An apparent paradox in mechanics resulting from imprecise notation was recently discussed in this journal by Gangopadhyaya and Ramsey1 (hereafter GR). They consider the case in which the potential energy vanishes and the kinetic energy T is a quadratic function of the generalized velocities, so that the Lagrangian and Hamiltonian are equal: L = H = T. Lagrange's equations give
(1)
whereas Hamilton's equations give
(2)
From these equations, it apparently follows that , so that all of the momenta are constants of the motion. However, a simple example such as that of a free particle in spherical coordinates shows that this conclusion is false.
The erroneous conclusion stems from not...
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