How to determine what is conserved given a hamilitonian

[AverageJoePHSX]

Homework Statement

Given the Hamiltonian's determine what quantities are conserved. These are labeled H1 and H2 in the picture.

Homework Equations

The relevant equations are q'=dH/dp and p'=-dH/dq

The Attempt at a Solution

My attempt is below

ignore the negative sign for q' in H2

 

[Daeho Ro]

Well, looks fine. But, in second problem, one of the \dot{P}_i should be positive and so the total momentum variation will be zero.