Solve Hamilton-Jacobi Equation for Hamiltonian w/ Mixed Terms

[jamaicanking]

how would you solve the hamilton - jacobi equation for something with a hamiltonian with mixed terms like 1/2(p1q2 + 2p1p2 + (q1)^2)

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 can't 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.

 

[jamaicanking]

I guess I did not phrase the question well.

The issue is I have a given hamiltonian H = 1/2(p1q2 + 2p1p2 + (q1)^2)

I need to solve this and I chose to begin by using the hamilton - jacobi equation and since we have no time dependence . If S is the hamilton action function then

((ds/dq1)*q1 + 2(ds/dq1)(ds/dq2) + (q1)^2) = E where E is now energy.

How would you go about solving this differential equation?

 

[jamaicanking]

Actually this is quite trivial..thank you anyway.