Unformatted text preview: 8 Physics Formulary by ir. J.C.A. Wevers If the equation of continuity, t + ( v ) = 0 holds, this can be written as: { , H} + =0 t For an arbitrary quantity A holds: dA A = {A, H} + dt t Liouville's theorem can than be written as: d = 0 ; or: dt pdq = constant 1.7.5 Generating functions
Starting with the coordinate transformation: Qi = Qi (qi , pi , t) Pi = Pi (qi , pi , t) one can derive the following Hamilton equations with the new Hamiltonian K: dQi K = ; dt Pi Now, a distinction between 4 cases can be made: 1. If pi qi  H = Pi Qi  K(Pi , Qi , t)  pi = dF1 (qi , Qi , t) , the coordinates follow from: dt K dPi = dt Qi F1 F1 F1 ; Pi =  ; K =H+ qi Qi t dF2 (qi , Pi , t) , the coordinates follow from: 2. If pi qi  H = Pi Qi  K(Pi , Qi , t) + dt pi = F2 F2 F2 ; Qi = ; K=H+ qi Pi t dF3 (pi , Qi , t) , the coordinates follow from: dt 3. If pi qi  H = Pi Qi  K(Pi , Qi , t) + qi =  F3 F3 F3 ; Pi =  ; K =H+ pi Qi t dF4 (pi , Pi , t) , the coordinates follow from: dt 4. If pi qi  H = Pi Qi  K(Pi , Qi , t) + qi =  F4 F4 F4 ; Qi = ; K=H+ pi Pi t The functions F 1 , F2 , F3 and F4 are called generating functions. ...
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This note was uploaded on 11/16/2010 for the course ENGR 201 taught by Professor Elder during the Spring '10 term at Blinn College.
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