Editing Noether's theorem (section)

=== Example 1: Conservation of energy ===
Looking at the specific case of a Newtonian particle of mass ''m'', coordinate ''x'', moving under the influence of a potential ''V'', coordinatized by time ''t''. The [[action (physics)|action]], ''S'', is:

:<math>\begin{align}
  \mathcal{S}[x] & = \int L\left[x(t),\dot{x}(t)\right] \, dt \\
                 & = \int \left(\frac m 2 \sum_{i=1}^3\dot{x}_i^2 - V(x(t))\right) \, dt.
\end{align}</math>

The first term in the brackets is the [[kinetic energy]] of the particle, while the second is its [[potential energy]]. Consider the generator of [[time translation]]s <math>Q = \frac{d}{dt}</math>. In other words, <math>Q[x(t)] = \dot{x}(t)</math>. The coordinate ''x'' has an explicit dependence on time, whilst ''V'' does not; consequently:

:<math>Q[L] =
  \frac{d}{dt}\left[\frac{m}{2}\sum_i\dot{x}_i^2 - V(x)\right] =
  m \sum_i\dot{x}_i\ddot{x}_i - \sum_i\frac{\partial V(x)}{\partial x_i}\dot{x}_i
</math>

so we can set

:<math>L = \frac{m}{2} \sum_i\dot{x}_i^2 - V(x).</math>

Then,

:<math>\begin{align}
  j & = \sum_{i=1}^3\frac{\partial L}{\partial \dot{x}_i}Q[x_i] - L \\
    & = m \sum_i\dot{x}_i^2 - \left[\frac{m}{2}\sum_i\dot{x}_i^2 - V(x)\right] \\[3pt]
    & = \frac{m}{2}\sum_i\dot{x}_i^2 + V(x).
\end{align}</math>

The right hand side is the energy, and Noether's theorem states that <math>dj/dt = 0</math> (i.e. the principle of conservation of energy is a consequence of invariance under time translations).

More generally, if the Lagrangian does not depend explicitly on time, the quantity

:<math>\sum_{i=1}^3 \frac{\partial L}{\partial \dot{x}_i}\dot{x_i} - L</math>

(called the [[Hamiltonian mechanics|Hamiltonian]]) is conserved.