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well its quite trivial obtaining the HJ equation since there is no time dependence,

1/2( (ds/dq1)q2 + 2(ds/dq1)(ds/dq2) + (q1)^2 ) = E

I cant see how youw would separate the variables otherwise we could simple set

H(q1,p1) = E1 amd H(q2,p2) = E2 .

However I am stumped on how to do it for the above equation with mixed terms.