Scientific law

Scientific laws or laws of science are statements, based on repeated experiments or observations, that describe or predict a range of natural phenomena. The term "law" has diverse usage in many cases (approximate, accurate, broad, or narrow) across all fields of natural science (physics, chemistry, astronomy, geoscience, biology). Laws are developed from data and can be further developed through mathematics; in all cases they are directly or indirectly based on empirical evidence. It is generally understood that they implicitly reflect, though they do not explicitly assert, causal relationships fundamental to reality, and are discovered rather than invented.
Scientific laws summarize the results of experiments or observations, usually within a certain range of application. In general, the accuracy of a law does not change when a new theory of the relevant phenomenon is worked out, but rather the scope of the law's application, since the mathematics or statement representing the law does not change. As with other kinds of scientific knowledge, scientific laws do not express absolute certainty, as mathematical theorems or identities do. A scientific law may be contradicted, restricted, or extended by future observations. 
A law can usually be formulated as one or several statements or equations, so that it can predict the outcome of an experiment. Laws differ from hypotheses and postulates, which are proposed during the scientific process before and during validation by experiment and observation. Hypotheses and postulates are not laws, since they have not been verified to the same degree, although they may lead to the formulation of laws. Laws are narrower in scope than scientific theories, which may entail one or several laws. Science distinguishes a law or theory from facts. Calling a law a fact is ambiguous, an overstatement, or an equivocation. The nature of scientific laws has been much discussed in philosophy, but in essence scientific laws are simply empirical conclusions reached by scientific method; they are intended to be neither laden with ontological commitments nor statements of logical absolutes.
Overview.
A scientific law always applies to a physical system under repeated conditions, and it implies that there is a causal relationship involving the elements of the system. Factual and well-confirmed statements like "Mercury is liquid at standard temperature and pressure" are considered too specific to qualify as scientific laws. A central problem in the philosophy of science, going back to David Hume, is that of distinguishing causal relationships (such as those implied by laws) from principles that arise due to constant conjunction.
Laws differ from scientific theories in that they do not posit a mechanism or explanation of phenomena: they are merely distillations of the results of repeated observation. As such, the applicability of a law is limited to circumstances resembling those already observed, and the law may be found to be false when extrapolated. Ohm's law only applies to linear networks; Newton's law of universal gravitation only applies in weak gravitational fields; the early laws of aerodynamics, such as Bernoulli's principle, do not apply in the case of compressible flow such as occurs in transonic and supersonic flight; Hooke's law only applies to strain below the elastic limit; Boyle's law applies with perfect accuracy only to the ideal gas, etc. These laws remain useful, but only under the specified conditions where they apply.
Many laws take mathematical forms, and thus can be stated as an equation; for example, the law of conservation of energy can be written as formula_1, where formula_2 is the total amount of energy in the universe. Similarly, the first law of thermodynamics can be written as formula_3, and Newton's second law can be written as formula_4 . While these scientific laws explain what our senses perceive, they are still empirical (acquired by observation or scientific experiment) and so are not like mathematical theorems which can be proved purely by mathematics.
Like theories and hypotheses, laws make predictions; specifically, they predict that new observations will conform to the given law. Laws can be falsified if they are found in contradiction with new data.
Some laws are only approximations of other more general laws, and are good approximations with a restricted domain of applicability. For example, Newtonian dynamics (which is based on Galilean transformations) is the low-speed limit of special relativity (since the Galilean transformation is the low-speed approximation to the Lorentz transformation). Similarly, the Newtonian gravitation law is a low-mass approximation of general relativity, and Coulomb's law is an approximation to quantum electrodynamics at large distances (compared to the range of weak interactions). In such cases it is common to use the simpler, approximate versions of the laws, instead of the more accurate general laws.
Laws are constantly being tested experimentally to increasing degrees of precision, which is one of the main goals of science. The fact that laws have never been observed to be violated does not preclude testing them at increased accuracy or in new kinds of conditions to confirm whether they continue to hold, or whether they break, and what can be discovered in the process. It is always possible for laws to be invalidated or proven to have limitations, by repeatable experimental evidence, should any be observed. Well-established laws have indeed been invalidated in some special cases, but the new formulations created to explain the discrepancies generalize upon, rather than overthrow, the originals. That is, the invalidated laws have been found to be only close approximations, to which other terms or factors must be added to cover previously unaccounted-for conditions, e.g. very large or very small scales of time or space, enormous speeds or masses, etc. Thus, rather than unchanging knowledge, physical laws are better viewed as a series of improving and more precise generalizations.
Properties.
