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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 ...
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
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 laws of physics are invariant with respect to orientation—for example, floating in outer space, there is no measurement you can do to say "which way is up"; the laws of physics are the same regardless of how you are oriented. This symmetry leads to the continuity equation for conservation of angular momentum.
As another example, if a physical process exhibits the same outcomes regardless of place or time, then its Lagrangian is symmetric under continuous translations in space and time respectively: by Noether's theorem, these symmetries account for the conservation laws of linear momentum and energy within this system, respectively.
Bicycles and motorcycles, flying discs, [1] rifled bullets, and gyroscopes owe their useful properties to conservation of angular momentum. Conservation of angular momentum is also why hurricanes [2] form spirals and neutron stars have high rotational rates. In general, conservation limits the possible motion of a system, but it does not ...
The moment of force, or torque, is a first moment: =, or, more generally, .; Similarly, angular momentum is the 1st moment of momentum: =.Momentum itself is not a moment.; The electric dipole moment is also a 1st moment: = for two opposite point charges or () for a distributed charge with charge density ().