Invariance of a volume element in phase space, What does it mean?

[Maumas]

Homework Statement

I have been reading the third edition of Classical Mechanics by Goldstein, in particular, chapter 9 Poisson Brackets and Other canonical invariants. And it is shown that the magnitude of a volume element is invariant. I can understand what it means mathematically, but what physical implications does it have?

I have been reading similar questions and noticed that this invariance is related to Liouville’s Theorem. But I do not understand the underlying physics.

Maybe someone can shed light on this issue

Relevant Equations

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The invariance of this volume element is shown by writing the infinitesimal volume elements $$d\eta$$ and $$d\rho$$

$$d\eta=dq_1.....dq_ndp_1......dp_n$$

$$d\rho=dQ_1.......dQ_ndP_1....dP_n$$

and we know that both of them are related to each other by the absolute value of the determinant of the Jacobian matrix. So I do understand that if we have a canonical transformation $$d\eta=|M|d\rho$$ is $$d\eta=d\rho$$ but i do not know what it means physically.

 

[wrobel]

but what physical implications does it have?

A phase flow of a Hamiltonian system consists of canonical transformations. Now see the Poincare recurrence theorem.