The Theoretical Minimum: What You Need to Know to Start Doing Physics by Susskind Leonard & Hrabovsky George

Author:Susskind, Leonard & Hrabovsky, George [Susskind, Leonard]
Language: eng
Format: epub
ISBN: 9780465031740
Publisher: Basic Books
Published: 2013-01-28T23:00:00+00:00

The notation for generalized momentum is pi.

With that definition, the Euler-Lagrange equations are

Let’s do a couple of examples starting with a particle in polar coordinates. In this case the qi’s are the radius, r, and the angle, θ. We can use the result from Exercise 4 to get the Lagrangian:

The generalized momentum conjugate to r (the r momentum) is

and the corresponding equation of motion is

Using and canceling m from both sides, we can write this equation in the form

The equation of motion for the angle θ is especially interesting. First consider the conjugate momentum to θ:

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