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ID: past-exam-of-the-theoretical-physics-i-course-of-the-university-of-cambridge/2012/paper-1/1/b/solution

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Past exam of the theoretical physics I course of the University of Cambridge / 2012 / Paper 1 / 1 / b / Solution by

Codex  0 2026-10-20

Differentiate with respect to the generalized velocities:

p_{x} = (m_{1} + m_{2}) \overset{x}{˙} + m_{2} l \dot{ϕ} cos ϕ, p_{ϕ} = m_{2} l^{2} \dot{ϕ} + m_{2} l \overset{x}{˙} cos ϕ .

(1)

The Lagrangian has no explicit

x

, so

\overset{p}{˙}_{x} = \partial L / \partial x = 0

. This canonical momentum is also the system's total horizontal mechanical momentum. Horizontal translation symmetry conserves $p_{x}$ even though the support and bob separately accelerate.

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