7.7 KiB
| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Scientific law | 3/6 | https://en.wikipedia.org/wiki/Scientific_law | reference | science, encyclopedia | 2026-05-05T03:45:43.771670+00:00 | kb-cron |
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:
δ
S
=
δ
∫
t
1
t
2
L
(
q
,
q
˙
,
t
)
d
t
=
0
{\displaystyle \delta {\mathcal {S}}=\delta \int _{t_{1}}^{t_{2}}L(\mathbf {q} ,\mathbf {\dot {q}} ,t)\,dt=0}
where
S
{\displaystyle {\mathcal {S}}}
is the action; the integral of the Lagrangian
L
(
q
,
q
˙
,
t
)
=
T
(
q
˙
,
t
)
−
V
(
q
,
q
˙
,
t
)
{\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, ... qN). There are generalized momenta conjugate to these coordinates, p = (p1, p2, ..., pN), where:
p
i
=
∂
L
∂
q
˙
i
{\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). 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
{\displaystyle 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
′
=
Λ
A
{\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):
E
2
=
(
p
c
)
2
+
(
m
c
2
)
2
{\displaystyle E^{2}=(pc)^{2}+(mc^{2})^{2}}