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:

${\displaystyle {\frac {\partial \rho }{\partial t}}=-\nabla \cdot \mathbf {J} }$

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.

Physics, conserved quantity	Conserved quantity q	Volume density ρ (of q)	Flux J (of q)	Equation
Hydrodynamics, fluids	m = mass (kg)	ρ = volume mass density (kg m−3)	ρ u, where u = velocity field of fluid (m s−1)	${\displaystyle {\frac {\partial \rho }{\partial t}}=-\nabla \cdot (\rho \mathbf {u} )}$
Electromagnetism, electric charge	q = electric charge (C)	ρ = volume electric charge density (C m−3)	J = electric current density (A m−2)	${\displaystyle {\frac {\partial \rho }{\partial t}}=-\nabla \cdot \mathbf {J} }$
Thermodynamics, energy	E = energy (J)	u = volume energy density (J m−3)	q = heat flux (W m−2)	${\displaystyle {\frac {\partial u}{\partial t}}=-\nabla \cdot \mathbf {q} }$
Quantum mechanics, probability	P = (r, t) = ∫|Ψ|2d3r = probability distribution	ρ = ρ(r, t) = |Ψ|2 = probability density function (m−3), Ψ = wavefunction of quantum system	j = probability current/flux	${\displaystyle {\frac {\partial |\Psi |^{2}}{\partial t}}=-\nabla \cdot \mathbf {j} }$

More general equations are the convection–diffusion equation and Boltzmann transport equation, which have their roots in the continuity equation.

Principle of least action

Classical mechanics, including Newton's laws, Lagrange's equations, Hamilton's equations, etc., can be derived from the following principle:

${\displaystyle \delta {\mathcal {S}}=\delta \int _{t_{1}}^{t_{2}}L(\mathbf {q} ,\mathbf {\dot {q}} ,t)\,dt=0}$

where ${\displaystyle {\mathcal {S}}}$ is the action; the integral of the Lagrangian

${\displaystyle L(\mathbf {q} ,\mathbf {\dot {q}} ,t)=T(\mathbf {\dot {q}} ,t)-V(\mathbf {q} ,\mathbf {\dot {q}} ,t)}$

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, ... q_N).

There are generalized momenta conjugate to these coordinates, p = (p₁, p₂, ..., p_N), where:

${\displaystyle p_{i}={\frac {\partial L}{\partial {\dot {q}}_{i}}}}$

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 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 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).^[12]

Notice L is not the total energy E of the system due to the difference, rather than the sum:

${\displaystyle E=T+V}$

The following^[13][14] 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.

Laws of motion
Principle of least action: ${\displaystyle {\mathcal {S}}=\int _{t_{1}}^{t_{2}}L\,\mathrm {d} t\,\!}$
The Euler–Lagrange equations are: ${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}\left({\frac {\partial L}{\partial {\dot {q}}_{i}}}\right)={\frac {\partial L}{\partial q_{i}}}}$ Using the definition of generalized momentum, there is the symmetry: ${\displaystyle p_{i}={\frac {\partial L}{\partial {\dot {q}}_{i}}}\quad {\dot {p}}_{i}={\frac {\partial L}{\partial {q}_{i}}}}$	Hamilton's equations ${\displaystyle {\dfrac {\partial \mathbf {p} }{\partial t}}=-{\dfrac {\partial H}{\partial \mathbf {q} }}}$ ${\displaystyle {\dfrac {\partial \mathbf {q} }{\partial t}}={\dfrac {\partial H}{\partial \mathbf {p} }}}$ The Hamiltonian as a function of generalized coordinates and momenta has the general form: ${\displaystyle H(\mathbf {q} ,\mathbf {p} ,t)=\mathbf {p} \cdot \mathbf {\dot {q}} -L}$
	Hamilton–Jacobi equation ${\displaystyle H\left(\mathbf {q} ,{\frac {\partial S}{\partial \mathbf {q} }},t\right)=-{\frac {\partial S}{\partial t}}}$
Newton's laws Newton's laws of motion They are low-limit solutions to relativity. Alternative formulations of Newtonian mechanics are Lagrangian and Hamiltonian mechanics. The laws can be summarized by two equations (since the 1st is a special case of the 2nd, zero resultant acceleration): ${\displaystyle \mathbf {F} ={\frac {\mathrm {d} \mathbf {p} }{\mathrm {d} t}},\quad \mathbf {F} _{ij}=-\mathbf {F} _{ji}}$ where p = momentum of body, Fij = force on body i by body j, Fji = force on body j by body i. For a dynamical system the two equations (effectively) combine into one: ${\displaystyle {\frac {\mathrm {d} \mathbf {p} _{\mathrm {i} }}{\mathrm {d} t}}=\mathbf {F} _{E}+\sum _{\mathrm {i} \neq \mathrm {j} }\mathbf {F} _{\mathrm {ij} }\,\!}$ in which FE = resultant external force (due to any agent not part of system). Body i does not exert a force on itself.

