Calculus of Variations, Conjugate Points and Morse Index

Abstract

Let us re-examine the classical conditions of Calculus of Variations geared to obtain a strong minimum (in the topology of the uniform convergence) for an arbitrary Lagrangian function $L(t,q,\dot{q})$, convex in the velocities $\dot{q}$. The result can be obtained in the geometric setup of symplectic geometry using the Hamiltonian description of the problem: the joint use of the theory of Poincaré-Cartan Integral Invariant and Young Inequality rapidly leads to the thesis. The Morse index of stationary curves is discussed in the mechanical case.

Appendix

5.1.1 Strong and Weak Minima

Consider the set (typical¹¹) of the curves in the calculus of variations for the functional of Hamilton’s problem $J[q(\cdot )] =\int _{ 0}^{T}L(q(t),\dot{q}(t),t)\mathit{dt}$:

$$\displaystyle{\varGamma = \left \{q(\cdot ) \in W^{1,2}([0,T]; \mathbb{R}^{n}) = H^{1}([0,T]; \mathbb{R}^{n}): \quad q(0) = q_{ 0},\ q(T) = q_{T}\right \}}$$

$$\displaystyle{\left \{\begin{array}{lcl} \vert \vert q(\cdot )\vert \vert _{H^{1}} &:=&\left ( \frac{1} {T}\int _{0}^{T}\vert q(t)\vert ^{2}\mathit{dt}\right )^{1/2} + \left ( \frac{1} {T}\int _{0}^{T}\vert \dot{q}(t)\vert ^{2}\mathit{dt}\right )^{1/2} \\ & \ & \\ \vert \vert q(\cdot )\vert \vert _{\mathcal{C}^{1}} &:=&\sup _{t\in [0,T]}\vert q(t)\vert +\sup _{t\in [0,T]}\vert \dot{q}(t)\vert,\\ &\ & \\ \vert \vert q(\cdot )\vert \vert _{\mathcal{C}^{0}} &:=&\sup _{t\in [0,T]}\vert q(t)\vert.\\ \end{array} \right.}$$

By definition, the Sobolev space H ¹ is precisely composed by the completion of the $\mathcal{C}^{1}$ curves with respect to the above norm $\vert \vert \cdot \vert \vert _{H^{1}}$; it comes out that they are not necessarily differentiable: for the Sobolev immersion theorem, $H^{1}\hookrightarrow \mathcal{C}^{0}$, one has that the curves are continuous. For every curve $q(\cdot ) \in \varGamma \cap \mathcal{C}^{1}$ one has that

$$\displaystyle{\vert \vert q(\cdot )\vert \vert _{\mathcal{C}^{0}} \leq \vert \vert q(\cdot )\vert \vert _{\mathcal{C}^{1}},}$$

and hence, the $q(\cdot ) \in \varGamma \cap \mathcal{C}^{1}$ such that $\vert \vert q(\cdot )\vert \vert _{\mathcal{C}^{1}} \leq r$ and such that q(⋅ ) belongs to B ₁(r), are all in B ₀(r): this means that the $\mathcal{C}^{1}$ topology is finer (or stronger) than the $\mathcal{C}^{0}$ topology.

Despite this,

• (i) If q ^∗(⋅ ) is such that for some $\varepsilon > 0$:

  $$\displaystyle{J[q^{{\ast}}(\cdot )] \leq J[q(\cdot )],\qquad \forall q(\cdot ) - q^{{\ast}}(\cdot ) \in B_{ 0}(\varepsilon ) (\star )}$$

  we will say that q ^∗(⋅ ) is a strong minimum,

• (ii) If q ^∗(⋅ ) is such that for some $\varepsilon > 0$:

  $$\displaystyle{J[q^{{\ast}}(\cdot )] \leq J[q(\cdot )],\qquad \forall q(\cdot ) - q^{{\ast}}(\cdot ) \in B_{ 1}(\varepsilon ) (\star \star )}$$

  we will say that q ^∗(⋅ ) is a weak minimum.

