Comment:

6.2: Show that the characteristic frequencies for m₁ << m₂ are those you would expect from elementary considerations.

Hint: Start by finding the Lagrangian of the system in the limit of small angles. The upper and lower masses are m1 and m2, respectively. Call the characteristic angular frequencies . The actual construction of the solution for the given initial conditions can be messy. It will help to express the elementary solutions in terms of the trigonometric functions .

Comment and question: It is easy to decide what the normal modes and eigenfrequencies should be in (a) before solving the problem analytically. This gives a check on your solution. Could you use the results in (a) and (b) and measurements of the molecular vibrational frequencies to determine which atom was replaced by its isotope? Explain using physical arguments.

Comment: In its present guise, LD10 is a typical small oscillation problem. However, as remarked in connection with Problem Set 3, this general problem appears in many different settings with dynamically broken symmetries. For example, an expansion of the exact potential to fourth order in the angular displacement from the bottom of the hoop gives the potential or free energy used in the Landau theory of second order phase transitions. The oscillations correspond in that context to fluctuations of the order parameter around its equilibrium value, and the condition you are asked to develop corresponds to the condition that the fluctuations to be small enough that the theory is reasonable.

Caution: Be very careful with the expansion in (a), and in getting the equations of motion. It is easy to leave out terms!   The matrix method for finding eigenfrequencies in small oscillation problems does not work directly in (b) because the Lagrangian contains linear as well as quadratic terms in the angular velocity. The appearance of "gyroscopic terms" of this sort is characteristic of small oscillations around steady motions. A determinantal condition similar to that in the matrix method will appear when you solve the equations of motion using the substitution given.

Send comments or questions to: ldurand@hep.wisc.edu

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© 1997, 1998, Loyal Durand