{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

# p09fl29 - Lecture 29 Hamilton-Jacobi Poisson brackets(9 Nov...

This preview shows pages 1–3. Sign up to view the full content.

Lecture 29: Hamilton-Jacobi Poisson brackets (9 Nov 09) A. Canonical transformations 1. FW Secs. 34,35; LL Secs. 46, 47 2. Transform to new coordinates Q j , P j p j = p j ( P i , Q i , t ); q j = q j ( P i , Q i , t ) 3. Ask for transformations that keep the form of Hamilton’s equations for the new Hamiltonian ˜ H : ˙ Q j = ˜ H ∂P j ; ˙ P j = - ˜ H ∂Q j 4. This will be the situation if the modified Hamilton’s principle is avail- able for the new coordinates. That allows a bit of generality: X j ˙ q j p j - H = X j ˙ Q j P j - ˜ H + dF 1 ( q, Q, t ) dt p j = ∂F 1 ∂q j ; P j = - ∂F 1 ∂Q j ; ˜ H ( Q, P, t ) = H ( p, q, t ) + ∂F 1 ∂t 5. For Hamilton-Jacobi theory, one changes the variables in the added function F to q, P by the Legendre transformation F 2 = X j Q j P j + F 1 X j ˙ q j p j - H = X j ˙ Q j P j - ˜ H + dF 1 ( q, Q, t ) dt = - X j Q j ˙ P j - ˜ H + dF 2 ( q, P, t ) dt Then set up the total time derivative of F 2 : dF 2 dt = X j ∂F 2 ∂q j ˙ q j + ∂F 2 ∂P j ˙ P j + ∂F 2 ∂t p j = ∂F 2 ∂q j ; Q j = ∂F 2 ∂P j ; ˜ H ( Q, P, t ) = H ( p, q, t ) + ∂F 2 ∂t 1

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
B. Hamilton-Jacobi theory
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 4

p09fl29 - Lecture 29 Hamilton-Jacobi Poisson brackets(9 Nov...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online