Observe that, if q ^∗(⋅ ) is a strong minimum, then it also is a weak minimum, but the vice-versa is not true, in fact:

$$\displaystyle{B_{1}(\varepsilon ) \subset B_{0}(\varepsilon ).}$$

5.1.2 Variational Theory in H ¹ = W ^1, 2

Lemma 5.3 (DuBois-Reymond Lemma)

Let $\varphi \in L^{2}([a,b], \mathbb{R}^{n})$ and suppose that ¹²

$$\displaystyle{\int _{a}^{b}\varphi (t)\dot{h}(t)\mathit{dt} = 0\quad \forall h \in H_{ 0}^{1}([a,b], \mathbb{R}^{n})}$$

Then

$$\displaystyle{\varphi (t) \equiv \text{const.}\ \ a.e.}$$

Proof

Define $\mu:= \frac{1} {b-a}\int _{a}^{b}\varphi (t)\mathit{dt}$, it is meaningful since $\varphi \in L^{2} \subset L^{1}$. Let $h(t):=\int _{ a}^{t}(\varphi (s)-\mu )ds$, we see that this choice of h is in H ₀ ¹: h(a) = 0 = h(b), $\dot{h} =\varphi -\mu \in L^{2}$ and $\vert \vert h\vert \vert _{L^{2}}^{2} =\int _{ a}^{b}(\int _{a}^{t}(\varphi (s)-\mu )ds)\cdot (\int _{a}^{t}(\varphi (s)-\mu )\mathit{ds})\mathit{dt} \leq (b-a)\sup _{t\in [a,b]}\vert \int _{a}^{t}(\varphi (s)-\mu )ds\vert ^{2} < +\infty$ because $\varphi \in L^{1}$.

$$\displaystyle{0 =\int _{ a}^{b}\varphi (t)\dot{h}(t)\mathit{dt} =\int _{ a}^{b}\varphi (t)(\varphi (t)-\mu )\mathit{dt} =\int _{ a}^{b}(\varphi (t)-\mu )^{2}\mathit{dt},}$$

since $\varphi -\mu \in L^{2}$, then: $\varphi (t) \equiv \mu \ \ a.e.$ □ 

Lemma 5.4 (First Variation Lemma)

Suppose

$$\displaystyle{\mathit{dJ}(q)h =\int _{ a}^{b}[L_{ q}(q(t),\dot{q}(t))h(t) + L_{\dot{q}}(q(t),\dot{q}(t))\dot{h}(t)]\mathit{dt} = 0}$$

$\forall h \in H_{0}^{1}([a,b], \mathbb{R}^{n})$ and

$$\displaystyle{q \in \varGamma _{a,b}^{q_{0},q_{1} }:= \left \{q(\cdot ) \in H^{1}([a,b], \mathbb{R}^{n}):\ q(a) = q_{ 0},\ q(b) = q_{1}\right \}}$$

If $L: \mathbb{R}^{n} \times \mathbb{R}^{n} \rightarrow \mathbb{R}$ is such that the above Gateaux differential there exists and

• (i) $P(t):=\int _{ a}^{t}L_{q}(q(s),\dot{q}(s))ds \in L^{2}$ ,

• (ii) $L_{\dot{q}}(q(t),\dot{q}(t)) \in L^{2}$ ,

then:

$$\displaystyle{ P - L_{\dot{q}} = \text{const.}\quad a.e. }$$

(5.55)

Remark 5.2

Note that the above relation is precisely the integral version of the Euler-Lagrange equations.

Proof

$0 =\int _{ a}^{b}[L_{q}(q(t),\dot{q}(t))h(t)+L_{\dot{q}}(q(t),\dot{q}(t))\dot{h}(t)]\mathit{dt} = P(t)h(t)\vert _{a}^{b}-\int _{a}^{b}(P-L_{\dot{q}})\dot{h}\mathit{dt}$, the above Lemma leads us to the thesis. □ 

The mechanical case Let $L = \frac{1} {2}m\vert \dot{q}\vert ^{2} - V (q)$, $L_{\dot{q}} = m\dot{q}$ and if q ∈ H ¹ then $m\dot{q} \in L^{2}$. If $V: \mathbb{R}^{n} \rightarrow \mathbb{R}$ is $\mathcal{C}^{1}$, then

$$\displaystyle{P(t) = -\int _{a}^{t}\nabla U(q(s))\mathit{ds},}$$

is a continuous function on the compact set [a, b], then P ∈ L ².

