The total energy may be given by the hamiltonian in terms of the coordinates and linear momenta in Cartesian coordinates (that is, the kinetic energy term is split into the familiar pi2/2m. When transformed to spherical coordinates, however, two terms are angular momentum terms.
Why do these angular componenets arise, and how come it is not possible to derive this expression in terms of linear momentum in spherical coordinates. In other words, if we resolve the velocity in terms of, say, θ by taking the time derivative of the position vector in spherical coordinates and dotting it with the θ unit vector, and then multiply this velocity by m and do the same for the r and phi components, the outcome is not the same as that of the Hamiltonian transformed in terms of the angular momenta.
Can anyone elaborate on this peculiarity?