Continuous symmetries of the mechanical Lagrangian of a potential presentation: symmetry data, the Noether identity, the charge and its dimensions

Adopt one force model of Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance: $N\ge 1$ particle labels with fixed masses $m_1,\dots ,m_N>0$ (clause (i)), an open nondegenerate time window $I_0\subseteq \mathbb{R}$, an open configuration domain $Q\subseteq {\mathbb{R}}^{3N}$ and a force map $F$ with the standing continuity and state-local-Lipschitz hypotheses of clause (iii) of that postulate, and fix one inertial frame with adapted coordinates $(t,x)$ (Configuration domain of $N$ labeled particles: open admissible sets, regular holonomic constraints and the mass operator). Adopt in addition a potential presentation $(U,d)$ of Potential presentations of a force model, potential and mechanical energy, closed versus exact force fields, local potentials and equipotential level sets: a function $U:I_0\times Q\to \mathbb{R}$ of class $C^2$ of dimension $\mu {\mathrm{\ell }}^2{\tau }^{-2}$ with $F_i(t,x,v)=-{\mathrm{\nabla }}_{x_i}U(t,x)+d_i(t,x,v)$, with residual forces $d=(d_1,\dots ,d_N)$ of the standing regularity. Let $L=K-U$, $O:=I_0\times Q\times {\mathbb{R}}^{3N}$ and $\mathsf{M}$ be the mechanical Lagrangian of the presentation, the chart domain and the mass operator of The action functional of a force model with a potential presentation: the mechanical Lagrangian, its chart instance and its dimensions, so that $L$ is of class $C^2$ on the open set $O$. When $d\equiv ̸0$, $L$ represents only the potential part and excludes the residual forces; its symmetry identity places no condition on $d$ and does not assert a symmetry of the full force map. When $d\equiv 0$, $L$ is a Lagrangian of the model itself. When $Q\mathrm{\ne }\mathrm{\varnothing }$, $(O,L)$ is a time-dependent Lagrangian chart over every $[a,b]\times Q$ with $a<b$ and $[a,b]\subseteq I_0$ (A time-dependent Lagrangian chart, its action functional and its fixed-endpoint first variations). Dimensions are read by Dimensions and SI unit conventions of the non-relativistic classical mechanics framework, inner products and norms are Euclidean (The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$), and differentiability classes are those of $C^k$ Euclidean maps and diffeomorphisms. For a $C^1$ function $\phi$ on an open subset of $\mathbb{R}\times {\mathbb{R}}^{3N}$ and $(t,x,v)\in O$ write

$D_t\phi (t,x,v):={\phi }_t(t,x)+{\phi }_x(t,x)\cdot v,$

the derivative of $s\mapsto \phi (s,x+v(s-t))$ at $s=t$; the last two slots of $L$ are $(x,v)$, and $L_t$, $L_x$, $L_v$ are the gradients of $L$ in its three slots (The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case, Directional derivatives and partial derivatives of a map $U\subseteq {\mathbb{R}}^m\to {\mathbb{R}}^n$).

(a) Symmetry datum. A symmetry datum of the mechanical Lagrangian of the declared potential presentation is a triple $(\tau ,\xi ,B)$ of maps

$\tau :I_0\times Q\to \mathbb{R},\xi :I_0\times Q\to {\mathbb{R}}^{3N},B:I_0\times Q\to \mathbb{R},$

of class $C^2$, that satisfies for every $(t,x,v)\in O$ the Noether identity of the model:

$L_t(t,x,v)\tau (t,x)+L_x(t,x,v)\cdot \xi (t,x)+L_v(t,x,v)\cdot (D_t\xi (t,x,v)-v{D}_t\tau (t,x,v))+L(t,x,v)D_t\tau (t,x,v)=D_{t}B(t,x,v)\mathrm{.}$

Because $K(v)=\frac{1}{2}v\cdot \mathsf{M}v$ does not depend on $t$ or $x$ and $U$ does not depend on $v$, the identity is the following identity among the model's data:

$-U_t(t,x)\tau (t,x)-{\mathrm{\nabla }}_{x}U(t,x)\cdot \xi (t,x)+\left(\mathsf{M}v\right)\cdot (D_t\xi (t,x,v)-v{D}_t\tau (t,x,v))+(K(v)-U(t,x))D_t\tau (t,x,v)=D_{t}B(t,x,v)\mathrm{.}$

It involves the declared potential $U$, the masses through $\mathsf{M}$, and the boundary function $B$; it is an identity imposed on the model's declared data on the whole state domain $O$.

(b) Strict data and point symmetries. A symmetry datum is strict when $\tau \equiv 0$; its identity reads $L_x\cdot \xi +L_v\cdot D_t\xi =D_{t}B$. The following chart identification is for $Q\mathrm{\ne }\mathrm{\varnothing }$. If a strict datum has $\xi$ independent of $t$, then the pair $(\xi ,B)$ is an infinitesimal point symmetry of the chart $(O,L)$ in the sense of Infinitesimal point symmetry of a time-dependent Lagrangian and its conserved charge: that definition's identity, instantiated with $n=3N$ in the configuration slot, is verbatim $L_x(t,x,v)\cdot \xi (x)+L_v(t,x,v)\cdot D\xi (x)v=B_t(t,x)+B_x(t,x)\cdot v$. Conversely every infinitesimal point symmetry of the chart $(O,L)$ whose generator and boundary function are of class $C^2$, with the boundary function defined on $I_0\times Q$, is a strict datum with $\xi$ independent of $t$. A general datum is a point symmetry with time reparametrisation: the family

$x_i⟼x_i+s{\xi }_i(t,x),t⟼t+s\tau (t,x)$

changes the coordinate velocity to first order by $D_t\xi -v{D}_t\tau$, and the identity says that the first-order change of the action density $Ldt$, expressed per unit of the original $dt$, is $D_{t}B$. The term $L{D}_t\tau$ accounts for the change of the time differential; the variation of $L$ alone equals $D_{t}B$ only when this term vanishes.

