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Unformatted text preview: PHYS 3202 Classical Mechanics II - Homework #5 Due at 12:05pm, Friday March 4, 2011 in the class In homework, quizzes, and final exam, please show intermediate steps of your calculations. Whenever appropriate, you may draw diagram(s). Problem 1 (10 points): A transformation from ( q 1 , ,q s ,p 1 , ,p s ) to ( Q 1 , ,Q s ,P 1 , ,P s ) is called a canonical transformation if I X j P j Q j = I X j p j q j for all loops in the phase space. Equivalently, if j P j Q j- j p j q j is a total differential: X j P j Q j- X j p j q j = S, the transformation is canonical. Now consider the following transformations: Q =- p/ 2 , P = 2 q- p 2 . (1) Q = q cos + p sin , P =- q sin + p cos . ( = constant) . (2) Q 1 = 2 q 1 , P 1 = cp 1 , Q 2 =- p 2 , P 2 = q 2 , (3) where c is a constant (may or may not be 1 / 2). Which of the transformations are canonical? Find the function S for each canonical transformation. S is called the generating function of the canonical transformation....
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