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The swing without the tension

• A pendulum's string tension changes all through the swing. The Lagrangian method gets the angular equation without ever finding it.
• GRE Lagrangian questions test constraints, Euler–Lagrange 欧拉–拉格朗日 equations, cyclic coordinates 循环坐标 and small oscillations.

Coordinates that respect constraints

• Fixed length: $x = l\sin\theta$, $y = -l\cos\theta$, so $v^2 = l^2\dot\theta^2$. Then $T = \tfrac12 ml^2\dot\theta^2$, $V = mgl(1 - \cos\theta)$.
• A holonomic 完整的 constraint (a relation among coordinates) removes a coordinate. Rolling constraints on velocities need separate care.

Euler–Lagrange

• $\dfrac{d}{dt}\dfrac{\partial L}{\partial\dot q} - \dfrac{\partial L}{\partial q} = 0$, $L = T - V$. Pendulum: $ml^2\ddot\theta + mgl\sin\theta = 0$.
• Differentiate every time-dependent factor: $\frac{d}{dt}(mr^2\dot\theta) = mr^2\ddot\theta + 2mr\dot r\dot\theta$.

Cyclic coordinates

• Central force: $L = \tfrac12 m(\dot r^2 + r^2\dot\theta^2) - V(r)$. No $\theta$ in $L$ (only $\dot\theta$), so $p_\theta = mr^2\dot\theta$ is conserved.
• Halving $r$ multiplies $\dot\theta$ by 4. Radial equation: $m\ddot r = mr\dot\theta^2 - V'(r)$.

Small oscillations

• Near a stable $q_0$: $\omega = \sqrt{V''(q_0)/M}$ with $M$ the local inertia. Pendulum: $\omega = \sqrt{g/l}$.
• $V'' < 0$ is an instability, not an imaginary frequency to report.

Recap