Scientific laws are typically conclusions based on repeated scientific experiments and observations over many years and which have become accepted universally within the scientific community. A scientific law is "inferred from particular facts, applicable to a defined group or class of phenomena, and expressible by the statement that a particular phenomenon always occurs if certain conditions be present." The production of a summary description of our environment in the form of such laws is a fundamental aim of science.
Several general properties of scientific laws, particularly when referring to laws in physics, have been identified. Scientific laws are:
The term "scientific law" is traditionally associated with the natural sciences, though the social sciences also contain laws. For example, Zipf's law is a law in the social sciences which is based on mathematical statistics. In these cases, laws may describe general trends or expected behaviors rather than being absolutes.
In natural science, impossibility assertions come to be widely accepted as overwhelmingly probable rather than considered proved to the point of being unchallengeable. The basis for this strong acceptance is a combination of extensive evidence of something not occurring, combined with an underlying theory, very successful in making predictions, whose assumptions lead logically to the conclusion that something is impossible. While an impossibility assertion in natural science can never be absolutely proved, it could be refuted by the observation of a single counterexample. Such a counterexample would require that the assumptions underlying the theory that implied the impossibility be re-examined.
Some examples of widely accepted impossibilities in physics are perpetual motion machines, which violate the law of conservation of energy, exceeding the speed of light, which violates the implications of special relativity, the uncertainty principle of quantum mechanics, which asserts the impossibility of simultaneously knowing both the position and the momentum of a particle, and Bell's theorem: no physical theory of local hidden variables can ever reproduce all of the predictions of quantum mechanics.
Laws as consequences of mathematical symmetries.
Some laws reflect mathematical symmetries found in Nature (e.g. the Pauli exclusion principle reflects identity of electrons, conservation laws reflect homogeneity of space, time, and Lorentz transformations reflect rotational symmetry of spacetime). Many fundamental physical laws are mathematical consequences of various symmetries of space, time, or other aspects of nature. Specifically, Noether's theorem connects some conservation laws to certain symmetries. For example, conservation of energy is a consequence of the shift symmetry of time (no moment of time is different from any other), while conservation of momentum is a consequence of the symmetry (homogeneity) of space (no place in space is special, or different than any other). The indistinguishability of all particles of each fundamental type (say, electrons, or photons) results in the Dirac and Bose quantum statistics which in turn result in the Pauli exclusion principle for fermions and in Bose–Einstein condensation for bosons. The rotational symmetry between time and space coordinate axes (when one is taken as imaginary, another as real) results in Lorentz transformations which in turn result in special relativity theory. Symmetry between inertial and gravitational mass results in general relativity.
The inverse square law of interactions mediated by massless bosons is the mathematical consequence of the 3-dimensionality of space.
One strategy in the search for the most fundamental laws of nature is to search for the most general mathematical symmetry group that can be applied to the fundamental interactions.
Laws of physics.
Conservation laws.
Conservation and symmetry.
Conservation laws are fundamental laws that follow from the homogeneity of space, time and phase, in other words "symmetry".
Continuity and transfer.
Conservation laws can be expressed using the general continuity equation (for a conserved quantity) can be written in differential form as:
where ρ is some quantity per unit volume, J is the flux of that quantity (change in quantity per unit time per unit area). Intuitively, the divergence (denoted ∇•) of a vector field is a measure of flux diverging radially outwards from a point, so the negative is the amount piling up at a point, hence the rate of change of density in a region of space must be the amount of flux leaving or collecting in some region (see main article for details). In the table below, the fluxes, flows for various physical quantities in transport, and their associated continuity equations, are collected for comparison.
More general equations are the convection–diffusion equation and Boltzmann transport equation, which have their roots in the continuity equation.
Laws of classical mechanics.
Principle of least action.
Classical mechanics, including Newton's laws, Lagrange's equations, Hamilton's equations, etc., can be derived from the following principle:
where formula_7 is the action; the integral of the Lagrangian
of the physical system between two times "t"1 and "t"2. The kinetic energy of the system is "T" (a function of the rate of change of the configuration of the system), and potential energy is "V" (a function of the configuration and its rate of change). The configuration of a system which has "N" degrees of freedom is defined by generalized coordinates q = ("q"1, "q"2, ... "qN").
There are generalized momenta conjugate to these coordinates, p = ("p"1, "p"2, ..., "pN"), where:
The action and Lagrangian both contain the dynamics of the system for all times. The term "path" simply refers to a curve traced out by the system in terms of the generalized coordinates in the configuration space, i.e. the curve q("t"), parameterized by time (see also parametric equation for this concept).
