## Noether’s theorem

In connection with classical mechanics and its formulation in terms of a least action principle there exists a remarkable theorem, the so called Noether theorem. It states:

For every continuous symmetry of a physical system, there exists a conserved quantity.

The reverse of this statement is also true as we will see later.

$$S[\vec{q}(t)] = \int_{t_1}^{t_2} L(\vec{q}, \dot{\vec{q}}, t) ~ dt$$
which is a functional $S: F^n \rightarrow \mathbb{R}$, i.e. it maps the $n$ functions $\vec{q}(t) = (q_1(t),\dots,q_n(t)) \in F^n$ of some function space $F$ onto the real numbers. Continue reading “The Euler-Lagrange equation and the principle of least action”