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's law
• Navier–Stokes equations
• Faxén's law

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

${\displaystyle A'=\Lambda A}$

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):

${\displaystyle E^{2}=(pc)^{2}+(mc^{2})^{2}}$

in which the (more famous) mass–energy equivalence E = mc² 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.

Gravitomagnetism

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.^[15]

Einstein field equations (EFE): ${\displaystyle R_{\mu \nu }+\left(\Lambda -{\frac {R}{2}}\right)g_{\mu \nu }={\frac {8\pi G}{c^{4}}}T_{\mu \nu }\,\!}$ where Λ = cosmological constant, Rμν = Ricci curvature tensor, Tμν = Stress–energy tensor, gμν = metric tensor	Geodesic equation: ${\displaystyle {\frac {{\rm {d}}^{2}x^{\lambda }}{{\rm {d}}t^{2}}}+\Gamma _{\mu \nu }^{\lambda }{\frac {{\rm {d}}x^{\mu }}{{\rm {d}}t}}{\frac {{\rm {d}}x^{\nu }}{{\rm {d}}t}}=0\ ,}$ where Γ is a Christoffel symbol of the second kind, containing the metric.
GEM Equations If g the gravitational field and H the gravitomagnetic field, the solutions in these limits are: ${\displaystyle \nabla \cdot \mathbf {g} =-4\pi G\rho \,\!}$ ${\displaystyle \nabla \cdot \mathbf {H} =\mathbf {0} \,\!}$ ${\displaystyle \nabla \times \mathbf {g} =-{\frac {\partial \mathbf {H} }{\partial t}}\,\!}$ ${\displaystyle \nabla \times \mathbf {H} ={\frac {4}{c^{2}}}\left(-4\pi G\mathbf {J} +{\frac {\partial \mathbf {g} }{\partial t}}\right)\,\!}$ where ρ is the mass density and J is the mass current density or mass flux.
In addition there is the gravitomagnetic Lorentz force: ${\displaystyle \mathbf {F} =\gamma (\mathbf {v} )m\left(\mathbf {g} +\mathbf {v} \times \mathbf {H} \right)}$ where m is the rest mass of the particlce and γ is the Lorentz factor.

Classical laws

Kepler's Laws, though originally discovered from planetary observations (also due to Tycho Brahe), are true for any central forces.^[16]

Newton's law of universal gravitation: For two point masses: ${\displaystyle \mathbf {F} ={\frac {Gm_{1}m_{2}}{\left|\mathbf {r} \right|^{2}}}\mathbf {\hat {r}} \,\!}$ For a non uniform mass distribution of local mass density ρ (r) of body of Volume V, this becomes: ${\displaystyle \mathbf {g} =G\int _{V}{\frac {\mathbf {r} \rho \,\mathrm {d} {V}}{\left|\mathbf {r} \right|^{3}}}\,\!}$	Gauss' law for gravity: An equivalent statement to Newton's law is: ${\displaystyle \nabla \cdot \mathbf {g} =4\pi G\rho \,\!}$
Kepler's 1st Law: Planets move in an ellipse, with the star at a focus ${\displaystyle r={\frac {\ell }{1+e\cos \theta }}\,\!}$ where ${\displaystyle e={\sqrt {1-(b/a)^{2}}}}$ is the eccentricity of the elliptic orbit, of semi-major axis a and semi-minor axis b, and ℓ is the semi-latus rectum. This equation in itself is nothing physically fundamental; simply the polar equation of an ellipse in which the pole (origin of polar coordinate system) is positioned at a focus of the ellipse, where the orbited star is.
Kepler's 2nd Law: equal areas are swept out in equal times (area bounded by two radial distances and the orbital circumference): ${\displaystyle {\frac {\mathrm {d} A}{\mathrm {d} t}}={\frac {\left|\mathbf {L} \right|}{2m}}\,\!}$ where L is the orbital angular momentum of the particle (i.e. planet) of mass m about the focus of orbit,
Kepler's 3rd Law: The square of the orbital time period T is proportional to the cube of the semi-major axis a: ${\displaystyle T^{2}={\frac {4\pi ^{2}}{G\left(m+M\right)}}a^{3}\,\!}$ where M is the mass of the central body (i.e. star).