Conservation laws

Classical mechanics has several well-known conservation laws. Here we show how these laws can be derived using Lagrangian formalism.

For an system without external interactions, the Lagrangian cannot depend on time.

\begin{align*} \frac{\d L}{\d t} &= \sum_i \left( \frac{\partial L}{\partial q_i}\frac{\d q_i}{\d t} + \frac{\partial L}{\partial \dot q_i} \frac{\d \dot q_i}{\d t}\right) \\ \dot L &= \dot q_i \partial_i L + \ddot q_i \dot \partial_i L = 0. \end{align*}

From the Euler-Lagrange equation we can substitute $\partial_i L$

\begin{align*} \left(\frac{\d}{\d t} \dot \partial_i L \right) \dot q_i + \ddot q_i \dot\partial_i L - \dot L = \frac{\d}{\dt} \big(\dot q_i \dot\partial_i L - L \big) = 0. \end{align*}

Assuming $L = \frac12 m \dot q_i \dot q_i - U(q)$ , the conserved quantity is

\begin{align*} m\dot q_i \dot q_i - \left(\frac12 m \dot q_i \dot q_i - U(q)\right) = \frac12 m \dot q_i \dot q_i + U(q) =: E. \end{align*}

So energy is conserved when there are no time-dependent interactions (varying external potentials).

Another symmetry that follows from postulate 2 of classical mechanics is spacial invariance. If a system is translated some small amount $\dx$ ,

Abuse of notation

I’m using Einstein notation here, without caring about up/down indices.