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The difference between a conservative and a non-conservative force is that when a conservative force moves an object from one point to another, the work done by the conservative force is independent of the path. On the contrary, when a non-conservative force acts upon an object, the work done by the non-conservative force is dependent of the path.
Gravitational force is an example of a conservative force, while frictional force is an example of a non-conservative force. Other examples of conservative forces are: force in elastic spring, electrostatic force between two electric charges, and magnetic force between two magnetic poles. The last two forces are called central forces as they ...
The Hopf decomposition states that dynamical systems can be decomposed into a conservative and a dissipative part; more precisely, it states that every measure space with a non-singular transformation can be decomposed into an invariant conservative set and an invariant dissipative set.
The conservative force acts to give beads a chemical identity, while the dissipative and random forces together form a thermostat that keeps the mean temperature of the system constant. A key property of all of the non-bonded forces is that they conserve momentum locally, so that hydrodynamic modes of the fluid emerge even for small particle ...
A non-singular dynamical system is conservative if, for every set of positive measure and for every , one has some integer > such that () >. Informally, this can be interpreted as saying that the current state of the system revisits or comes arbitrarily close to a prior state; see Poincaré recurrence for more.
In Cartesian coordinates the Lagrangian of a non-relativistic classical particle in an electromagnetic field is (in SI Units): = ˙ + ˙, where q is the electric charge of the particle, φ is the electric scalar potential, and the A i are the components of the magnetic vector potential that may all explicitly depend on and .
A local conservation law is usually expressed mathematically as a continuity equation, a partial differential equation which gives a relation between the amount of the quantity and the "transport" of that quantity. It states that the amount of the conserved quantity at a point or within a volume can only change by the amount of the quantity ...
It does not always work for non-conservative forces or dissipative forces like friction, in which case one may revert to Newtonian mechanics. Two dominant branches of analytical mechanics are Lagrangian mechanics (using generalized coordinates and corresponding generalized velocities in configuration space ) and Hamiltonian mechanics (using ...