Physics 110A: Hint on 5.18



You may benefit from a hint on Problem 5.18. If you express the potential energy in the stretched springs exactly, you'll end up staring at a radical that looks like: sqrt(a² + x² + y² + 2ax). Factor an a out, leaving something of the form sqrt(1 + &epsilon), where &epsilon is a small quantity relative to 1 (because x and y are small compared to a, this is true). Nevermind that ε is made up of three terms.

You'll need to Taylor expand and keep terms to second order in epsilon in order to capture all terms having x² and y². You may throw out higher orders.

The Taylor expansion to second order is: sqrt(1 + ε) = 1+ ε/2 − ε²/8.

If done correctly, you will end up with an anisotropic oscillator exhibiting the expected instability when a < l0 (in what direction?).


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