Conservation laws Conservation and symmetry Conservation laws are fundamental laws that follow from the homogeneity of space, time and
phase, in other words
symmetry. • '''
Noether's theorem''': Any quantity with a continuously differentiable symmetry in the action has an associated conservation law. •
Conservation of mass was the first law to be understood since most macroscopic physical processes involving masses, for example, collisions of massive particles or fluid flow, provide the apparent belief that mass is conserved. Mass conservation was observed to be true for all chemical reactions. In general, this is only approximative because with the advent of relativity and experiments in nuclear and particle physics: mass can be transformed into energy and vice versa, so mass is not always conserved but part of the more general conservation of
mass–energy. •
Conservation of energy,
momentum and
angular momentum for isolated systems can be found to be
symmetries in time, translation, and rotation. •
Conservation of charge was also realized since charge has never been observed to be created or destroyed and only found to move from place to place.
Continuity and transfer Conservation laws can be expressed using the general
continuity equation (for a conserved quantity) can be written in differential form as: : \frac{\partial \rho}{\partial t}=-\nabla \cdot \mathbf{J} 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 the 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: : \delta \mathcal{S} = \delta\int_{t_1}^{t_2} L(\mathbf{q}, \mathbf{\dot{q}}, t) \, dt = 0 where \mathcal{S} is the
action; the integral of the
Lagrangian : L(\mathbf{q}, \mathbf{\dot{q}}, t) = T(\mathbf{\dot{q}}, t)-V(\mathbf{q}, \mathbf{\dot{q}}, t) of the physical system between two times
t1 and
t2. 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 = (
q1,
q2, ...
qN). There are
generalized momenta conjugate to these coordinates,
p = (
p1,
p2, ...,
pN), where: : p_i = \frac{\partial L}{\partial \dot{q}_i} 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). 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
t1 to
t2). 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: : E=T+V 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.
Corollaries in mechanics: •
Euler's laws of motion •
Euler's equations (rigid body dynamics) Corollaries in fluid mechanics: 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. •
Archimedes' principle •
Bernoulli's principle •
Poiseuille's law •
Stokes' law •
Navier–Stokes equations •
Faxén's law 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 Special relativity: The two postulates of special relativity are not "laws" in themselves, but assumptions of their nature in terms of
relative motion. They can be stated as "the laws of physics are the same in all
inertial frames" and "the
speed of light is constant and has the same value in all inertial frames". The said postulates lead to the
Lorentz transformations – the transformation law between two
frame of references moving relative to each other. For any
4-vector : A' =\Lambda A 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): : E^2 = (pc)^2 + (mc^2)^2 in which the (more famous)
mass–energy equivalence is a special case.
General relativity: 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.
Gravitoelectromagnetism: 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. :
Classical laws Kepler's laws, though originally discovered from planetary observations (also due to
Tycho Brahe), are true for any
central forces. : {2m} \,\! where
L is the orbital angular momentum of the particle (i.e. planet) of mass
m about the focus of orbit, : T^2 = \frac{4\pi^2}{G \left ( m + M \right ) } a^3\,\! where
M is the mass of the central body (i.e. star).
Thermodynamics : •
Newton's law of cooling •
Fourier's law •
Ideal gas law, combines a number of separately developed gas laws; •
Boyle's law •
Charles's law •
Gay-Lussac's law •
Avogadro's law, into one : now improved by other
equations of state •
Dalton's law (of partial pressures) •
Boltzmann equation •
Carnot's theorem •
Kopp's law Electromagnetism Maxwell's equations give the time-evolution of the
electric and
magnetic fields due to
electric charge and
current distributions. Given the fields, the
Lorentz force law is the
equation of motion for charges in the fields. : 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.
Pre-Maxwell laws: 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's law (electrostatic form) and the Biot–Savart law can be deduced from Ampere's law (magnetostatic form). Lenz's law and Faraday's law can be incorporated into the Maxwell–Faraday equation. Nonetheless, they are still very effective for simple calculations. •
Lenz's law •
Coulomb's law •
Biot–Savart law Other laws: •
Ohm's law •
Kirchhoff's laws •
Joule's law Photonics Classically,
optics is based on a
variational principle: light travels from one point in space to another in the shortest time. •
Fermat's principle In
geometric optics laws are based on approximations in Euclidean geometry (such as the
paraxial approximation). •
Law of reflection •
Law of refraction,
Snell's law In
physical optics, laws are based on physical properties of materials. •
Brewster's angle •
Malus's law •
Beer–Lambert law 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. These postulates can be summarized as follows: • The state of a physical system, be it a particle or a system of many particles, is described by a
wavefunction. • Every physical quantity is described by an
operator acting on the system; the measured quantity has a
probabilistic nature. • The
wavefunction obeys the
Schrödinger equation. Solving this wave equation predicts the time-evolution of the system's behavior, analogous to solving Newton's laws in classical mechanics. • Two
identical particles, such as two electrons, cannot be distinguished from one another by any means. Physical systems are classified by their symmetry properties. These postulates in turn imply many other phenomena, e.g.,
uncertainty principles and the
Pauli exclusion principle. :
Radiation laws Applying electromagnetism, thermodynamics, and quantum mechanics, to atoms and molecules, some laws of
electromagnetic radiation and light are as follows. •
Stefan–Boltzmann law •
Planck's law of black-body radiation •
Wien's displacement law •
Radioactive decay law == Laws of chemistry ==