Integral of motion

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In physics, an integral of motion is a term for a conserved quantity that does not change as time evolves, cf. also constant of motion. In the case of classical mechanics, an integral of motion is a function defined on the phase space (a function of coordinates and their canonical momenta) that is invariant under the Hamiltonian flow. In other words, the Poisson bracket of the Hamiltonian with integrals of motion must vanish.

In nonintegrable models, integrals of motion are rare -- with the exception of Noether charges like the total momentum or angular momentum --because such systems are typically ergodic/chaotic but whenever they are found, they allow one to simplify the differential equations and effectively remove some of the coordinates and their momenta.