Hamiltonian mechanics

Hamiltonian mechanics is a mathematically sophisticated formulation of classical mechanics. Historically, it contributed to the formulation of statistical mechanics and quantum mechanics. Hamiltonian mechanics was first formulated by William Rowan Hamilton in 1833, starting from Lagrangian mechanics, a previous reformulation of classical mechanics introduced by Joseph Louis Lagrange in 1788. Like Lagrangian mechanics, Hamiltonian mechanics is equivalent to Newton's laws of motion in the framework of classical mechanics.
Overview.
In Hamiltonian mechanics, a classical physical system is described by a set of canonical coordinates , where each component of the coordinate is indexed to the frame of reference of the system. The are called generalized coordinates, and are chosen so as to eliminate the constraints or to take advantage of the symmetries of the problem, and are their conjugate momenta.
The time evolution of the system is uniquely defined by Hamilton's equations:
</math>
where formula_1 is the Hamiltonian, which often corresponds to the total energy of the system. For a closed system, it is the sum of the kinetic and potential energy in the system.
In Newtonian mechanics, the time evolution is obtained by computing the total force being exerted on each particle of the system, and from Newton's second law, the time evolutions of both position and velocity are computed. In contrast, in Hamiltonian mechanics, the time evolution is obtained by computing the Hamiltonian of the system in the generalized coordinates and inserting it into Hamilton's equations. This approach is equivalent to the one used in Lagrangian mechanics. The Hamiltonian is the Legendre transform of the Lagrangian when holding and fixed and defining as the dual variable, and thus both approaches give the same equations for the same generalized momentum. The main motivation to use Hamiltonian mechanics instead of Lagrangian mechanics comes from the symplectic structure of Hamiltonian systems.
While Hamiltonian mechanics can be used to describe simple systems such as a bouncing ball, a pendulum or an oscillating spring in which energy changes from kinetic to potential and back again over time, its strength is shown in more complex dynamic systems, such as planetary orbits in celestial mechanics. The more degrees of freedom the system has, the more complicated its time evolution is and, in most cases, it becomes chaotic.
Basic physical interpretation.
A simple interpretation of Hamiltonian mechanics comes from its application on a one-dimensional system consisting of one particle of mass . The Hamiltonian can represent the total energy of the system, which is the sum of kinetic and potential energy, traditionally denoted and , respectively. Here is the momentum and is the space coordinate. Then
 is a function of alone, while is a function of alone (i.e., and are scleronomic).
In this example, the time derivative of the momentum equals the "Newtonian force", and so the first Hamilton equation means that the force equals the negative gradient of potential energy. The time derivative of is the velocity, and so the second Hamilton equation means that the particle's velocity equals the derivative of its kinetic energy with respect to its momentum.
Calculating a Hamiltonian from a Lagrangian.
Given a Lagrangian in terms of the generalized coordinates and generalized velocities and time,
Example.
Spherical pendulum consists of a mass "m" moving without friction on the surface of a sphere. The only forces acting on the mass are the reaction from the sphere and gravity. Spherical coordinates are used to describe the position of the mass in terms of ("r", "θ", "φ"), where "r" is fixed, "r"="l".
The Lagrangian for this system is
Thus the Hamiltonian is
where
and
In terms of coordinates and momenta, the Hamiltonian reads
Hamilton's equations give the time evolution of coordinates and conjugate momenta in four first-order differential equations,
Momentum formula_14, which corresponds to the vertical component of angular momentum formula_15, is a constant of motion. That is a consequence of the rotational symmetry of the system around the vertical axis. Being absent from the Hamiltonian, azimuth formula_16 is a cyclic coordinate, which implies conservation of its conjugate momentum.
Deriving Hamilton's equations.
Hamilton's equations can be derived by looking at how the total differential of the Lagrangian depends on time, generalized positions , and generalized velocities :
The generalized momenta were defined as
If this is substituted into the total differential of the Lagrangian, one gets
This can be rewritten as
which after rearranging leads to
The term on the left-hand side is just the Hamiltonian that was defined before, therefore
It is also possible to calculate the total differential of the Hamiltonian with respect to time directly, similar to what was carried on with the Lagrangian above, yielding:
It follows from the previous two independent equations that their right-hand sides are equal with each other. The result is
Since this calculation was done off-shell , one can associate corresponding terms from both sides of this equation to yield:
On-shell, Lagrange's equations indicate that
A rearrangement of this yields
Thus Hamilton's equations are
Hamilton's equations consist of first-order differential equations, while Lagrange's equations consist of second-order equations. Hamilton's equations usually do not reduce the difficulty of finding explicit solutions, but they still offer some advantages: Important theoretical results can be derived, because coordinates and momenta are independent variables with nearly symmetric roles.
Hamilton's equations have another advantage over Lagrange's equations: if a system has a symmetry, such that a coordinate does not occur in the Hamiltonian, the corresponding momentum is conserved, and that coordinate can be ignored in the other equations of the set. This effectively reduces the problem from coordinates to coordinates. In the Lagrangian framework, the result that the corresponding momentum is conserved still follows immediately, but all the generalized velocities still occur in the Lagrangian. A system of equations in n coordinates still has to be solved. The Lagrangian and Hamiltonian approaches provide the groundwork for deeper results in the theory of classical mechanics, and for formulations of quantum mechanics.
