(a) For a one-dimensional system with the Hamiltonian show that there is a constant of the...

(a) For a one-dimensional system with the Hamiltonian

 

show that there is a constant of the motion

 

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(b) As a generalization of part (a), for motion in a plane with the Hamiltonian

 

where p is the vector of the momenta conjugate to the Cartesian coordinates, show that there is a constant of the motion

 

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(c) The transformation Q = λq, p = λP is obviously canonical. However, the same transformation with t time dilatation, Q = λq, p = λP, t′ = λ2t, is not. Show that, however, the equations of motion for q and p for the Hamiltonian in part (a) are invariant under this transformation. The constant of the motion D is said to be associated with this invariance.