Canonical Transformation

2. Recalling that the four-momentum of the particle is

$$p_\alpha=\frac{\partial W(u,v,x,y;\mathbf{I})}{\partial x^\alpha} ~~~~\alpha=u,v,x,y~,$$ (52)

one has the fact that these two sets of equations, (52) and (53), define implicitly the phase space transformation

$$(u,v,x,y,;p_u,p_v,p_x,p_y)\longrightarrow (\phi^u ,\phi^v ,\phi^x ,\phi^y; I_u,I_v,I_x,I_y)$$

Being generated from the scalar function $W(x^\alpha;\mathbf{I})$, this transformationis canonical, i.e. it leaves invariant the representation of the antisymmetric tensor, Eq.(8):

$$dx^\alpha\wedge dp_\alpha=d\phi^\beta\wedge dI_\beta ~.$$