Take a canonical transformation with generating function

PHYSICS · Multi-part · Fermi University Physics · Mechanics & Dynamical Systems · Hamiltonian mechanics and canonical structure · Hamiltonian mechanics and canonical structure

Action and Symmetry, Ch Canonical transformations, generating functions and symplectic structure

Problem

Take a canonical transformation with generating function F1(q,Q,t)F_1(q,Q,t), so that pk=∂F1/∂qkp_k=\partial F_1/\partial q_k, Pk=−∂F1/∂QkP_k=-\partial F_1/\partial Q_k and K=H+∂F1/∂tK=H+\partial F_1/\partial t. ((a)) For a system with two degrees of freedom x,yx,y, consider the infinitesimal transformation generated by F2=xPx+yPy+ϵGF_2=xP_x+yP_y+\epsilon G with G=px+2pyG=p_x+2p_y. Derive δx,δy,δpx,δpy\delta x,\delta y,\delta p_x,\delta p_y to first order in ϵ\epsilon, and describe the geometrical effect on a point of the xx-yy plane. ((b)) The Jacobian matrix M\mathsf M of the transformation in part (a) satisfies MTJM=J\mathsf M^{\mathrm T}\mathsf J\mathsf M=\mathsf J, as it must for any canonical transformation. Explain why this guarantees that the phase-space volume element dx dy dpx dpy\mathrm dx\,\mathrm dy\,\mathrm dp_x\,\mathrm dp_y is unchanged by the transformation of part (a), and why this fact alone does not tell you whether K=HK=H.

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