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p09fl12 - Lecture 12 Lagrangian and action integral exs(30...

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Lecture 12: Lagrangian and action integral exs. (30 Sep 09) 0. 28 Sep homework A. Review: Hamilton’s 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 σ L T - Φ d dt ∂L ˙ q σ - ∂L ∂q σ = 0 , σ = 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. Hamilton’s principle (“principle of least action”) is: δS = 0 i.e., a calculus of variations problem subject to δq σ ( t 2 ) = δq σ ( t 1 ) = 0 . B. Symmetry principles 1. FW Sec. 20; LL Sec. 6 2. Define generalized/canonical momentum p σ = ∂L ˙ q σ 1
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3. Example: N-particles, mechanical momentum L = X j m j ˙ r 2 j - Φ( { 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 Φ 4. Now: assume that the Lagrangian has no explicit time dependence, L = L ( { q σ } , { ˙ q σ }
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