Noether's theorem

Euler-Langrange Equation

∂L∂φ−∂μ∂L∂φ′=0\frac{\partial L}{\partial \varphi} - \partial_\mu \frac{\partial L}{\partial \varphi'} = 0

Action

S=∫L(φ,∂μφ,xμ)d4x\mathcal S = \int \mathcal L(\varphi, \partial_\mu\varphi, x^\mu)\,d^4x

transition

φ↦φ+δ\varphi \mapsto \varphi + \delta

L↦L+∂μJ\mathcal L \mapsto \mathcal L + \partial_\mu J

L difference

dL=∂L∂φ∂φ+∂L∂φ′∂φ′d\mathcal L = \frac{\partial\mathcal L}{\partial\varphi}\partial\varphi + \frac{\partial\mathcal L}{\partial\varphi'}\partial\varphi'

=∂L∂φ∂φ−∂μ(∂L∂φ′)∂φ+∂μ(∂L∂φ′)∂φ+∂L∂φ′∂φ′= \frac{\partial\mathcal L}{\partial\varphi}\partial\varphi - \partial_\mu(\frac{\partial\mathcal L}{\partial\varphi'})\partial\varphi + \partial_\mu(\frac{\partial\mathcal L}{\partial\varphi'})\partial\varphi + \frac{\partial\mathcal L}{\partial\varphi'}\partial\varphi'

=∂L∂φ∂φ−∂μ(∂L∂φ′)∂φ+∂μ(∂L∂φ′∂φ)= \frac{\partial\mathcal L}{\partial\varphi}\partial\varphi - \partial_\mu(\frac{\partial\mathcal L}{\partial\varphi'})\partial\varphi + \partial_\mu(\frac{\partial\mathcal L}{\partial\varphi'}\partial\varphi)

=(∂L∂φ−∂μ∂L∂φ′)∂φ+∂μ(∂L∂φ′∂φ)= (\frac{\partial\mathcal L}{\partial\varphi} - \partial_\mu\frac{\partial\mathcal L}{\partial\varphi'})\partial\varphi + \partial_\mu(\frac{\partial\mathcal L}{\partial\varphi'}\partial\varphi)

=∂μ(∂L∂φ′∂φ)= \partial_\mu(\frac{\partial\mathcal L}{\partial\varphi'}\partial\varphi)


∂μJ=∂μ(∂L∂φ′∂φ)\partial_\mu J = \partial_\mu(\frac{\partial\mathcal L}{\partial\varphi'}\partial\varphi)


∂μ(∂L∂φ′∂φ−J)=0\partial_\mu (\frac{\partial\mathcal L}{\partial\varphi'}\partial\varphi - J) = 0

Noether's current

j=∂L∂φ′∂φ−J is constj = \frac{\partial\mathcal L}{\partial\varphi'}\partial\varphi - J \text{ is const}

Examples

Euclidean translations

The Langrangian is

L=∑a12max^˙a2−V(x^1,x^2,...)\mathcal L = \sum_a \frac{1}{2}m_a\dot{\hat x}_a^2 - V(\hat x_1, \hat x_2, ...)

Transition is

x^↦x^+δ^,x^˙↦x˙,t↦t\hat x \mapsto \hat x + \hat \delta,\quad \dot {\hat x} \mapsto \dot x, \quad t \mapsto t

Because potential depends on the relative positions of each particles,

L↦L\mathcal L \mapsto \mathcal{L}

Noether's current is

j=∂L∂x^˙∂x^=∑amax^˙aδ^=constj = \frac{\partial\mathcal L}{\partial \dot {\hat x}}\partial \hat x = \sum_a m_a\dot {\hat x}_a\hat\delta=\text{const}

because of δ^\hat \delta is arbitary direction,

∑amax^˙a=const\sum_a m_a\dot {\hat x}_a=\text{const}

Euclidean rotation

The Langrangian is

L=∑a12max^˙a2−V(x^1,x^2,...)\mathcal L = \sum_a \frac{1}{2}m_a\dot{\hat x}_a^2 - V(\hat x_1, \hat x_2, ...)

Transition is

x^↦x^+δ^×x^,x^˙↦x^˙+δ^×x^˙,t↦t\hat x \mapsto \hat x + \hat \delta \times \hat x,\quad \dot {\hat x} \mapsto \dot {\hat x} + \hat \delta \times \dot {\hat x}, \quad t \mapsto t

Because potential depends on the relative positions of each particles,

L↦L\mathcal L \mapsto \mathcal{L}

Noether's current is

j=∂L∂x^˙∂x^=∑amax^˙a⋅(δ^×x^)=∑aδ^⋅(x^×max^˙a)=constj = \frac{\partial\mathcal L}{\partial \dot {\hat x}}\partial \hat x = \sum_a m_a\dot {\hat x}_a\cdot(\hat\delta \times {\hat x})=\sum_a\hat\delta \cdot({\hat x} \times m_a\dot {\hat x}_a )=\text{const}

because of δ^\hat \delta is arbitary direction,

∑a(x^×max^˙a)\sum_a({\hat x} \times m_a\dot {\hat x}_a )

time translations

The Langrangian is

L=∑a12max^˙a2−V(x^1,x^2,...)\mathcal L = \sum_a \frac{1}{2}m_a\dot{\hat x}_a^2 - V(\hat x_1, \hat x_2, ...)

Transition is

x^↦x^+∂x^∂tϵ,x^˙↦x^˙+∂x^˙∂tϵ,t↦t+ϵ\hat x \mapsto \hat x + \frac{\partial\hat x}{\partial t}\epsilon ,\quad \dot {\hat x} \mapsto \dot {\hat x} + \frac{\partial\dot {\hat x}}{\partial t}\epsilon, \quad t \mapsto t + \epsilon

Because potential depends on the relative positions of each particles,

L↦L+∂L∂tϵ\mathcal L \mapsto \mathcal{L} + \frac{\partial\mathcal{L}}{\partial t}\epsilon

Noether's current is

j=∂L∂x^˙∂x^∂tϵ−Lϵ=(∂L∂x^˙x^˙−L)ϵ=constj =\frac{\partial\mathcal{L}}{\partial \dot {\hat x}} \frac{\partial\hat x}{\partial t}\epsilon - \mathcal L\epsilon = (\frac{\partial\mathcal{L}}{\partial \dot {\hat x}}\dot {\hat x}- \mathcal L)\epsilon = \text{const}

because of δ^\hat \delta is arbitary direction,

∑a∂L∂x^˙x^˙−L=const\sum_a \frac{\partial\mathcal{L}}{\partial \dot {\hat x}}\dot {\hat x}- \mathcal L=\text{const}

called Hamiltonian is conserved