 Solution

Let $p_i=\partial L/\partial {\dot{q}}_i$ and consider the infinitesimal fixed-time transformation $\delta q_i=ϵK_i(\mathbf{q})$. It induces $\delta {\dot{q}}_i=ϵd{K}_i/dt$. Expanding the Lagrangian givesOn a solution of the Euler-Lagrange equations, $\partial L/\partial q_i={\dot{p}}_i$, so this equals $d({\sum }_i{p}_i{K}_i)/dt$. Exact invariance makes the left side zero, proving the fixed-time form of Noether's theorem:The symmetry direction need not coincide with one original coordinate axis; $\mathbf{K}$ can depend on the generalized coordinates. The conclusion is evaluated on physical trajectories, while the assumed invariance is a property of the Lagrangian under the transformation.