5.1.3 Local (for Short Time Interval) Minimum in the Calculus of Variations

First, we recall a general fact on the role of the coercivity in calculus of variations

Proposition 5.1 (Local coercivity induces local minimum)

• (i) Let $(X,\|\cdot \|)$ be a normed affine space of curves between two fixed configurations, $X = X_{t_{0},t_{1}}^{q_{0},q_{1}} = q + X_{t_{ 0},t_{1}}^{0,0}$ for a fixed q ∈ X,

• (ii) Let $J: (X,\|\cdot \|) \rightarrow \mathbb{R}$ be a real-valued functional of class $\mathcal{C}^{2}(X; \mathbb{R})$ ,

• (iii) Let q ∈ X be a stationary curve ¹³ for J:

  $$\displaystyle{J'[q]h = 0,\ \forall h \in X_{t_{0},t_{1}}^{0,0},}$$

• (iv) Let suppose there exists a positive constant α > 0 such that (local coercivity):

  $$\displaystyle{J''[q](h,h) \geq \alpha \| h\|^{2}\quad \text{for any}\ h \in X_{ t_{0},t_{1}}^{0,0}}$$

Then q is a strict local minimum for J in the induced topology from $\|\cdot \|$ .

Proof

From Taylor expansion,

$$\displaystyle{J[q + h] - J[q] = \frac{1} {2!}J''[q](h,h) +\| h\|^{2}g(q,h),}$$

where g(q, h) is an infinitesimal function for $\|h\| \rightarrow 0$. Thus,

$$\displaystyle{J[q + h] - J[q] \geq \big (\frac{1} {2}\alpha + g(q,h)\big)\|h\|^{2}}$$

and for small h (in the choosen norm $\|\cdot \|$) and different from zero (i.e., the curve h(t) ≡ 0), surely $(\frac{1} {2}\alpha + g(q,h)) > 0$ and hence

$$\displaystyle{J[q + h] > J[q].}$$

    □ 

Recently, it has been proved that the functional J is $\mathcal{C}^{2}$ in H ¹ if and only if the restriction $\dot{q}\mapsto L(t,q,\dot{q})$ is polynomial at most of degree 2,¹⁴ just it precisely happens in the mechanical and geodesic cases!

For general Lagrangians with $\dot{q}\mapsto L(t,q,\dot{q})$ convex and quadratically growing at infinity we can simply say that J is $\mathcal{C}^{1,1}$ in H ¹, that is $\mathcal{C}^{1}$ with first derivative Lipschitz.¹⁵

We are ready to the following

Proposition 5.2 (Local minimum in H ¹, the mechanical case)

Let $J(q) =\int _{ t_{0}}^{t_{1}}[\frac{1} {2}m\vert \dot{q}(t)\vert ^{2} - V (q(t))]\mathit{dt}$ and ¹⁶

$$\displaystyle{J'[q]h = 0\quad \forall h \in H_{0}^{1}([t_{ 0},t_{1}], \mathbb{R}^{n})}$$

Under the hypothesis that the interval [t ₀ ,t ₁ ] is small enough,

$$\displaystyle{t_{1} - t_{0} < \sqrt{2m \left (\max _{t\in [t_{0 },t_{1 } ] } \left (\mathrm{max }\vert \mathrm{spec }\nabla ^{2 } V \left (q(t) \right ) \vert \right ) \right ) ^{-1}},}$$

there exists a suitable constant α > 0 such that

$$\displaystyle{J''[q](h,h) = \frac{d^{2}} {d\lambda ^{2}}J[q +\lambda h]\vert _{\lambda =0} \geq \alpha \| h\|_{H^{1}}^{2},\qquad \forall h \in H_{ 0}^{1},}$$

so that q is a local minimum for J in H ¹ .

Proof

$$\displaystyle{\begin{array}{rcl} \frac{d^{2}} {d\lambda ^{2}} J[q +\lambda h]\vert _{\lambda =0} & =&\frac{d} {d\lambda }\{\frac{d} {d\lambda }J[q +\lambda h]\}\vert _{\lambda =0}\\ \\ & =&\frac{d} {d\lambda }\{\int _{t_{0}}^{t_{1}}[m(\dot{q} +\lambda \dot{ h}) \cdot \dot{ h} -\nabla V (q +\lambda h)] \cdot h\,\mathit{dt}\}\vert _{\lambda =0} \\ \\ & =&\int _{t_{0}}^{t_{1}}m\vert \dot{h}\vert ^{2}\mathit{dt} -\int _{t_{ 0}}^{t_{1}}\nabla _{\mathit{ ij}}^{2}V (q)h^{i}h^{j}\mathit{dt}\\ \\ & \geq &\int _{t_{0}}^{t_{1}}m\vert \dot{h}(t)\vert ^{2}\mathit{dt} -\int _{t_{ 0}}^{t_{1}}\mathrm{max}\vert \mathrm{spec}\nabla ^{2}V (q(t))\vert \cdot \vert h(t)\vert ^{2}\mathit{dt}, \end{array} }$$

