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Liouville's Phase Space Theorem

States that for a nondissipative Hamiltonian system, phase space density (the area between phase space contours) is constant. This requires that, given a small time increment dt,

q_1=q(t_0+dt)
(1)
=q_0+(partialH(q_0,p_0,t))/(partialp_0)dt+O(dt^2)
(2)
p_1=p(t_0+dt)
(3)
=p_0-(partialH(q_0,p_0,t))/(partialq_0)dt+O(dt^2),
(4)

the Jacobian be equal to one:

(partial(q_1,p_1))/(partial(q_0,p_0))=|(partialq_1)/(partialq_0) (partialp_1)/(partialq_0); (partialq_1)/(partialp_0) (partialp_1)/(partialp_0)|
(5)
=|1+(partial^2H)/(partialq_0partialp_0)dt -(partial^2H)/(partialq_0^2)dt; (partial^2H)/(partialp_0^2)dt 1-(partial^2H)/(partialq_0partialp_0)dt|+O(dt^2)
(6)
(7)
=1+O(dt^2).
(8)

Expressed in another form, the integral of the Liouville measure,

product_(i=1)^Nintdp_idq_i,
(9)

is a constant of motion. Symplectic maps of Hamiltonian systems must therefore be area preserving (and have determinants equal to 1).

See also

Liouville Measure, Phase Space

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References

Chavel, I. Riemannian Geometry: A Modern Introduction. New York: Cambridge University Press, 1994.

Referenced on Wolfram|Alpha

Liouville's Phase Space Theorem

Cite this as:

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

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