The fixed-endpoint second variation of a time-dependent action at a curve

Let $n\ge 1$, let $a<b$ be reals, let $U\subseteq {\mathbb{R}}^n$ be open and nonempty, and let $(O,\mathcal{L})$ be a time-dependent Lagrangian chart over $[a,b]\times U$ in the sense of A time-dependent Lagrangian chart, its action functional and its fixed-endpoint first variations: $O\subseteq \mathbb{R}\times {\mathbb{R}}^n\times {\mathbb{R}}^n$ is open with

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

and $\mathcal{L}:O\to \mathbb{R}$ is of class $C^2$. Recall from that definition that a $C^2$ curve $q:[a,b]\to U$ is the restriction to $[a,b]$ of a $C^2$ map on an open interval, that a variation of $q$ is such a map $\eta :[a,b]\to {\mathbb{R}}^n$ with

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

and that the action of a $C^2$ curve is

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

The fixed-endpoint family. Let $q:[a,b]\to U$ be a $C^2$ curve and let $\eta$ be a variation of $q$. For $\epsilon \in \mathbb{R}$ put

$q_{\epsilon }(t):=q(t)+\epsilon \eta (t)(t\in [a,b])\mathrm{.}$

By clause 1 of The first variation of a time-dependent action on a finite interval: admissibility, the integral formula and continuity of the Euler-Lagrange coefficient there is ${\epsilon }_0>0$ such that $q_{\epsilon }$ is a $C^2$ admissible curve with values in $U$ for every $\epsilon \in (-{\epsilon }_0,{\epsilon }_0)$, so that the action $S(q_{\epsilon })$ is defined for those $\epsilon$.

The second variation. The second variation of $S$ at $q$ in the direction $\eta$ is the second derivative at $\epsilon =0$ of the function $\epsilon \mapsto S(q+\epsilon \eta )$:

${\delta }^{2}S(q)[\eta ,\eta ]:=\frac{d^2}{d{\epsilon }^2}{\mid }_{\epsilon =0}S(q+\epsilon \eta )\mathrm{.}$

This is a well-defined real number under the present hypotheses: the map $\epsilon \mapsto S(q+\epsilon \eta )$ is twice differentiable at $0$, with the explicit integral formula, by The second variation of a time-dependent action: twice differentiability of the fixed-endpoint family and the Hessian integral formula ↗, which depends on this definition only for the notation ${\delta }^{2}S(q)[\eta ,\eta ]$. The value is a single real number determined by the pair $(q,\eta )$ and does not depend on the choice of the $C^2$ extensions of $q$ and $\eta$, since the values of $q_{\epsilon }$ and $q_{\epsilon }^{\mathrm{\prime }}$ on $[a,b]$ are those of any such extension and the derivative on $[a,b]$ is extension-independent (A time-dependent Lagrangian chart, its action functional and its fixed-endpoint first variations).

Scope. The definition introduces the one-parameter family and its second derivative at $\epsilon =0$, nothing else. It asserts nothing about minimising or maximising $S$, nothing about the sign of ${\delta }^{2}S$, and no physical interpretation of $\mathcal{L}$; stationarity of a curve is the separate first-order condition recorded in A time-dependent Lagrangian chart, its action functional and its fixed-endpoint first variations, and no equivalence with the vanishing of an Euler-Lagrange expression is used at this point.

• Normalisation. The symbol ${\delta }^{2}S(q)[\eta ,\eta ]$ denotes the second derivative itself, not one half of it. The companion propositions of this page use exactly this normalisation when they compute second-variation values for the oscillator Lagrangian.

• First and second variation together. The first variation $\delta S(q)[\eta ]=\frac{d}{d\epsilon }{\mid }_{0}S(q+\epsilon \eta )$ of A time-dependent Lagrangian chart, its action functional and its fixed-endpoint first variations and the second variation defined here are the two fixed-endpoint derivatives of the family $q+\epsilon \eta$. For a stationary $q$ the first one vanishes and ${\delta }^{2}S(q)[\eta ,\eta ]$ is the leading second-order quantity computed by The second variation of a time-dependent action: twice differentiability of the fixed-endpoint family and the Hessian integral formula ↗.