Liouville's Phase Space Theorem

Liouville's phase space theorem states that the phase flow of a Hamiltonian system preserves phase space volume. On a -dimensional phase space with position coordinates and momentum coordinates , the symplectic form is

(1)

and the associated differential 2n-form is

(2)

If denotes the phase flow, then the pullback map satisfies and therefore . Equivalently, for every region in phase space,

(3)

This volume integral is the top-dimensional Poincaré integral invariant (Arnold 1989, pp. 237-239).

For one degree of freedom, a small time increment gives

			(4)
			(5)
			(6)
			(7)

The Jacobian is then equal to one to first order:

			(8)
			(9)
			(10)

In the continuous-time limit, the Jacobian of the phase flow is 1. In particular, symplectic maps preserve the Liouville measure, and their Jacobians have determinant 1.

See also

Liouville Measure, Phase Space, Poincaré Integral Invariant, Symplectic Form, Symplectic Map

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References

Arnold, V. I. Mathematical Methods of Classical Mechanics, 2nd ed. New York: Springer-Verlag, pp. 237-239, 1989. https://doi.org/10.1007/978-1-4757-2063-1.Chavel, I. Riemannian Geometry: A Modern Introduction. New York: Cambridge University Press, 1994.

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Liouville's Phase Space Theorem

Cite this as:

Weisstein, Eric W. "Liouville's Phase Space Theorem." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/LiouvillesPhaseSpaceTheorem.html

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