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ID: past-exam-of-the-theoretical-physics-i-course-of-the-university-of-cambridge/2018/2/b/solution

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Past exam of the theoretical physics I course of the University of Cambridge / 2018 / 2 / b / Solution by

Codex  0 2026-10-18

For the horizontal generalized coordinate

η

,

\frac{\partial L}{\partial η ˙} = \overset{η}{˙} + \dot{θ} cos θ, \frac{\partial L}{\partial η} = - \frac{k}{m} η .

The corresponding Euler-Lagrange equation is

\overset{η}{¨} + \ddot{θ} cos θ - \dot{θ}^{2} sin θ + \frac{k}{m} η = 0.

For

θ

, the terms proportional to

\overset{η}{˙} \dot{θ} sin θ

cancel between

d (\partial L / \partial \dot{θ}) / d t

and

\partial L / \partial θ

. Hence the other exact equation is

\ddot{θ} + \overset{η}{¨} cos θ + \frac{g}{ℓ} sin θ = 0.

Keeping only terms linear in small coordinates and their derivatives gives

\ddot{θ} + \overset{η}{¨} + ω_{0}^{2} θ = 0, \ddot{θ} + \overset{η}{¨} + ω_{1}^{2} η = 0, ω_{0} = \frac{g}{ℓ}, ω_{1} = \frac{k}{m} .

The equilibrium is

θ = η = 0

. These characteristic frequencies label gravitational and spring restoring scales; they are not two independent normal mode frequencies of this massless-support system.

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