyond , all of them are nonintegrable .  in this review we have tried to report on this striking consequence of classical mechanics . in a world of uncertaintiesit all began with quantum mechanics we can add , as poincar envisaged , another one : the classical uncertainty .  partial financial support from the ministerio de ciencia e innovacin under contract no .fis 2009 - 09689 is acknowledged .we thank miquel montero , josep perell and nitin rughoonauth for a careful reading of the manuscript .we are indebted to prof .luis navarro for useful discussions and suggestions .we know that , by definition , a system is integrable when there exist integrals of motion that are independent and in involution : =0.\ ] ] this property allows us to use the s as the transformed momenta of some canonical transformation .indeed , let us recall from ( [ canonical ] ) that any set of variables can be considered as the momenta associated to some hamiltonian if =0 \theta'\theta\theta$}\right)+{\rm o}(\epsilon^2),\ ] ] which is equation ( [ h ] ) .99 arnold v i 1961 _ soviet math .doklady _ * 2 * 501 arnold v i 1979 _ mathematical methods of classical mechanics _( berlin : springer ) barrar r 1970 _ celestial mech . _ * 2 * 494 bergia s and navarro l 2000 _ archives internationales dhistoire des sciences _ * 50 * 321 bohm d 1989 _ quantum theory _( new york : dover ) ( first edition , 1951 ) born m 1960 _ the mechanics of the atom _( new york : ungar ) ( original german version , 1924 ) born m 2005 _ the born - 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Abstract: in recent years there has been a considerable increase in the publishing of textbooks and monographs covering what was formerly known as random or irregular deterministic motion , now named by the more fashionable term of deterministic chaos . there is still substantial interest in a matter that is included in many graduate and even undergraduate courses on classical mechanics . based on the hamiltonian formalism , the main objective of this article is to provide , from the physicist s point of view , an overall and intuitive review of this broad subject ( with some emphasis on the kam theorem and the stability of planetary motions ) which may be useful to both students and instructors .