Hamiltonian in spherical coordinates

[Syrus]

Homework Statement

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 p_i²/2m. 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?

Homework Equations

The Attempt at a Solution

 

[dextercioby]

My point, hopefully correct and helpful, is:

That the space (R^3) or the dynamics of a free pointlike particle in such an infinite space is rotationally invariant (or isotropic) is best exhibited in spherical coordinates, a generalization to 3 dimensions of the plane polar coordinates which can be used to describe R^2 or free dynamics in R^2. It's just the connection: rotation -> angle and point -> circle -> sphere -> hypersphere (for at least 4 dimensions). R^2 is the disk of infinite radius, R^3 is the ball with infinite radius. This fact sets the Hamiltonian to be rotationally invariant, hence it commutes (in the Poisson bracket) with the generator of rotations, the orbital angular momentum. Under rotations of coordinate axis (or rotation of inertial observers) energy is conserved.

There's no coincidence that a space (time) with rotational symmetry has 2,3,... observables hidden in it. We don't see them in cartesian coordinates, since there's no natural description of rotations in cartesian coordinates.

 

[Syrus]

Dex, thank you for your response. Can you tell me what specific area of mathetmatics or physics I should search for more detail?

Perhaps the question I wish to satisfy is: what is the theta (extend this to include r and phi) component of the linear velocity of a particle in spherical coordinates? Once we have these velocities, why is the kinetic energy in the hamiltonian not expressed in terms of these linear momenta (rather than their angular analogues)?