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Let a two-degree-of-freedom system be described by the Hamiltonian
1
H =(P +P)+ V(x, y)
and suppose the potential energy V is a homogenous function of degree -2:
V(xx, ày) = 2-2v(x,y) VÀ > 0.
Show that
$ = (xpy – yp.)² +2(x² + y²)V(x, y)
is a second constant of the motion independent of the Hamiltonian (Yoshida, 1987).
Therefore, this system is integrable.

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