The action is a "functional" rather than a "function", since it depends on the Lagrangian, and the Lagrangian depends on the path q("t"), so the action depends on the "entire" "shape" of the path for all times (in the time interval from "t"1 to "t"2). Between two instants of time, there are infinitely many paths, but one for which the action is stationary (to the first order) is the true path. The stationary value for the "entire continuum" of Lagrangian values corresponding to some path, "not just one value" of the Lagrangian, is required (in other words it is "not" as simple as "differentiating a function and setting it to zero, then solving the equations to find the points of maxima and minima etc", rather this idea is applied to the entire "shape" of the function, see calculus of variations for more details on this procedure).
Notice "L" is "not" the total energy "E" of the system due to the difference, rather than the sum:
The following general approaches to classical mechanics are summarized below in the order of establishment. They are equivalent formulations. Newton's is commonly used due to simplicity, but Hamilton's and Lagrange's equations are more general, and their range can extend into other branches of physics with suitable modifications.
From the above, any equation of motion in classical mechanics can be derived.
Equations describing fluid flow in various situations can be derived, using the above classical equations of motion and often conservation of mass, energy and momentum. Some elementary examples follow.
Laws of gravitation and relativity.
Some of the more famous laws of nature are found in Isaac Newton's theories of (now) classical mechanics, presented in his "Philosophiae Naturalis Principia Mathematica", and in Albert Einstein's theory of relativity. 
Modern laws.
Postulates of special relativity are not "laws" in themselves, but assumptions of their nature in terms of "relative motion".
Often two are stated as "the laws of physics are the same in all inertial frames" and "the speed of light is constant". However the second is redundant, since the speed of light is predicted by Maxwell's equations. Essentially there is only one.
The said postulate leads to the Lorentz transformations – the transformation law between two frame of references moving relative to each other. For any 4-vector
this replaces the Galilean transformation law from classical mechanics. The Lorentz transformations reduce to the Galilean transformations for low velocities much less than the speed of light "c".
The magnitudes of 4-vectors are invariants - "not" "conserved", but the same for all inertial frames (i.e. every observer in an inertial frame will agree on the same value), in particular if "A" is the four-momentum, the magnitude can derive the famous invariant equation for mass-energy and momentum conservation (see invariant mass):
in which the (more famous) mass-energy equivalence "E" = "mc"2 is a special case.
General relativity is governed by the Einstein field equations, which describe the curvature of space-time due to mass-energy equivalent to the gravitational field. Solving the equation for the geometry of space warped due to the mass distribution gives the metric tensor. Using the geodesic equation, the motion of masses falling along the geodesics can be calculated.
In a relatively flat spacetime due to weak gravitational fields, gravitational analogues of Maxwell's equations can be found; the GEM equations, to describe an analogous "gravitomagnetic field". They are well established by the theory, and experimental tests form ongoing research.
These equations can be modified to include magnetic monopoles, and are consistent with our observations of monopoles either existing or not existing; if they do not exist, the generalized equations reduce to the ones above, if they do, the equations become fully symmetric in electric and magnetic charges and currents. Indeed, there is a duality transformation where electric and magnetic charges can be "rotated into one another", and still satisfy Maxwell's equations.
These laws were found before the formulation of Maxwell's equations. They are not fundamental, since they can be derived from Maxwell's Equations. Coulomb's Law can be found from Gauss' Law (electrostatic form) and the Biot–Savart Law can be deduced from Ampere's Law (magnetostatic form). Lenz' Law and Faraday's Law can be incorporated into the Maxwell-Faraday equation. Nonetheless they are still very effective for simple calculations.
Photonics.
Classically, optics is based on a variational principle: light travels from one point in space to another in the shortest time.
In geometric optics laws are based on approximations in Euclidean geometry (such as the paraxial approximation).
In physical optics, laws are based on physical properties of materials.
In actuality, optical properties of matter are significantly more complex and require quantum mechanics.
Laws of quantum mechanics.
Quantum mechanics has its roots in postulates. This leads to results which are not usually called "laws", but hold the same status, in that all of quantum mechanics follows from them.
One postulate that a particle (or a system of many particles) is described by a wavefunction, and this satisfies a quantum wave equation: namely the Schrödinger equation (which can be written as a non-relativistic wave equation, or a relativistic wave equation). Solving this wave equation predicts the time-evolution of the system's behaviour, analogous to solving Newton's laws in classical mechanics.
Other postulates change the idea of physical observables; using quantum operators; some measurements can't be made at the same instant of time (Uncertainty principles), particles are fundamentally indistinguishable. Another postulate; the wavefunction collapse postulate, counters the usual idea of a measurement in science.
Radiation laws.
Applying electromagnetism, thermodynamics, and quantum mechanics, to atoms and molecules, some laws of electromagnetic radiation and light are as follows.
