It's Obvious... Not: Charged Particle Motion in a Magnetic Field and Third Order Homgeneous Differential Equations

This installment of “It’s Obvious. Not!” looks at:

Book: “Classical Dynamics of Particles and Systems”

Edition: third

Authors: Jerry B. Marion and Stephen T. Thornton

Publisher: Harcourt Brace Jovanovich

Page: 68

This post looks at Example 2.10 that investigates the motion of a charged particle in a magnetic field. The example is fairly straightforward with one exception. When determining the equations of motion, the authors propose a solution for the system of differential equations discussed below and reference example C.2 of Appendix C. It’s not immediately apparent how to use Example C.2 to arrive at the authors’ solution, so the steps are outlined in detail here. If you have questions, or suggestions, all comments are always welcome!

The original system of coupled differential equations is:

First, the authors’ differentiate both equations and then substitute the results into the other:

at this point, the book suggests using the technique of example C.2, (finding the roots of the auxiliary equations), to find the solutions of this system of third order differential equations, and then states that the solutions will be of the form:

Now, let’s follow the actual steps required by example C2 and verify that the solution for x is the same. The solution for z will follow the same steps.

For:

the auxiliary equation is:

First, factor out r to get:

the first root is

The second two roots can be obtained by factoring the second term as the sum of two squares producing:

which provides the roots:

The solution can then be written as:

Using Euler’s Formula, we arrive at:

removing the negative signs from the inside of the trigonometric functions gives:

which can be re-arranged as:

or

which fits the authors’ solution using:

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