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hw4 - Florida State University Department of Physics PHY...

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Florida State University Department of Physics PHY 5246 Assignment # 4 (Due Friday, 23rd October, 2009) (1) (10 points) If L is a Lagrangian for a system of n degrees of freedom satisfying Lagrange’s equations, show by direct substitution that L = L + dF ( q 1 ,...,q n ,t ) dt also satisfies Lagrange’s equations, where F is any arbitrary, but differentiable, function of its arguments. (2) (10 points) Let q 1 ,...,q n be a set of independent generalized coordinates for a system of n degrees of freedom, with a Lagrangian L ( q, ˙ q,t ). Suppose we transform to another set of independent coordinates s 1 ,...,s n , by means of transformation equations q i = q i ( s 1 ,...,s n ,t ) , i = 1 ,...,n. (Such a transformation is called a point transformation .) Show that if the Lagrangian function is expressed as a function of s j , ˙ s j and t through the equations of transformation, then L satisfies Lagrange’s equations with respect to the s coordinates: d dt parenleftBigg ∂L ˙ s j parenrightBigg ∂L ∂s j = 0 .
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