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The time-evolution of a physical system with one coordinate q is described by the La- grangian L = ? + aġ sin q sint+ b cos q, where a and b are constants. (a) Show that the corresponding Hamiltonian is H (p - 2 a sin q sin t)? – b cos q. Is H a constant of the motion? (b) Obtain a type 2 generating function, F2(q, P, t), for the canonical transformation Q = q, P = p – a sin q sin t. [ ƏF2 p = ƏF2 Q 1 | Definition of a type 2 generating function: (c) Use K = H+ ƏF2/ðt to find the new Hamiltonian, K(Q, P, t), obtained by applying the transformation from part (b) to the Hamiltonian given in part (a).

The time-evolution of a physical system with one coordinate q is described by the La- grangian L = ? + aġ sin q sint+ b cos q, where a and b are constants. (a) Show that the corresponding Hamiltonian is H (p - 2 a sin q sin t)? – b cos q. Is H a constant of the motion? (b) Obtain a type 2 generating function, F2(q, P, t), for the canonical transformation Q = q, P = p – a sin q sin t. [ ƏF2 p = ƏF2 Q 1 | Definition of a type 2 generating function: (c) Use K = H+ ƏF2/ðt to find the new Hamiltonian, K(Q, P, t), obtained by applying the transformation from part (b) to the Hamiltonian given in part (a).

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The time-evolution of a physical system with one coordinate q is described by the La- grangian 1 L ģ² + aġ sin q sin t + b cos q, 2 where a and b are constants. (a) Show that the corresponding Hamiltonian is 1 (p – a sin q sin t)² – b cos q. 2 H Is H a constant of the motion? (b) Obtain a type 2 generating function, F2(q, P,t), for the canonical transformation Q = q, P = p – a sin q sin t. ƏF2 dq Q= OF2 Definition of a type 2 generating function: (c) Use K = H + ƏF2/ðt to find the new Hamiltonian, K(Q, P,t), obtained by applying the transformation from part (b) to the Hamiltonian given in part (a). (d) Using your result from part (c), or otherwise, derive the equation of motion Q = -(a cos t + b) sin Q.
Transcribed Image Text:The time-evolution of a physical system with one coordinate q is described by the La- grangian 1 L ģ² + aġ sin q sin t + b cos q, 2 where a and b are constants. (a) Show that the corresponding Hamiltonian is 1 (p – a sin q sin t)² – b cos q. 2 H Is H a constant of the motion? (b) Obtain a type 2 generating function, F2(q, P,t), for the canonical transformation Q = q, P = p – a sin q sin t. ƏF2 dq Q= OF2 Definition of a type 2 generating function: (c) Use K = H + ƏF2/ðt to find the new Hamiltonian, K(Q, P,t), obtained by applying the transformation from part (b) to the Hamiltonian given in part (a). (d) Using your result from part (c), or otherwise, derive the equation of motion Q = -(a cos t + b) sin Q.
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