Past exam of the physics 1 paper of the University of Oxford / 2019 / 9 / a

For a Lagrangian $L(q_i,{\dot{q}}_i,t)$ expressed in independent generalized coordinates, stationary action gives the Euler-Lagrange equationsIdeal constraints can either be incorporated into the independent coordinates or treated using Lagrange multipliers, as in solution.

A cyclic coordinate is absent from the Lagrangian itself: $\partial L/\partial q_i=0$. Its conjugate momentum $p_i=\partial L/\partial {\dot{q}}_i$ therefore obeys ${\dot{p}}_i=0$. In the rolling-wedge calculation, the cyclic horizontal translation gives conservation of the system's horizontal momentum.

A holonomic constraint can be expressed as $f(q_1,\dots ,q_n,t)=0$, rather than as a genuinely nonintegrable relation among velocities. A fixed rod length is an example. Rolling along a single straight incline also gives an integrable relation $s+R\theta =\text{constant}$ with the sign convention used below; arbitrary rolling in more dimensions need not be holonomic.