Is a canonical transformation equivalent to a transformation that preserves volume and orientation?

Question

We have seen the reverse statement: Lioville's Theorem states that canonical transformations preserve volume (and orientation as well). Is the reverse true? If I demand a map from the phase space to the phase space to preserve volume, is it necessarilly a canoncial transformation? I couldn't come up with a counter example, that's why I ask.

Answer 1

In dimension $2n>2$ they are not equivalent since (for time-independent transformations) canonical is equivalent to $$\sum_{k=1}^n dq^k\wedge dp_k = \sum_{k=1}^n dQ^k\wedge dP_k\tag{1}$$ whereas conservation of oriented volume means $$dq^1\wedge \cdots \wedge dq^n \wedge dp_1\cdots \wedge dp_n = dQ^1\wedge \cdots \wedge dQ^n \wedge dP_1\cdots \wedge dP_n\:.\tag{2}$$ The former is much more restrictive. The latter only requires that the Jacobian matrix has determinant $1$. Already with $4\times 4$ matrices there are easy conterexamples.

$Q^1 = aq^1$,

$Q^2= b q^2$,

$P_1 = b^{-1} p_1$,

$P_2 = a^{-1}p_2$

where coordinates are over $\mathbb R^4$ and with constants $a,b>0$ satisfying $a\neq b$. This transformation satisfies (2) but not (1).

Instead, for $2n=2$, (1) and (2) are evidently equivalent.

Answer 2

1. Counterexample: The transformation $$Q^1~=~2q^1 ,\qquad P_1~=~p_1,\qquad Q^2~=~\frac{1}{2}q^2 ,\qquad P_2~=~p_2 $$ preserves phase space volume & orientation, but is not a symplectomorphism.$^1$

2. For 2D phase space, the canonical phase space volume form $$\Omega~=~\frac{1}{n!}\omega^{\wedge n}$$ is the symplectic 2-form $\omega$ itself, so that the orientation & volume preserving transformations are the symplectomorphisms.

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$^1$ Here we will assume that OP defines a canonical transformation (CT) as a symplectomorphism. Be aware that several non-equivalent definitions of CTs appear in the literature, cf. e.g. this Phys.SE post.

Answer 3

It boils down to the structure of the Phase Space. You know that the Phase Space must have a two-differential form, which is represented by your Poisson Brackets. This two-form "keeps track" of your orientation thanks to its intrinsic skew-symmetric nature, and due to it being the external product of your canonical momentum and position variable, it defines "hypervolumes" in your Phase Space. Any space equipped with Poisson Brackets is a symplectic space and the Poisson Bracket will be a symplectic form. When you change variables, you want to keep this structure, hence you want to apply a symplectomorphism (a.k.a. a canonical transformation). It is demonstrable that the Jacobian determinant of a symplectomorphism is always $1$ so it does not deform volumes in the phase space. Due to the definition of symplectomorphism, it will conserve your orientation too, since it keeps the symplectic structure unchanged. So, is a transformation with $\left|\underline{J}\right|=1$ a symplectomorphism? In general, it isn't, since you don't know if it keeps the symplectic form on your space unchanged.