$$\displaystyle{ \begin{array}{rcl} \frac{d^{2}} {d\lambda ^{2}} J[q +\lambda h]\vert _{\lambda =0} & \geq &\int _{t_{0}}^{t_{1}}m\vert \dot{h}\vert ^{2}\mathit{dt} - c\int _{t_{ 0}}^{t_{1}}\vert h\vert ^{2}\mathit{dt},\end{array} }$$

(5.56)

where $c:=\max _{t\in [t_{0},t_{1}]}(\mathrm{max}\vert \mathrm{spec}\nabla ^{2}V (q(t))\vert )$. In order to estimate $\int _{t_{0}}^{t_{1}}\vert \dot{h}\vert ^{2}\mathit{dt}$ with $\int _{t_{0}}^{t_{1}}\vert h\vert ^{2}\mathit{dt}$ we recall the Cauchy-Schwarz inequality:

$$\displaystyle{\left (\int _{a}^{b}\mathit{fgdt}\right )^{2} \leq \int _{ a}^{b}f^{2}\mathit{dt}\int _{ a}^{b}g^{2}\mathit{dt}.}$$

Using the fact that h(t ₀) = 0, we have that

$$\displaystyle{\int _{t_{0}}^{t_{1} }\vert h(t)\vert ^{2}\mathit{dt} =\int _{ t_{0}}^{t_{1} }\sum _{i=1}^{3}(h^{i}(t))^{2}\mathit{dt} =\int _{ t_{0}}^{t_{1} }\sum _{i=1}^{3}\left (\int _{ t_{0}}^{t}\dot{h}^{i}(\tau )d\tau \right )^{2}\mathit{dt}.}$$

We estimate this last term by means of Cauchy-Schwarz,

$$\displaystyle{\sum _{i=1}^{n}\left (\int _{ t_{0}}^{t}1 \cdot \dot{ h}^{i}(\tau )d\tau \right )^{2} \leq \sum _{ i=1}^{n}\int _{ t_{0}}^{t}d\tau \int _{ t_{0}}^{t}\left (\dot{h}^{i}(\tau )\right )^{2}d\tau = (t - t_{ 0})\int _{t_{0}}^{t}\vert \dot{h}(\tau )\vert ^{2}d\tau.}$$

Hence

$$\displaystyle\begin{array}{rcl} & \int _{t_{0}}^{t_{1}}\vert h(t)\vert ^{2}\mathit{dt} \leq \int _{t_{ 0}}^{t_{1}}(t - t_{ 0})\left (\int _{t_{0}}^{t}\vert \dot{h}(\tau )\vert ^{2}d\tau \right )\mathit{dt} & {}\\ & \leq \int _{t_{0}}^{t_{1}}(t - t_{0})\mathit{dt}\int _{t_{ 0}}^{t_{1}}\vert \dot{h}(\tau )\vert ^{2}d\tau = \frac{1} {2}(t_{1} - t_{0})^{2}\int _{ t_{0}}^{t_{1}}\vert \dot{h}\vert ^{2}\mathit{dt}.& {}\\ \end{array}$$

Then we have that

$$\displaystyle{ \|h\|_{L^{2}}^{2} \leq \frac{1} {2}(t_{1} - t_{0})^{2}\|\dot{h}\|_{ L^{2}}^{2}. }$$

(5.57)

We obtain for (5.56):

$$\displaystyle\begin{array}{rcl} & J''[q](h,h) = \frac{d^{2}} {d\lambda ^{2}} J[q +\lambda h]\vert _{\lambda =0} \geq m\|\dot{h}\|_{L^{2}}^{2} - c\|h\|_{L^{2}}^{2}& {}\\ & \geq \left ( \frac{2m} {(t_{1}-t_{0})^{2}} - c\right )\|h\|_{L^{2}}^{2}. & {}\\ \end{array}$$

