Noether identity for point transformations with time reparametrisation: the conserved charge $p\cdot \xi -E\tau -B$

Let $(O,\mathcal{L})$ be a time-dependent Lagrangian chart over $[a,b]\times U$ (A time-dependent Lagrangian chart, its action functional and its fixed-endpoint first variations). Let $A\subseteq \mathbb{R}\times {\mathbb{R}}^n$ be open and contain the time-configuration projection of $O$. Let $\tau ,\xi :A\to \mathbb{R},{\mathbb{R}}^n$ and $B:A\to \mathbb{R}$ be of class $C^2$, and for a $C^1$ function $\phi (t,q)$ with scalar values write $D_t\phi (t,q,v):={\phi }_t(t,q)+{\phi }_q(t,q)\cdot v$; for vector-valued $\xi$ set $D_t\xi :={\xi }_t+D_q\xi v$, equivalently $(D_t\xi {)}_j=D_t({\xi }_j)$. Assume the Noether identity

${\mathcal{L}}_t\tau +{\mathcal{L}}_q\cdot \xi +{\mathcal{L}}_v\cdot (D_t\xi -v{D}_t\tau )+\mathcal{L}{D}_t\tau =D_{t}B$

on $O$, where all arguments of $\mathcal{L}$ and its derivatives are $(t,q,v)$. Write $p:={\mathcal{L}}_v$ for the momentum and put

$E(t,q,v):={\mathcal{L}}_v(t,q,v)\cdot v-\mathcal{L}(t,q,v),C(t,q,v):={\mathcal{L}}_v(t,q,v)\cdot \xi (t,q)-E(t,q,v)\tau (t,q)-B(t,q)\mathrm{.}$

Then for every $C^2$ curve $q:I\to U$ on an open interval $I\supseteq [a,b]$ with $(t,q(t),q^{\mathrm{\prime }}(t))\in O$ that solves the Euler–Lagrange equation $E_{\mathcal{L}}(q)(t)=0$, the function $t\mapsto C(t,q(t),q^{\mathrm{\prime }}(t))$ is constant on $I$. Moreover:

1. at $\tau =0$ and with $\xi$ independent of $t$ the Noether identity is exactly the infinitesimal point symmetry identity of Infinitesimal point symmetry of a time-dependent Lagrangian and its conserved charge for the pair $(\xi ,B)$, and the charge reduces to $C={\mathcal{L}}_v\cdot \xi -B$, so the present theorem specialises to The charge of an infinitesimal point symmetry is constant along Euler-Lagrange solutions;
2. at $\xi =0$, $\tau =1$, $B=0$ the Noether identity holds if and only if ${\mathcal{L}}_t=0$ on $O$, and the charge is $C=-E$; thus the theorem contains the conservation of the energy in the time-independent case, in agreement with clause 3 of The Hamiltonian on an inverse Legendre branch: differential, equivalence with the Euler-Lagrange equation, and the energy identity.

• Dimensions of the generators. If a one-parameter family of transformations is parametrised by a parameter $s$ of physical dimension $[s]$, then $\tau$ has dimension time$\mathrm{/}[s]$, $\xi$ has dimension $[q]\mathrm{/}[s]$, $B$ has dimension action$\mathrm{/}[s]$, and the charge $C=p\cdot \xi -E\tau -B$ has dimension action$\mathrm{/}[s]$. Rescaling the generator by a constant rescales the charge by the same constant. These dimensional remarks are bookkeeping for applications; the theorem itself is the unconditional statement proved above, and no physical interpretation of the dimensions is asserted.

• Orientation. The local transformations are meant to act on the same oriented path space, which for an infinitesimal family with time change requires $\mathrm{d}{t}_{\text{new}}\mathrm{/}\mathrm{d}t>0$; the present theorem only uses the infinitesimal identity and the charge, and the global interpretation of the family is not needed for the conservation statement.

• Scope. The identity is a hypothesis on the chart data. It does not follow from the existence of a kinematic transformation of the configuration space; a chosen potential may break it. The concrete charges and the failure of the symmetry for a quadratic potential are recorded in Conserved charges for translation, rotation and boost symmetries of a mechanical Lagrangian and Scope of point symmetries: symmetries belong to the Lagrangian, not to the configuration space, and time translation is the energy case.