Solutions_HW4 - Theoretical Dynamics October 01, 2010...

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Unformatted text preview: Theoretical Dynamics October 01, 2010 Homework 4 Instructor: Dr. Thomas Cohen Submitted by: Vivek Saxena Goldstein 9.7 Part (a) F 1 ( q,Q,t )- F 2 ( q,P,t )- P i = F 1 Q i (1) F 2 ( q,P,t ) = F 1 ( q,Q,t ) + P i Q i (2) F 1 ( q,Q,t )- F 3 ( p,Q,t ) p i = F 1 q i (3) F 3 ( p,Q,t ) = F 1 ( q,Q,t )- p i q i (4) F 1 ( q,Q,t )- F 4 ( p,P,t ) p i = F 1 q i (5) P i =- F 1 Q i (6) F 4 ( p,P,t ) = F 1 ( q,Q,t )- p i q i + P i Q i (7) F 2 ( q,P,t )- F 3 ( p,Q,t ) p i = F 2 q i (8) Q i = F 2 P i (9) F 3 ( p,Q,t ) = F 2 ( q,P,t )- p i q i- P i Q i (10) F 2 ( q,P,t )- F 4 ( p,P,t ) p i = F 2 q i (11) F 4 ( p,P,t ) = F 2 ( q,P,t )- p i q i (12) 4- 1 F 3 ( p,Q,t )- F 4 ( p,P,t ) P i =- F 3 Q i (13) F 4 ( p,P,t ) = F 3 ( p,Q,t ) + P i Q i (14) Part (b) For an identity transformation, F 2 = q i P i and by equation (7), the type 4 generating function is F 4 ( p,P,t ) = F 2 ( q,P,t )- p i q i (15) = q i P i- p i q i = 0 as p i = F 2 q i = P i For an exchange transformation, F 1 = q i Q i and by equation (4), the type 3 generating function is F 3 ( p,Q,t ) = F 1 ( q,Q,t )- p i q i (16) = q i Q i- p i q i = 0 as p i = F 1 q i = Q i (17) Part (c) Consider a type 2 generating function F 2 ( q,P,t ) of the old coordinates and the new mo- menta, of the form F 2 ( q,P,t ) = f i ( q 1 ,...,q n ; t ) P i- g ( q 1 ,...,q n ; t ) (18) where f i s are a set of independent functions, and g i s are differentiable functions of the old coordinates and time. The new coordinates Q i are given by Q i = F 2 P i = f i ( q 1 ,...,q n ; t ) (19) In particular, the function f i ( q 1 ,...,q n ; t ) = R ij q j (20) where R ij is the ( i,j )-th element of a N N orthogonal matrix, generates an orthogonal transformation of the coordinates. Now, p j = F 2 q j = f i q j P i- g q j = R ij P i- g q j (21) This equation can be written in matrix form, as p = f q P- g q (22) 4- 2 where p denotes the N 1 column vector ( p 1 ,...,p N ) T , g/ q denotes the N 1 column vector ( g/q 1 ,...,g/q n ) T , and f q denotes the N N matrix with entries f q ij = f i q j = R ij (23) From (22), the new momenta are given by P = f q- 1 p + g q (24) = R- 1 p + g q (25) = R- 1 ( p + q g ) (26) As R is an orthogonal matrix, RR T = R T R = I , so R- 1 = R T is also an orthogonal transformation. This gives the required result: the new momenta are given by the orthogonal transformation ( R- 1 ) of an n-dimensional vector ( p + q g ), whose components are the old momenta ( p ) plus a gradient in configuration space ( q g )....
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This note was uploaded on 12/29/2011 for the course PHYSICS 601 taught by Professor Hassam during the Fall '11 term at Maryland.

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Solutions_HW4 - Theoretical Dynamics October 01, 2010...

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