A time-dependent Lagrangian chart, its action functional and its fixed-endpoint first variations

Chart data. Let $n\ge 1$, let $a<b$ be reals and let $U\subseteq {\mathbb{R}}^n$ be open and nonempty. Let $O\subseteq \mathbb{R}\times {\mathbb{R}}^n\times {\mathbb{R}}^n$ be an open set with

$[a,b]\times U\times {\mathbb{R}}^n\subseteq O,$

and let $\mathcal{L}:O\to \mathbb{R}$ be of class $C^2$ ($C^k$ Euclidean maps and diffeomorphisms). The pair $(O,\mathcal{L})$ is a time-dependent Lagrangian chart over $[a,b]\times U$. The three slots of $\mathcal{L}$ are written $(t,q,v)$, so that $t$ is the time, $q$ the configuration and $v$ the velocity; the open set $O$ has the three coordinate blocks of sizes $1,n,n$.

Admissible curves and variations. A map $q:[a,b]\to U$ is of class $C^2$ when there are $\epsilon >0$ and a $C^2$ map $\tilde{q}:(a-\epsilon ,b+\epsilon )\to {\mathbb{R}}^n$ with $\tilde{q}{\mathrm{\mid }}_{[a,b]}=q$ and $\tilde{q}(t)\in U$ for every $t\in [a,b]$; the derivative $q^{\mathrm{\prime }}$ and the second derivative $q^{\mathrm{\prime }\mathrm{\prime }}$ below are those of any such extension, and they are independent of the extension on $[a,b]$. A variation of a $C^2$ curve $q$ is a map $\eta :[a,b]\to {\mathbb{R}}^n$ of class $C^2$ in the same sense with

$\eta (a)=\eta (b)=0.$

Thus the admissible curves and variations are the restrictions to $[a,b]$ of $C^2$ maps on a common open interval. This closed-interval extension convention is stipulated here; on the open interval each component has continuous derivatives through order two in the sense of $C^k$ maps and multi-index derivative notation in Euclidean space.

The action. For a $C^2$ curve $q:[a,b]\to U$ the map $t\mapsto \mathcal{L}(t,q(t),q^{\mathrm{\prime }}(t))$ is continuous on $[a,b]$, because $q,q^{\mathrm{\prime }}$ are continuous and $\mathcal{L}$ is continuous (The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral); it is therefore Darboux integrable (The lower and upper Darboux integrals of a bounded $f$ on $[a,b]$ as ${\sup }_{P}L(f,P)$ and ${\inf }_{P}U(f,P)$, Darboux integrability as their equality, and the notation ${\int }_a^{b}f$). The action of $q$ is the real number

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

First variation. For a $C^2$ curve $q$ and a variation $\eta$ the curve $q_{\epsilon }:=q+\epsilon \eta$ takes values in $U$ for all sufficiently small $\mathrm{\mid }\epsilon \mathrm{\mid }$; this is proved in The first variation of a time-dependent action on a finite interval: admissibility, the integral formula and continuity of the Euler-Lagrange coefficient ↗ and is not part of the present definition. The first variation of $S$ at $q$ in the direction $\eta$ is the ordinary two-sided derivative at $\epsilon =0$ of the function $\epsilon \mapsto S(q+\epsilon \eta )$, written

$\delta S(q)[\eta ]:=\frac{d}{d\epsilon }{\mid }_{\epsilon =0}S(q+\epsilon \eta ),$

whenever this derivative exists; The first variation of a time-dependent action on a finite interval: admissibility, the integral formula and continuity of the Euler-Lagrange coefficient ↗ proves that it does exist under the present hypotheses and computes it, so the notation denotes a single real number for every pair $(q,\eta )$ arising here.

Euler–Lagrange expression. Let ${\mathcal{L}}_q(t,q,v)\in {\mathbb{R}}^n$ and ${\mathcal{L}}_v(t,q,v)\in {\mathbb{R}}^n$ be the gradients of $\mathcal{L}$ in the second and third slots (The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case, Directional derivatives and partial derivatives of a map $U\subseteq {\mathbb{R}}^m\to {\mathbb{R}}^n$). Since $\mathcal{L}$ is $C^2$, the maps $t\mapsto {\mathcal{L}}_v(t,q(t),q^{\mathrm{\prime }}(t))$ are $C^1$ and $t\mapsto {\mathcal{L}}_q(t,q(t),q^{\mathrm{\prime }}(t))$ is continuous on $[a,b]$. The Euler–Lagrange expression of $\mathcal{L}$ along $q$ is the continuous map

$E_{\mathcal{L}}(q)(t):={\mathcal{L}}_q(t,q(t),q^{\mathrm{\prime }}(t))-\frac{d}{dt}\left({\mathcal{L}}_v\right(t,q(t),q^{\mathrm{\prime }}(t)\left)\right)\in {\mathbb{R}}^n\mathrm{.}$

Stationarity. A $C^2$ curve $q$ is stationary for $S$ when

$\delta S(q)[\eta ]=0\text{for every variation }\eta \mathrm{.}$

Scope. The definition introduces only chart data, the action and its first variation. It asserts nothing about minimising $S$, nothing about the existence of stationary curves and nothing about any physical interpretation of $\mathcal{L}$; the equivalence of stationarity with the vanishing of $E_{\mathcal{L}}(q)$ is the theorem Stationarity of the time-dependent action is equivalent to the vanishing of the Euler-Lagrange expression, and the distinction between stationary and minimising curves is recorded in Scope of stationarity: stationary is not minimum, and no second-variation theory is developed here.

• The Lagrangian is allowed to depend on $t$, which is what makes the chart suitable for systems with prescribed time-dependent constraints, moving frames or external drifts; the chart formulation with the extension convention above is the one used by the consuming classical mechanics pages.

• If $U={\mathbb{R}}^n$ and $\mathcal{L}$ is the restriction of a $C^2$ function on $\mathbb{R}\times {\mathbb{R}}^n\times {\mathbb{R}}^n$, then $O$ may be taken to be all of ${\mathbb{R}}^{1+2n}$ and no extension discussion is needed. The general formulation keeps $O$ open so that the Legendre map of The Legendre map of a time-dependent Lagrangian: regularity and hyperregularity can be studied on the same domain.