Solutions_hw3 - Theoretical Dynamics September 24, 2010...

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Unformatted text preview: Theoretical Dynamics September 24, 2010 Homework 3 Instructor: Dr. Thomas Cohen Submitted by: Vivek Saxena 1 Goldstein 8.1 1.1 Part (a) The Hamiltonian is given by H ( q i ,p i ,t ) = p i q i- L ( q i , q i ,t ) (1) where all the q i s on the RHS are to be expressed in terms of q i , p i and t . Now, dH = H q i dq i + H p i dp i + H t dt (2) From (1), dH = p i d q i + q i dp i- dL = p i d q i + q i dp i- L q i dq i + L q i d q i + L t dt =- L q i dq i + q i dp i + p i- L q i d q i- L t dt (3) Comparing (2) and (3) we get H q i =- L q i =- p i (2nd equality from Hamiltons equation) (4) q i = H q i (also Hamiltons equation) (5) p i- L q i = 0 (H is not explicitly dependent on q i ) (6)- L t = H t (7) From (4) and (6) we have d dt L q i- L q i = 0 , i = 1 , 2 ,...,n (8) which are the Euler-Lagrange equations. 3- 1 1.2 Part (b) L ( p, p,t ) =- p i q i- H ( q,p,t ) (9) = p i q i- H ( q,p,t )- d dt ( p i q i ) (10) = L ( q, q,t )- d dt ( p i q i ) (11) = L ( q, q,t )- p i q i- p i q i (12) So, dL = L p i dp i + L p i d p i + L t dt (13) =- q i dp i- q i d p i + L t dt (from (9)) (14) Comparing (12) and (13) we get q i =- L p i (15) q i =- L p i (16) Thus the equations of motion are d dt L p i- L p i = 0 , i = 1 , 2 ,...,n (17) 2 Goldstein 8.6 Hamiltons principle is Z Ldt = 0 (18) or equivalently Z 2 Ldt = 0 (19) We can subtract the total time derivative of a function whose variation vanishes at the end points of the path, from the integrand, without invalidating the variational principle. This is because such a function will only contribute to boundary terms involving the variation of q i and p i at the end points of the path, which vanish by assumption. Such a function is p i q i . So, the modified Hamiltons principle is Z 2 L- d dt ( p i q i ) dt = 0 (20) 3- 2 Using the Legendre transformation, this becomes Z (2 p i q i- 2 H- p i q i- p i q i ) dt = 0 (21) = Z (2 H + p i q i- p i q i ) dt = 0 (22) now, p i q i- p i q i = q 1 ... q n | p 1 ... p n 1 2 n n n 1 n n- 1 n n n n 2 n 2 n q 1 . . q n-- p 1 . . p n 2 n 1 (23) = T J (24) So (22) becomes Z ( 2 H + T J ) dt = 0 (25) which is the required form of Hamiltons principle. 3 Goldstein 8.9 The constraints can be incorporated into the Lagrangian L by defining a constrained Lagrangian L c , as L c ( q, q,t ) = L ( q, q,t )- X k k k ( q,p,t ) (26) Applying Hamiltons principle, and using the Legendre transformation for L , we get Z p i q i- H ( q,p,t )- X k k k ( q,p,t ) !...
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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_hw3 - Theoretical Dynamics September 24, 2010...

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