Hamiltonian solving

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 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.
 
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?
 
Actually this is quite trivial..thank you anyway.
 

The Physics Forums Way

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving
Top