Noether’s theorem

The integral conserved quantities are corresponding to an infinitesimal transformation under which the action remains invariant.

Notes

In action, there are several integral conserved quantities: Time invariant of the action:

$$\begin{array}{rl}\frac{\partial L}{\partial t}& =0\\ \frac{dL}{dt}& =\frac{\partial L}{\partial q}\dot{q}+\frac{\partial L}{\partial \dot{q}}\ddot{q}\end{array}$$

Use Euler-Lagrange equation:

$$\begin{array}{rl}\frac{dL}{dt}& =\frac{d}{dt}(\frac{\partial L}{\partial \dot{q}}\dot{q})\\ \frac{d}{dt}(\frac{\partial L}{\partial \dot{q}}\dot{q}-L)& =0\\ E& =\frac{\partial L}{\partial \dot{q}}\dot{q}-L\end{array}\text{\hspace{1em}(1)(2)}$$

Space invariant of the action, combined with Euler-Lagrange equation

$$\frac{d}{dt}(\frac{\partial L}{\partial \dot{q}})=0\text{\hspace{1em}(3)}$$

This also defines generalized Momentum, also named canonical momentum.

Examples

Related

• Integrable systems
• gauge symmetries
• Killing tensor