\gdef\∂{\partial} \gdef\∫{\int} \gdef\d{\operatorname{d}\!} \gdef\g{\mathrm{g}} \gdef\vec#1{\mathbf #1}

To every differentiable symmetry generated by local actions, there corresponds a conserved current.

Discrete Langrangian Mechanics

In Langrangian mechanics a system is described by a set of generalized parameters \vec x(t), a time depended vector in the state space of the system.

The physics of the system is contained in a Lagrangian function:

L(t, \vec x, \vec v): ℝ × \mathrm{T}\mkern2mu X → ℝ

where \vec x ∈ X and \vec v ∈ \mathrm{T}_{\vec x} X.

Hamilton’s principle than states that:

δ \vec x(t) \∫ t L(\vec x(t), (\d t {\vec x}) (t), t) = 0

this can be restated as the Euler-Lagrange equations:

\d t \∂ {v_k} L = \∂{x_k} L

Noether’s theorem

Given a symmetry T_r, \vec Q_r, the quantity

\g{ \∂ {\vec v} L · \vec v - L } T_r - \∂ {\vec v} L · \vec Q_r

is conserved.

In reality

The standard model

Symmetry Generators Conserved quantity

Time shift 1 Energy Translation 3 Momentum Rotation 3 Angular Momentum Boosts 3 ?? Gauge 6 Color charge, Weak isospin, Electric charge, Weak hypercharge Phase 4 Baryon, electron, muon and tau numbers

General Relativity

The Einstein-Hilbert Langrangian:

L = \frac{c^4}{16 π G} R_j^j \sqrt{-\det g_{μν}}

Killing vector fields are symmetries of the metric tensor.