(c) Charge. The charge of the datum is the function

$C:O\to \mathbb{R},C(t,x,v):=L_v(t,x,v)\cdot \xi (t,x)-E(t,x,v)\tau (t,x)-B(t,x),$

where

$E(t,x,v):=L_v(t,x,v)\cdot v-L(t,x,v)=K(v)+U(t,x)$

is the mechanical energy function of the chart (Potential presentations of a force model, potential and mechanical energy, closed versus exact force fields, local potentials and equipotential level sets). Since $L_v=\mathsf{M}v$,

$C(t,x,v)=\left(\mathsf{M}v\right)\cdot \xi (t,x)-(K(v)+U(t,x))\tau (t,x)-B(t,x)\mathrm{.}$

(d) Dimensions and rescaling. Read the family parameter $s$ as carrying a declared unit $\sigma$ (for instance radians for a rotation family, seconds for a time-reparametrisation family, or a dimensionless parameter). Then $\tau$ has dimension $\tau {\sigma }^{-1}$, every component ${\xi }_i$ has dimension $\mathrm{\ell }{\sigma }^{-1}$, $B$ has dimension $\mu {\mathrm{\ell }}^2{\tau }^{-1}{\sigma }^{-1}$ (action per parameter unit) and $C$ has the same dimension as $B$ (Dimensions and SI unit conventions of the non-relativistic classical mechanics framework). Replacing $(\tau ,\xi ,B)$ by $(c\tau ,c\xi ,cB)$ with a real constant $c\mathrm{\ne }0$ — the same family reparametrised by $s\mapsto cs$ — rescales $C$ by $c$.

(e) Status. This item fixes the model's symmetry vocabulary: which data satisfy the model's Noether identity, and which function is their charge. It asserts neither that a given model admits a nonzero symmetry datum, nor that the charge of a datum is conserved. Conservation is proved under explicit hypotheses by the theorems of this page, and the hypotheses must be verified against the declared potential $U$ and the masses: a change of inertial frame, the affine structure of the spacetime of Adopted Newtonian spacetime structure: absolute time, Euclidean spatial slices and their smooth structure, or a relabelling of the particle labels is a kinematic or descriptive operation and is not by itself a symmetry datum of $L$. Every statement that uses a charge names the potential representative used (Potential presentations of a force model, potential and mechanical energy, closed versus exact force fields, local potentials and equipotential level sets clause (c)); the effect of changing the representative is recorded in the general Noether theorem of this page.

• Generator convention. The sign of $\xi$ is fixed by the family $x\mapsto x+s\xi$: with an inverse-family convention, in which the transformed coordinate is written as a function of the old one by $x\mapsto x-s\xi$, the same identity holds with $C$ replaced by $-C$. The present convention gives $a\cdot J$ for the rotation generator ${\xi }_i=a\times x_i$. In SICM Section 1.8.5, the rotation matrix in Eq. (1.163) has derivative $(-y,x,0)$ at zero, whereas Eq. (1.168) displays $(y,-x,0)$ and hence the negative angular momentum. Those displayed formulas have a sign inconsistency; the coordinate formula and the fixed generator, rather than that printed derivative, determine the sign used here.

• Relation to the mathematical track. The chart-level results Infinitesimal point symmetry of a time-dependent Lagrangian and its conserved charge, The charge of an infinitesimal point symmetry is constant along Euler-Lagrange solutions and Noether identity for point transformations with time reparametrisation: the conserved charge $p\cdot \xi -E\tau -B$ concern a general chart Lagrangian; this item only instantiates them at the mechanical Lagrangian of the declared presentation, and the identity above is exactly theirs written in model terms. The conservation theorems of this page are instances of those results, not independent postulates.

• What a symmetry is not. The identity constrains the potential: for the translation family with frozen generator $\xi =a$ and $B=0$ it fails at the state $(t,x,v)$ exactly where $a\cdot {\mathrm{\nabla }}_{x}U(t,x)\mathrm{\ne }0$, and for the quadratic potential $U=\frac{k}{2}\mid x{\mid }^2$ with $k>0$ it fails exactly where $a\cdot x\mathrm{\ne }0$, since $a\cdot {\mathrm{\nabla }}_{x}U=ka\cdot x$. A model may therefore possess the full kinematic group of Galilean frame changes and still admit no translation symmetry datum. The worked models of the companion page make this failure explicit.

• Not asserted. No converse is asserted: from the conservation of a quantity along the motions of a model it does not follow that the model has a symmetry datum producing it, and no claim is made that the conservation laws observed in nature are generated by symmetries. The item also does not treat discrete symmetries, gauge families whose generators are not $C^2$, field-theoretic actions, or global existence of the transformation families; only the identity and the charge are defined here.