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8 Physics Formulary by ir. J.C.A. Wevers If the equation of continuity, t + · ( v ) = 0 holds, this can be written as: { , H } + t = 0 For an arbitrary quantity A holds: dA dt = { A, H } + A t Liouville’s theorem can than be written as: d dt = 0 ; or: pdq = constant 1.7.5 Generating functions Starting with the coordinate transformation: Q i = Q i ( q i , p i , t ) P i = P i ( q i , p i , t ) one can derive the following Hamilton equations with the new Hamiltonian K : dQ i dt = K P i ; dP i dt = - K Q i Now, a distinction between 4 cases can be made: 1. If p i ˙ q i - H = P i Q i - K ( P i , Q i , t ) - dF 1 ( q i , Q i , t ) dt , the coordinates follow from: p i = F 1 q i ; P i = - F 1 Q i ; K = H + F 1 t 2. If p i ˙ q i - H = - ˙ P i Q i - K ( P i , Q i , t ) + dF 2 ( q i , P i , t ) dt , the coordinates follow from:
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  • Spring '10
  • elder
  • Geographic coordinate system, Hamiltonian mechanics, Liouville's theorem, Generalized coordinates, Symplectic manifold

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