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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 01/30/2011 for the course PHYSICS 208 taught by Professor Ye during the Spring '10 term at Blinn College.

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