The action functional of a force model with a potential presentation: the mechanical Lagrangian, its chart instance 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. Fix one inertial frame with adapted coordinates $(t,x)$ and identify position tuples with points of ${\mathbb{R}}^{3N}=({\mathbb{R}}^3{)}^N$ (Configuration domain of $N$ labeled particles: open admissible sets, regular holonomic constraints and the mass operator); 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. Let $\mathsf{M}$ be the mass operator of clause (d) of Configuration domain of $N$ labeled particles: open admissible sets, regular holonomic constraints and the mass operator, $\mathsf{M}(x_1,\dots ,x_N)=(m_1{x}_1,\dots ,m_N{x}_N)$, a symmetric positive definite linear map of ${\mathbb{R}}^{3N}$.

(a) Kinetic energy. The kinetic energy of the labels at coordinate velocity $v\in ({\mathbb{R}}^3{)}^N={\mathbb{R}}^{3N}$ is

$K(v):=\frac{1}{2}{\sum }_{i=1}^N{m}_i\mid v_i{\mid }_2^2=\frac{1}{2}v\cdot \mathsf{M}v\mathrm{.}$

$K$ is the restriction of a $C^{\mathrm{\infty }}$ function of $({\mathbb{R}}^3{)}^N$ and has dimension $\mu {\mathrm{\ell }}^2{\tau }^{-2}$ (energy). Its Euclidean gradient is $K^{\mathrm{\prime }}(v)=\mathsf{M}v$, because $\mathrm{\partial }K\mathrm{/}\mathrm{\partial }{v}_i=m_i{v}_i$ (The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case, Directional derivatives and partial derivatives of a map $U\subseteq {\mathbb{R}}^m\to {\mathbb{R}}^n$).

(b) Mechanical Lagrangian of a potential presentation. Let $(U,d)$ be a potential presentation of the model's force map in the sense of clause (a) 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 $C^2$ with dimension $\mu {\mathrm{\ell }}^2{\tau }^{-2}$ and residual forces $d=(d_1,\dots ,d_N)$ such that $F_i(t,x,v)=-{\mathrm{\nabla }}_{x_i}U(t,x)+d_i(t,x,v)$ on the state domain. The mechanical Lagrangian of the presentation is

$L:I_0\times Q\times {\mathbb{R}}^{3N}⟶\mathbb{R},L(t,x,v):=K(v)-U(t,x)\mathrm{.}$

$L$ is of class $C^2$ on its open domain and has dimension $\mu {\mathrm{\ell }}^2{\tau }^{-2}$. When $d\equiv 0$ the presentation is a global potential model and $L$ is a Lagrangian of the model itself; when $d\equiv ̸0$, $L$ carries the potential part only, and the residual force contribution $d$ is not included in $L$. The potential representative is part of the declared data: replacing $U$ by $U+c(t)$ for differentiable $c:I_0\to \mathbb{R}$ replaces $L$ by $L-c(t)$ (Potential presentations of a force model, potential and mechanical energy, closed versus exact force fields, local potentials and equipotential level sets clause (c)).

(c) Action, variations, stationarity. Let $a<b$ be reals with $[a,b]\subseteq I_0$. If $Q=\mathrm{\varnothing }$, there is no map $x:[a,b]\to Q$, so this clause has no action, variation or stationarity instance. Assume $Q\mathrm{\ne }\mathrm{\varnothing }$ for the chart construction below, and put $O:=I_0\times Q\times {\mathbb{R}}^{3N}$. The set $O$ is open and contains $[a,b]\times Q\times {\mathbb{R}}^{3N}$, so with $n:=3N$ the pair $(O,L)$ is a time-dependent Lagrangian chart over $[a,b]\times Q$ in the sense of A time-dependent Lagrangian chart, its action functional and its fixed-endpoint first variations. Admissible curves (the $C^2$ curves $x:[a,b]\to Q$ with the extension convention of that definition), variations $\eta$ with $\eta (a)=\eta (b)=0$, the action

$S(x):={\int }_a^{b}L(t,x(t),x^{\mathrm{\prime }}(t))dt,$

the first variation $\delta S(x)[\eta ]$ and stationarity of $x$ are the notions of A time-dependent Lagrangian chart, its action functional and its fixed-endpoint first variations instantiated at this chart; the derivative $x^{\mathrm{\prime }}$ is the coordinate velocity (Coordinate velocity and acceleration of a worldline in an inertial frame, and the spatial velocity and acceleration vectors). The action has dimension $\mu {\mathrm{\ell }}^2{\tau }^{-1}$ (energy multiplied by time). Write $E_L(x):[a,b]\to {\mathbb{R}}^{3N}$ for the Euler-Lagrange expression of the instantiated chart.

(d) Status. This item defines the action functional and the Lagrangian of a presentation from data already adopted in the model: no new force law, no equation of motion and no measured value are introduced. The first variation of $S$ exists for every admissible curve and variation and its formula, the equivalence of stationarity with the vanishing of $E_L(x)$, and the relation of both to the model's equation of motion are theorems of this page, not part of the definition.

• The Lagrangian is not unique. The value of $L$ depends on the declared potential representative and on the split of the force map into potential and residual parts. Two presentations of the same force map with different splits produce the same model motions but different Lagrangians; the effect of changing the potential representative is a total time derivative, and the equations of motion are insensitive to it (the boundary-gauge theorem of this page).

• Velocity and configuration slots. The three arguments of $L$ are $(t,x,v)$ with $x\in Q$ the configuration and $v$ the coordinate velocity in the fixed inertial frame; the kinetic energy uses the fixed masses and the coordinate velocity, not any generalized velocity. Generalized coordinates enter only through the substitution construction of this page.

• Fixed endpoints. Stationarity is defined with variations vanishing at both endpoints, exactly as in A time-dependent Lagrangian chart, its action functional and its fixed-endpoint first variations. The endpoint-action differential and free-endpoint problems belong to the Hamiltonian development and are not asserted here.