Laws of chemistry.
Chemical laws are those laws of nature relevant to chemistry. Historically, observations led to many empirical laws, though now it is known that chemistry has its foundations in quantum mechanics.
The most fundamental concept in chemistry is the law of conservation of mass, which states that there is no detectable change in the quantity of matter during an ordinary chemical reaction. Modern physics shows that it is actually energy that is conserved, and that energy and mass are related; a concept which becomes important in nuclear chemistry. Conservation of energy leads to the important concepts of equilibrium, thermodynamics, and kinetics.
Additional laws of chemistry elaborate on the law of conservation of mass. Joseph Proust's law of definite composition says that pure chemicals are composed of elements in a definite formulation; we now know that the structural arrangement of these elements is also important.
Dalton's law of multiple proportions says that these chemicals will present themselves in proportions that are small whole numbers; although in many systems (notably biomacromolecules and minerals) the ratios tend to require large numbers, and are frequently represented as a fraction.
The law of definite composition and the law of multiple proportions are the first two of the three laws of stoichiometry, the proportions by which the chemical elements combine to form chemical compounds. The third law of stoichiometry is the law of reciprocal proportions, which provides the basis for establishing equivalent weights for each chemical element. Elemental equivalent weights can then be used to derive atomic weights for each element.
More modern laws of chemistry define the relationship between energy and its transformations.
Other fields.
Some mathematical theorems and axioms are referred to as laws because they provide logical foundation to empirical laws.
Examples of other observed phenomena sometimes described as laws include the Titius–Bode law of planetary positions, Zipf's law of linguistics, and Moore's law of technological growth. Many of these laws fall within the scope of uncomfortable science. Other laws are pragmatic and observational, such as the law of unintended consequences. By analogy, principles in other fields of study are sometimes loosely referred to as "laws". These include Occam's razor as a principle of philosophy and the Pareto principle of economics.
History.
The observation and detection of underlying regularities in nature date from prehistoric times - the recognition of cause-and-effect relationships implicitly recognises the existence of laws of nature. The recognition of such regularities as independent scientific laws "per se", though, was limited by their entanglement in animism, and by the attribution of many effects that do not have readily obvious causes—such as physical phenomena—to the actions of gods, spirits, supernatural beings, etc. Observation and speculation about nature were intimately bound up with metaphysics and morality.
In Europe, systematic theorizing about nature ("physis") began with the early Greek philosophers and scientists and continued into the Hellenistic and Roman imperial periods, during which times the intellectual influence of Roman law increasingly became paramount.The formula "law of nature" first appears as "a live metaphor" favored by Latin poets Lucretius, Virgil, Ovid, Manilius, in time gaining a firm theoretical presence in the prose treatises of Seneca and Pliny. Why this Roman origin? According to [historian and classicist Daryn] Lehoux's persuasive narrative, the idea was made possible by the pivotal role of codified law and forensic argument in Roman life and culture.
For the Romans . . . the place par excellence where ethics, law, nature, religion and politics overlap is the law court. When we read Seneca's "Natural Questions", and watch again and again just how he applies standards of evidence, witness evaluation, argument and proof, we can recognize that we are reading one of the great Roman rhetoricians of the age, thoroughly immersed in forensic method. And not Seneca alone. Legal models of scientific judgment turn up all over the place, and for example prove equally integral to Ptolemy's approach to verification, where the mind is assigned the role of magistrate, the senses that of disclosure of evidence, and dialectical reason that of the law itself.
The precise formulation of what are now recognized as modern and valid statements of the laws of nature dates from the 17th century in Europe, with the beginning of accurate experimentation and the development of advanced forms of mathematics. During this period, natural philosophers such as Isaac Newton (1642-1727) were influenced by a religious view - stemming from medieval concepts of divine law - which held that God had instituted absolute, universal and immutable physical laws. In chapter 7 of "The World", René Descartes (1596-1650) described "nature" as matter itself, unchanging as created by God, thus changes in parts "are to be attributed to nature. The rules according to which these changes take place I call the 'laws of nature'." The modern scientific method which took shape at this time (with Francis Bacon (1561-1626) and Galileo (1564-1642)) contributed to a trend of separating science from theology, with minimal speculation about metaphysics and ethics. (Natural law in the political sense, conceived as universal (i.e., divorced from sectarian religion and accidents of place), was also elaborated in this period by scholars such as Grotius (1583-1645), Spinoza (1632-1677), and Hobbes (1588-1679).)
The distinction between natural law in the political-legal sense and law of nature or physical law in the scientific sense is a modern one, both concepts being equally derived from "physis", the Greek word (translated into Latin as "natura") for "nature".