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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.
A form of Newton's second law, that force is the rate of change of momentum, also holds, as does the conservation of momentum. However, the definition of momentum is modified. Among the consequences of this is the fact that the more quickly a body moves, the harder it is to accelerate, and so, no matter how much force is applied, a body cannot ...
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 ...
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
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 ...
A general momentum equation is obtained when the conservation relation is applied to momentum. When the intensive property φ is considered as the mass flux (also momentum density ), that is, the product of mass density and flow velocity ρ u , by substitution into the general continuity equation:
Examples of integrals of motion are the angular momentum vector, =, or a Hamiltonian without time dependence, such as (,) = + (). An example of a function that is a constant of motion but not an integral of motion would be the function C ( x , v , t ) = x − v t {\displaystyle C(x,v,t)=x-vt} for an object moving at a constant speed in one ...
There are two conservation laws at work when a gun is fired: conservation of momentum and conservation of energy. Recoil is explained by the law of conservation of momentum, and so it is easier to discuss it separately from energy. Momentum is simply mass multiplied by velocity. Velocity is speed in a particular direction (not just speed).
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