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Classical Mechanics | 0th Edition | ISBN: 9781891389221 | Authors: John R Taylor
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Consider a mass m moving in two dimensions, subject to a

Chapter 13, Problem 13.20

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QUESTION:

Consider a mass m moving in two dimensions, subject to a single force F that is independent of r and t. (a) Find the potential energy U(r) and the Hamiltonian a. (b) Show that if you use rectangular coordinates x, y with the x axis in the direction of F, then y is ignorable. (c) Show that if you use rectangular coordinates x, y with neither axis in the direction of F, then neither coordinate is ignorable. (Moral: Choose generalized coordinates carefully!)

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QUESTION:

Consider a mass m moving in two dimensions, subject to a single force F that is independent of r and t. (a) Find the potential energy U(r) and the Hamiltonian a. (b) Show that if you use rectangular coordinates x, y with the x axis in the direction of F, then y is ignorable. (c) Show that if you use rectangular coordinates x, y with neither axis in the direction of F, then neither coordinate is ignorable. (Moral: Choose generalized coordinates carefully!)

ANSWER:

Step 1 of 4

The potential energy and Hamiltonian are express as,

In 2D, the force F is independent of r and t. Assume the single force to be a constant vector is express as,

                                               

 Here is a constant vector.

The potential energy can be express as,

                                             

 Substitute all the values in the above equation.

                                             

Hence potential energy is, .

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