正準変換 (Canonical transformation)

Liouville's theorem

∫dQdP=∫Jdqdp\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≡∂(Q,P)∂(q,p)J \equiv \frac{\partial \left( \mathbf{Q} , \mathbf{P} \right)}{\partial \left( \mathbf{q} , \mathbf{p} \right)}