User:Direwolf202/sandbox

=DRAFT: Noether's theorem=

Noether's theorem (or Noether's first theorem) states that every differentiable symmetry of the action of a physical system has a corresponding conservation law. The theorem was proven by Emmy Noether in 1915 and published in 1918. The action of a system is the integral over time of a Lagrangian function, from which the behavior of the system can be determined using the principle of stationary action.

Noether's theorem is used in theoretical physics and in the calculus of variations, in order to identify conserved quantities. Conserved quantities are extremely useful tools for calculations (for example, the conservation of energy and momentum can be used to determine the velocities of two objects after they collide), and also can provide more direct insight into the behavior of the system.

Informally, a symmetry is a feature of a system that remains unaffected by some transformation. If a sphere is rotated about its center, it looks the same as before it was rotated and the sphere is therefore said to have spherical symmetry. Noether's theorem provides conservation laws (such as the conservation of energy) when these symmetries exist in a physical system.

Statements of the theorem
In Lagrangian mechanics, the action $$\mathcal{S}$$ is given as the integral of the Lagrangian


 * $$\mathcal{S}[\bold{q}(t)] = \int_{t_0}^{t_1} \operatorname{L}[\bold{q}(t), \dot{\bold{q}}(t), t] dt$$

If $$\mathcal{S}$$ is invariant under continuous transformations defined by:


 * $$\bold{q}(t) \rightarrow \bold{q}'(t) = \operatorname{F}[\bold{q}(t), \dot{\bold{q}}(t)]$$

where $$\rm{F}$$ is a functional which accepts the functions $$\bold{q}(t)$$ and $$\dot{\bold{q}}(t)$$ and returns a scalar, then the difference


 * $$\delta_{s} \bold{q}(t) = \bold{q}'(t) - \bold{q}(t)$$

is the symmetry variation, which has the general form:


 * $$\delta_{s} \bold{q}(t) = \varepsilon \Delta [\bold{q}(t), \dot{\bold{q}}(t), t]$$

where $$\Delta$$ is a functional like $$\rm{F}$$, but with the additional argument of time.