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Consider a particle whose motion is constrained to a plane. The motion can be described in...

Consider a particle whose motion is constrained to a plane. The motion can be described in terms of Cartesian coordinates and momenta (x, y, px, py) or in terms of polar coordinates and momenta (r, θ, pr, pθ). The differential volume elements dxdydpxdpy and drdθdprdpθ are related by dxdydpxdpy = Jdrdθdprdpθ. Show that J = 1.

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Answer #1

The results of this problem are in general true. The Jacobian of any canonical transformation is 1. This is known as Liouville equation.

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