Hamiltonian of a charged particle in an electromagnetic field.
A sufficient illustration of Hamiltonian mechanics is given by the Hamiltonian of a charged particle in an electromagnetic field. In Cartesian coordinates the Lagrangian of a non-relativistic classical particle in an electromagnetic field is (in SI Units):
where is the electric charge of the particle, is the electric scalar potential, and the are the components of the magnetic vector potential that may all explicitly depend on formula_30 and formula_31.
This Lagrangian, combined with Euler–Lagrange equation, produces the Lorentz force law
and is called minimal coupling.
Note that the values of scalar potential and vector potential would change during a gauge transformation, and the Lagrangian itself will pick up extra terms as well; But the extra terms in Lagrangian add up to a total time derivative of a scalar function, and therefore won't change the Euler–Lagrange equation.
The canonical momenta are given by:
Note that canonical momenta are not gauge invariant, and are not physically measurable. However, the kinetic momentum:
is gauge invariant and physically measurable.
The Hamiltonian, as the Legendre transformation of the Lagrangian, is therefore:
This equation is used frequently in quantum mechanics.
Under gauge transformation:
where "f"(r,t) is any scalar function of space and time, the aforementioned Lagrangian, canonical momenta, and Hamiltonian transform like:
which still produces the same Hamilton's equation:
In quantum mechanics, the wave function will also undergo a local U(1) group transformation during the Gauge Transformation, which implies that all physical results must be invariant under local U(1) transformations.
Relativistic charged particle in an electromagnetic field.
The relativistic Lagrangian for a particle (rest mass and charge ) is given by:
Thus the particle's canonical momentum is
that is, the sum of the kinetic momentum and the potential momentum.
Solving for the velocity, we get
So the Hamiltonian is
This results in the force equation (equivalent to the Euler–Lagrange equation)
from which one can derive
The above derivation makes use of the vector calculus identity:
An equivalent expression for the Hamiltonian as function of the relativistic (kinetic) momentum, , is
This has the advantage that kinetic momentum can be measured experimentally whereas canonical momentum cannot. Notice that the Hamiltonian (total energy) can be viewed as the sum of the relativistic energy (kinetic+rest), , plus the potential energy, .
Mathematical structures.
Geometry of Hamiltonian systems.
The Hamiltonian can induce a symplectic structure on a smooth even-dimensional manifold "M"2"n" in several different, but equivalent, ways the best known among which are the following:
As a closed nondegenerate symplectic 2-form ω. According to the Darboux's theorem, in a small neighbourhood around any point on "M" in suitable local coordinates formula_47 exists the symplectic form 
The local coordinates "p", "q" are then called "canonical" or "symplectic".
The form formula_49 allows to construct a natural isomorphism formula_50 of the tangent space formula_51 and the cotangent space formula_52 This is done by mapping a vector formula_53 to the 1-form formula_54 where formula_55 for an arbitrary formula_56 Due to the bilinearity and non-degeneracy of formula_57 and the fact that formula_58 the mapping formula_59 is indeed a linear isomorphism. This isomorphism is "natural" in that it does not change with change of coordinates on formula_60 Repeating for every formula_61 we end up with an isomorphism formula_62 between the infinite-dimensional space of smooth vector fields and that of smooth 1-forms. For every formula_63 and formula_64
(In algebraic terms, one would say that the formula_66-modules formula_67 and formula_68 are isomorphic). If formula_69 then, for every fixed formula_70 formula_71 and formula_72 formula_73 is known as a Hamiltonian vector field. The respective differential equation on formula_74
is called "Hamilton's equation". Here formula_76 and formula_77 is the (time-dependent) value of the vector field formula_73 at formula_79
A Hamiltonian system may be understood as a fiber bundle over time , with the fibers , , being the position space. The Lagrangian is thus a function on the jet bundle over ; taking the fiberwise Legendre transform of the Lagrangian produces a function on the dual bundle over time whose fiber at is the cotangent space , which comes equipped with a natural symplectic form, and this latter function is the Hamiltonian. The correspondence between Lagrangian and Hamiltonian mechanics is achieved with the tautological one-form.
Any smooth real-valued function on a symplectic manifold can be used to define a Hamiltonian system. The function is known as "the Hamiltonian" or "the energy function." The symplectic manifold is then called the phase space. The Hamiltonian induces a special vector field on the symplectic manifold, known as the Hamiltonian vector field.
The Hamiltonian vector field induces a Hamiltonian flow on the manifold. This is a one-parameter family of transformations of the manifold (the parameter of the curves is commonly called "the time"); in other words, an isotopy of symplectomorphisms, starting with the identity. By Liouville's theorem, each symplectomorphism preserves the volume form on the phase space. The collection of symplectomorphisms induced by the Hamiltonian flow is commonly called "the Hamiltonian mechanics" of the Hamiltonian system.
