Classical Mechanics/Central Field

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Consider a central potential V(r). A central potential is where the potential is dependent only on the field point's distance from the origin; in other words, the potential is isotropic.

The Lagrangian of the system can be written as

Since the potential is spherically symmetry, it makes sense to write the Lagrangian in spherical coordinates.

It can then be worked out that:

Hence the equation for the Lagrangian is

One can then extract three laws of motion from the Lagrangian using the Euler-Lagrange formula

This looks messy, but when we look at the Euler-Lagrange relation for , we have

Hence is a constant throughout the motion.