p09fl12 - Lecture 12: Lagrangian and action integral exs....

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Unformatted text preview: Lecture 12: Lagrangian and action integral exs. (30 Sep 09) 0. 28 Sep homework A. Review: Hamiltons principle 1. The Lagrange equations of motion are a transformation of the Newto- nian equations with k constraints on x i to a formulation with n- k generalized independent coordinates q L T- d dt L q - L q = 0 , = 1 ,...,n- k 2. These equations also come from a variational principle for the action S S = Z t 2 t 1 L [ q ( t ) , q ( t ) ,t ] dt where the q and q may be n- k component vectors. 3. Hamiltons principle (principle of least action) is: S = 0 i.e., a calculus of variations problem subject to q ( t 2 ) = q ( t 1 ) = 0 . B. Symmetry principles 1. FW Sec. 20; LL Sec. 6 2. Define generalized/canonical momentum p = L q 1 3. Example: N-particles, mechanical momentum L = X j m j r 2 j- ( { r j } ); p j = L/ r So the time derivative of the total momentum is d dt X j p j =- X j r j 4. Now: assume that the Lagrangian has no explicit time dependence,4....
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This note was uploaded on 09/29/2009 for the course PHYS 711 taught by Professor Bruch during the Fall '09 term at Wisconsin.

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p09fl12 - Lecture 12: Lagrangian and action integral exs....

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