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In continuum mechanics, the most general form of an exact conservation law is given by a continuity equation. For example, conservation of electric charge q is = where ∇⋅ is the divergence operator, ρ is the density of q (amount per unit volume), j is the flux of q (amount crossing a unit area in unit time), and t is time.
The continuity equation for the conserved current is a statement of a conservation law. Examples of canonical conjugate quantities are: Time and energy - the continuous translational symmetry of time implies the conservation of energy; Space and momentum - the continuous translational symmetry of space implies the conservation of momentum
Euler's second law states that the rate of change of angular momentum L about a point that is fixed in an inertial reference frame (often the center of mass of the body), is equal to the sum of the external moments of force acting on that body M about that point: [1] [4] [5]
Continuity equations are a stronger, local form of conservation laws. For example, a weak version of the law of conservation of energy states that energy can neither be created nor destroyed—i.e., the total amount of energy in the universe is fixed. This statement does not rule out the possibility that a quantity of energy could disappear ...
That is, conservation of momentum is a consequence of the fact that the laws of physics do not depend on position; this is a special case of Noether's theorem. [25] For systems that do not have this symmetry, it may not be possible to define conservation of momentum. Examples where conservation of momentum does not apply include curved ...
A continuity equation: Representing the conservation of mass. Conservation of momentum : Consisting of a form of the Navier–Stokes equations that describe hydrodynamical flow on the surface of a sphere under the assumption that vertical motion is much smaller than horizontal motion (hydrostasis) and that the fluid layer depth is small ...
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 =, where is the total amount of energy in the universe. Similarly, the first law of thermodynamics can be written as d U = δ Q − δ W {\displaystyle \mathrm {d} U=\delta Q-\delta W\,} , and Newton's ...
The advection equation is a first-order hyperbolic partial differential equation that governs the motion of a conserved scalar field as it is advected by a known velocity vector field. [1] It is derived using the scalar field's conservation law , together with Gauss's theorem , and taking the infinitesimal limit.