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Unformatted text preview: Homework 5 Classical Mechanics (Phys701) Due: October 10, 2011 Problem 17 Show that Lagrange’s equations can also be written as ∂ ˙ T ∂ ˙ q j − 2 ∂T ∂q j = Q j . These are sometimes known as the Nielsen form of the Lagrange equations. (Hint: start with writing a formula for ˙ T ). Solution : First, calculate d dt T ( ˙ q i , q i , t ) = ∂T ∂ ˙ q i ¨ q i + ∂T ∂q i ˙ q i + ∂T ∂t . Consequently ∂ ∂ ˙ q j dT dt = ∂ 2 T ∂ ˙ q j ∂ ˙ q i ¨ q i + ∂ 2 T ∂ ˙ q j ∂q i ˙ q i + ∂T ∂q j + ∂ 2 T ∂ ˙ q j ∂t . Then the Nielsen equation reads: ∂ 2 T ∂ ˙ q j ∂ ˙ q i ¨ q i + ∂ 2 T ∂ ˙ q j ∂q i ˙ q i + ∂ 2 T ∂ ˙ q j ∂t − ∂T ∂q j = 0 . However, the first three terms here can be identified as ∂ 2 T ∂ ˙ q j ∂ ˙ q i ¨ q i + ∂ 2 T ∂ ˙ q j ∂q i ˙ q i + ∂ 2 T ∂ ˙ q j ∂t = d dt ∂T ∂ ˙ q j (check it directly) and we get d dt ∂T ∂ ˙ q j − ∂T ∂q j = 0 , i.e., the Lagrange equation. Problem 19 The electromagnetic field is invariant under a gauge transformation of the scalar and vector potentials given by A → A + ∇ Ψ( r, t ) , ϕ → ϕ − 1 c ∂ Ψ ∂t , where Ψ is an arbitrary (but differentiable). What effect does this gauge transformation have on the Lagrangian of a particle moving in the electromagnetic field? Is the motion affected? Solution . 1 v eff + r _ r E eff Figure 1: Dependence V eff ( r ) and the graphic solution for the values of r ± . Problem 121 Two mass points of mass m 1 and m 2 are connected by a string passing through a hole in a smooth table so that m 1 rests on the table surface and m 2 hangs suspended. Assuming m 2 moves only only in a vertical line, what are the generalized coordinates for the system? Write the Lagrange equations for the systemline, what are the generalized coordinates for the system?...
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This note was uploaded on 12/13/2011 for the course PHYS 701 taught by Professor Bazaliy during the Fall '11 term at South Carolina.
 Fall '11
 Bazaliy
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