3 Noether’s theorem

Noether’s theorem elucidates how continuous symmetries result in conserved quantities; the converse will be provided by Hamiltonian mechanics. The key points in this section are:

  1. The definition of a continuous symmetry.
  2. Noether’s theorem.
  3. The conserved-quantity when \(\mathcal L\) is time-invariant: the Hamiltonian.

It’s convenient to now define a quantity we’ll be using later very often:

Definition 3.1 (canonical momentum) Given the Lagrangian \(\mathcal L(q, \dot q, t)\) of a system, its canonical momentum is \[ p = \nabla_{\dot q} \mathcal L(q, \dot q, t) \] In terms of the canonical momentum, the Euler-Lagrange equations read \[ \mathbb L_q = \dot p - \nabla_q \mathcal L = 0 \]

We will make extensive use of lemma 2.5, which we state here again using the canonical momentum for convenience:

Lemma 3.1 (ε-derivative of the perturbed Lagrangian) Given a path \(q:[a, b]\to \mathbb R\) and a perturbation path \(\eta:[a, b]\to \mathbb R\), define the parameterized family of paths \(q_\epsilon=q + \epsilon \eta\), then \[ d_\epsilon \mathcal L(q_\epsilon, \dot q_\epsilon) = -\eta \cdot \mathbb L_q + d_t \left(\eta \cdot p\right) \]

Definition 3.2 (continuous symmetry) Consider a family of maps \(q_\epsilon\) such that \(q_0\) is the classical trajectory w.r.t. the Lagrangian \(\mathcal L\). Define \[ \mathcal L_\epsilon = \mathcal L(q_\epsilon, \dot q_\epsilon, t) \] as well as the parameterized action function \[ S_\epsilon = \int_a^b \mathcal L(q_\epsilon, \dot q_\epsilon, t)\, dt \] The physical system has continuous symmetry under the transformation parameterized by \(q\mapsto q_\epsilon\) if \[ d_\epsilon \big|_0 S_\epsilon = C \] where \(C\) is a constant independent of the dynamical variables \(q, \dot q, t\). In other words, infinitesimal reparameterization by \(\epsilon\) leaves the equations of motion unchanged.

Proposition 3.1 (characterization of continuous symmetry via Lagrangian) \(\epsilon \mapsto q_\epsilon\) is a continuous symmetry if \[ d_\epsilon \big|_0 \mathcal L_\epsilon = d_t F(q, \dot q, t) \] Note that the right-hand side must be a total time derivative.

Proof: Direct computation: \[ d_\epsilon \big|_0 S_\epsilon = \int_a^b d_\epsilon \big|_0 \mathcal L_\epsilon \, dt = \int_a^b d_t(F) \, dt = F(b)-F(a) \]

Theorem 3.1 (Noether's theorem) Given a continuous symmetry \(q_\epsilon\) such that \[ d_\epsilon \big|_0 \mathcal L(q_\epsilon, \dot q_\epsilon, t) = d_t F(q, \dot q, t) \] The following Noether change is conserved along a classical trajectory obeying the Euler-Lagrange equations \[ Q = \left(d_\epsilon \big|_0 q_\epsilon \right) \cdot \nabla_{\dot q}\mathcal L(q, \dot q, t) - F(q, \dot q, t) \] Let \(q_\epsilon = q + \epsilon \eta\) for small \(\epsilon\) (this is effectively adapting vector field perspective), the Noether charge is \[ Q = \eta \cdot p - F(q, \dot q, t) \] One special case is for \(F\) to vanish, in which case \[ d_\epsilon \big|_0 \mathcal L(q_\epsilon, \dot q_\epsilon, t) = 0 \implies Q = \eta \cdot p \text{ conserved.} \]

Proof: Direct computation invoking lemma 3.1: the \(\mathbb L_q\) term vanishes on the classical trajectory \[\begin{align} d_t F &= d_\epsilon \big|_0 \mathcal L(q_\epsilon, \dot q_\epsilon) = - \eta \cdot \mathbb L_q + d_t(\eta \cdot p) = d_t(\eta \cdot p) \end{align}\] This implies that the time-derivative of the Noether charge is \(0\).

Example 3.1 (the Hamiltonian) Let \(\mathcal L\) be independent of time and denote the time-translation transform \[ q_\epsilon(t) = q(t+\epsilon) \] This is a continuous symmetry with \(F(q, \dot q) = \mathcal L(q, \dot q)\) \[ d_\epsilon \big|_0 \mathcal L(q_\epsilon, \dot q_\epsilon) = d_t \mathcal L \] The associated \(\eta = d_\epsilon \big|_0 q_\epsilon = d_\epsilon \big|_0 q(t+\epsilon) = \dot q\). Substituting into Noether’s theorem yields the conserved quantity \[ H = \dot q\cdot p - \mathcal L(q, \dot q) \]