= Solution
For an infinitesimal continuous symmetry $\delta q_i=\varepsilon\xi_i(q,t)$, allow the <Lagrangian> to change by a total derivative, $\delta L=\varepsilon\,dF/dt$. Direct variation gives
$$
\frac{\delta L}{\varepsilon}=\sum_i\left(\frac{\partial L}{\partial q_i}-\frac d{dt}\frac{\partial L}{\partial\dot q_i}\right)\xi_i+\frac d{dt}\sum_i p_i\xi_i.
$$
On a solution of the <Euler-Lagrange equations>, the first term vanishes. This proves <Noether's theorem> in the fixed-time form:
$$
\boxed{Q=\sum_i p_i\xi_i-F,\qquad\dot Q=0.}
$$
Spatial translations yield total <momentum>, and rotations yield <angular momentum>. Time translations yield <energy> when the Lagrangian has no explicit time dependence; this follows either from the extended time-transformation form of the theorem or by differentiating $H=\sum_ip_i\dot q_i-L$ to obtain $\dot H=-\partial L/\partial t$.
Back to article page
