On two degrees of freedom,

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

On two degrees of freedom, z=(x,y,px,py)\mathbf{z} = (x,y,p_x,p_y), a candidate map is X=3x,Y=y/3,Px=px,Py=py.X = 3x, \quad Y = y/3, \quad P_x = p_x, \quad P_y = p_y. ((a)) Write down its Jacobian matrix M\mathsf{M} and its determinant. ((b)) Test the symplectic condition MTJM=J\mathsf{M}^{\mathrm{T}}\mathsf{J}\mathsf{M} = \mathsf{J} directly, by computing {X,Px}\{X,P_x\} and {Y,Py}\{Y,P_y\}, and decide whether the map is canonical.

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