1.1Derive a Lagrange equation
For $L=\tfrac12m\dot x^2-V(x)$, derive the equation of motion.
Solution
$$
\frac{\partial L}{\partial\dot x}=m\dot x,
\qquad \frac{\partial L}{\partial x}=-V'(x),
$$
so Euler–Lagrange gives $m\ddot x=-V'(x)$.
Mathematical Mechanics Practice
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1.2Identify a cyclic coordinate
If $L$ does not contain $\theta$, what quantity is conserved?
Solution
$p_\theta=\partial L/\partial\dot\theta$ is conserved because $d p_\theta/dt=\partial L/\partial\theta=0$.
1.3Use canonical equations
For $H=p^2/(2m)+kx^2/2$, find $\dot x$ and $\dot p$.
Solution
$$
\dot x=\frac{\partial H}{\partial p}=\frac{p}{m},
\qquad \dot p=-\frac{\partial H}{\partial x}=-kx.
$$
1.4Compute a Poisson bracket
Compute $\{x,p^2\}$.
Solution
$$
\{x,p^2\}=\frac{\partial x}{\partial x}\frac{\partial p^2}{\partial p}=2p.
$$
1.5Separate a Hamilton–Jacobi equation
For a free particle with $H=p^2/(2m)$, find a separated principal function with constant momentum $p_0$.
Solution
Set $S=W(x)-Et$. The Hamilton–Jacobi equation gives $(W')^2/(2m)=E$. Choosing $W'=p_0$ gives $E=p_0^2/(2m)$ and
$$
S(x,t)=p_0x-\frac{p_0^2}{2m}t+C.
$$