The symplectic structure induces a Poisson bracket. The Poisson bracket gives the space of functions on the manifold the structure of a Lie algebra.
If "F" and "G" are smooth functions on "M" then the smooth function ω2("IdG", "IdF") is properly defined; it is called a "Poisson bracket" of functions "F" and "G" and is denoted {"F", "G"}. The Poisson bracket has the following properties:
Given a function 
if there is a probability distribution, , then (since the phase space velocity formula_84 has zero divergence and probability is conserved) its convective derivative can be shown to be zero and so
This is called Liouville's theorem. Every smooth function over the symplectic manifold generates a one-parameter family of symplectomorphisms and if , then is conserved and the symplectomorphisms are symmetry transformations.
A Hamiltonian may have multiple conserved quantities . If the symplectic manifold has dimension 2"n" and there are "n" functionally independent conserved quantities which are in involution (i.e., ), then the Hamiltonian is Liouville integrable. The Liouville–Arnold theorem says that, locally, any Liouville integrable Hamiltonian can be transformed via a symplectomorphism into a new Hamiltonian with the conserved quantities as coordinates; the new coordinates are called "action-angle coordinates". The transformed Hamiltonian depends only on the , and hence the equations of motion have the simple form
for some function . There is an entire field focusing on small deviations from integrable systems governed by the KAM theorem.
The integrability of Hamiltonian vector fields is an open question. In general, Hamiltonian systems are chaotic; concepts of measure, completeness, integrability and stability are poorly defined.
Riemannian manifolds.
An important special case consists of those Hamiltonians that are quadratic forms, that is, Hamiltonians that can be written as
where is a smoothly varying inner product on the fibers , the cotangent space to the point in the configuration space, sometimes called a cometric. This Hamiltonian consists entirely of the kinetic term.
If one considers a Riemannian manifold or a pseudo-Riemannian manifold, the Riemannian metric induces a linear isomorphism between the tangent and cotangent bundles. (See Musical isomorphism). Using this isomorphism, one can define a cometric. (In coordinates, the matrix defining the cometric is the inverse of the matrix defining the metric.) The solutions to the Hamilton–Jacobi equations for this Hamiltonian are then the same as the geodesics on the manifold. In particular, the Hamiltonian flow in this case is the same thing as the geodesic flow. The existence of such solutions, and the completeness of the set of solutions, are discussed in detail in the article on geodesics. See also Geodesics as Hamiltonian flows.
Sub-Riemannian manifolds.
When the cometric is degenerate, then it is not invertible. In this case, one does not have a Riemannian manifold, as one does not have a metric. However, the Hamiltonian still exists. In the case where the cometric is degenerate at every point of the configuration space manifold , so that the rank of the cometric is less than the dimension of the manifold , one has a sub-Riemannian manifold.
The Hamiltonian in this case is known as a sub-Riemannian Hamiltonian. Every such Hamiltonian uniquely determines the cometric, and vice versa. This implies that every sub-Riemannian manifold is uniquely determined by its sub-Riemannian Hamiltonian, and that the converse is true: every sub-Riemannian manifold has a unique sub-Riemannian Hamiltonian. The existence of sub-Riemannian geodesics is given by the Chow–Rashevskii theorem.
The continuous, real-valued Heisenberg group provides a simple example of a sub-Riemannian manifold. For the Heisenberg group, the Hamiltonian is given by
 is not involved in the Hamiltonian.
Poisson algebras.
Hamiltonian systems can be generalized in various ways. Instead of simply looking at the algebra of smooth functions over a symplectic manifold, Hamiltonian mechanics can be formulated on general commutative unital real Poisson algebras. A state is a continuous linear functional on the Poisson algebra (equipped with some suitable topology) such that for any element of the algebra, maps to a nonnegative real number.
A further generalization is given by Nambu dynamics.
Generalization to quantum mechanics through Poisson bracket.
Hamilton's equations above work well for classical mechanics, but not for quantum mechanics, since the differential equations discussed assume that one can specify the exact position and momentum of the particle simultaneously at any point in time. However, the equations can be further generalized to then be extended to apply to quantum mechanics as well as to classical mechanics, through the deformation of the Poisson algebra over and to the algebra of Moyal brackets.
Specifically, the more general form of the Hamilton's equation reads
where is some function of and , and is the Hamiltonian. To find out the rules for evaluating a Poisson bracket without resorting to differential equations, see Lie algebra; a Poisson bracket is the name for the Lie bracket in a Poisson algebra. These Poisson brackets can then be extended to Moyal brackets comporting to an inequivalent Lie algebra, as proven by Hilbrand J. Groenewold, and thereby describe quantum mechanical diffusion in phase space (See the phase space formulation and the Wigner-Weyl transform). This more algebraic approach not only permits ultimately extending probability distributions in phase space to Wigner quasi-probability distributions, but, at the mere Poisson bracket classical setting, also provides more power in helping analyze the relevant conserved quantities in a system.