Canonical Transformation - Liouville's Theorem

Liouville's Theorem

The direct conditions allow us to prove Liouville's theorem, which states that the volume in phase space is conserved under canonical transformations, i.e.,

$\int d\mathbf{q} d\mathbf{p} = \int d\mathbf{Q} d\mathbf{P}$

By calculus, the latter integral must equal the former times the Jacobian

$\int d\mathbf{Q} d\mathbf{P} = \int J d\mathbf{q} d\mathbf{p}$

where the Jacobian is the determinant of the matrix of partial derivatives, which we write as

$J \equiv \frac{\partial (\mathbf{Q}, \mathbf{P})}{\partial (\mathbf{q}, \mathbf{p})}$

Exploiting the "division" property of Jacobians yields

$J \equiv \frac{\partial (\mathbf{Q}, \mathbf{P})}{\partial (\mathbf{q}, \mathbf{P})} \left/ \frac{\partial (\mathbf{q}, \mathbf{p})}{\partial (\mathbf{q}, \mathbf{P})} \right.$

Eliminating the repeated variables gives

$J \equiv \frac{\partial (\mathbf{Q})}{\partial (\mathbf{q})} \left/ \frac{\partial (\mathbf{p})}{\partial (\mathbf{P})} \right.$

Application of the direct conditions above yields .

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