Well, I had a couple problems on my final I was hoping to go over- hope nobody minds. Here's the third.(adsbygoogle = window.adsbygoogle || []).push({});

1. The problem statement, all variables and given/known data

The Hamiltonian of the System is H = (1/2)*(p1*q2 + 2*p1*p2 + q1^2)

A.) Derive the Hamilton-Jacobi Equation

B.) Find the Solution.

2. Relevant equations

H = (q1, q1, . . ., qk , dS/dq1, dS/dq2, . . ., dS/dqk , t ) + dS/dt = 0 where S is the generator.

3. The attempt at a solution

Putting the Hamiltonian in proper formulation, it becomes.

(1/2)*(dS/dq1*q2 + 2*dS/dq1*dS/dq2 + q1^2)

So (1/2)*(dS/dq1*q2 + 2*dS/dq1*dS/dq2 + q1^2) + dS/dt = 0

B.

Let's try S = -q1^2 * t - q2^2 / 4

then (q2 / 2)*(-2*q1*t) + (-q2 / 2)*(-2*q1*t) + q1^2 + (-q1^2) = 0

Therefore S = -q1^2 * t - q2^2 / 4 + A is the generator, where A is a constant.

Let S = S2(q2) + S1(q1,t) + A as well.

Then (q2 / 2)*(dS1/dq1) + (dS1/dq1)*(dS2/dq2) + q1^2 + (dS1/dt) = 0

Dividing all terms by (dS1/dq1) creates:

(q2 / 2) + (dS2/dq2) + q1^2*(dS1/dq1)^-1 + (dS1/dt)*(dS1/dq1)^-1 = 0

(-q2 / 2) + (-dS2/dq2) = q1^2*(dS1/dq1)^-1 + (dS1/dt)*(dS1/dq1)^-1

Which can only be true in general if both sides are equal to a constant we'll call C.

(-q2 / 2) + (-dS2/dq2) = C implies dS2/dq2 = -(q2/2 + C). But from earlier we know that S2 = -q2^2 / 4.

So -q2 / 2 = -(q2 / 2 + C) and C = 0, q2 = q2

A similar argument with q1^2*(dS1/dq1)^-1 + (dS1/dt)*(dS1/dq1)^-1 = C creates:

q1^2*(dS1/dq1)^-1 + q1' = C implies (q1^2)/(C - q1') = dS1/dq1.

But from earlier we know S1 = -q1^2 * t so. . .

(q1^2)/(C - q1') = -2*q1*t, which is nasty- and I doubt it's separable.

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# Homework Help: Hamilton-Jacobi Derivation

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