This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Lecture 28: Hamiltonians (6 Nov 09) A. Hamilton equations 1. In Lagrangian formulation the independent variables are (generalized) coordinates q and velocities q . In the Hamiltonian formulation, they are (generalized) coordinates q and canonical momenta p . 2. Thus calculate the total time derivative of the Lagrangian (with no explicit time dependence) L = L ( { q j } , { q j } ) dL dt = X j L q j q j + L q j q j = X j q j d dt L q j + L q j q j = d dt X j q j p j using the Lagrange equations L q j = d dt L q j = d dt p j ; p j L q j 3. Thus H j q j p j L is conserved/constant (the energy): dH/dt = 0 4. The transformation from L to H is a Legendre transformation and the independent variables change from q , , q j to q j ,p j : dH = X j [ p j d q j + q j dp j ] X j [ L q j dq j + L q j d q j ] L t = X j [ q j dp j L q j dq j ] L t 5. Compare to the differential dH = X j [ H q j dq j + H p j dp j ] + H t 6. Then the identifications are (notice that the partials have different other coordinates held fixed for...
View
Full Document
 Fall '09
 BRUCH
 Hamiltonian mechanics, pj, Lagrangian mechanics

Click to edit the document details