We see that $\alpha _{1}:= \frac{2m} {(t_{1}-t_{0})^{2}} - c > 0$ if and only if $t_{1} - t_{0} < \sqrt{\frac{2m} {c}}$. On the other hand, still with (5.57), the following estimate is holding

$$\displaystyle{J''[q](h,h) \geq m\|\dot{h}\|_{L^{2}}^{2} - c\|h\|_{ L^{2}}^{2} \geq \left (m - c\frac{1} {2}(t_{1} - t_{0})^{2}\right )\|\dot{h}\|_{ L^{2}}^{2}}$$

where $\alpha _{2}:= m -\frac{1} {2}c(t_{1} - t_{0})^{2} > 0$ if and only if (as above) $t_{1} - t_{0} < \sqrt{\frac{2m} {c}}$. At the end, the obtain an estimate¹⁷ in the Sobolev H ¹ norm:

$$\displaystyle{J''[q](h,h) \geq \frac{1} {2}\mathrm{min}\{\alpha _{1},\alpha _{2}\}(\|h\|_{L^{2}}^{2} +\|\dot{ h}\|_{ L^{2}}^{2}) = \frac{1} {2}\mathrm{min}\{\alpha _{1},\alpha _{2}\}\|h\|_{H^{1}}^{2} =\alpha \| h\|_{ H^{1}}^{2}}$$

 □ 

Remark 5.3 (On the existence)

Our effort here has been to highlight the quality of the critical curves using terms involving the second derivatives of the functional; if we are interested simply in search of the minimum of the functional, the only condition of uniform convexity of L with respect to $\dot{q}$, without restriction on the size of the time interval, would be sufficient to ensure the existence of the minimum in H ¹: this is a Tonelli like problem, see e.g. [60], Th. 5, p. 453. An alternative way to capture the existence could be to observe that J, if sup | V ″(q) |  < +∞, can be finitely reduced to a Generating Function Quadratic at Infinity (GFQI, see Chap. 7) which is Palais-Smale, see [32].

5.1.4 A Regularity Result

By means of the Legendre diffeomorphism $\mathcal{T}$ – see (5.4) – we are able to establish the following main connection between the regularity of the solutions q(⋅ ) and the regularity of L: The H ¹ critical curves q(⋅ ) have the same regularity of the Lagrangian L.

Proposition 5.3

Let q(⋅) ∈ H ¹ be a curve satisfying the integral version  (5.55) of the Euler-Lagrange equations. Suppose that L is a C ^k (k ≥ 2) Tonelli Lagrangian, then q(⋅) is C ^k .

Proof

By Sobolev embedding theorem, we know that q ∈ C ⁰. We combine the integral form of the Euler-Lagrange equations, which are satisfied by q(⋅ ), with the conjugate momenta. We obtain:

$$\displaystyle{ p(t) = \frac{\partial L} {\partial \dot{q}} (t,q(t),\dot{q}(t)) =\int _{ 0}^{t} \frac{\partial } {\partial q}L(\tau,q(\tau ),\dot{q}(\tau ))d\tau + K }$$

(5.58)

So p(⋅ ), being equal to an integral function, is continuous. By the $\mathcal{C}^{k-1}$ diffeomorphism $\mathcal{T}$, which is in particular a homeomorphism, we obtain that

$$\displaystyle{ (t,q(t),\dot{q}(t)) = \mathcal{T}^{-1}(t,q(t),p(t)) }$$

(5.59)

is continuous, and in particular $\dot{q}(\cdot )$ is continuous, so $q(\cdot ) \in \mathcal{C}^{1}$. The same technique can be repeated as many times as the order of regularity of $\mathcal{T}$, until we obtain the thesis: inserting now $q(\cdot ) \in \mathcal{C}^{1}$ into (5.58) we see that $p(\cdot ) \in \mathcal{C}^{1}$, and, by (5.59), $\dot{q}(\cdot ) \in \mathcal{C}^{\mathrm{min}(1,k-1)}$ or $q(\cdot ) \in \mathcal{C}^{\mathrm{min}(2,k)}$. If k > 2, at the next step: $q(\cdot ) \in \mathcal{C}^{\mathrm{min}(3,k)}$. All this, until $q(\cdot ) \in \mathcal{C}^{k}$. □ 

5.1.5 On the Linearized Flow

Working in $\mathbb{R}^{n}$ for the sake of simplicity, we denote by

$$\displaystyle{ \tilde{x}(t,x) =\varPhi _{ X}^{t}(x) }$$

(5.60)

the flow relative to the vector field X:

$$\displaystyle{ \mathbb{R}^{n} \ni x\mapsto (x,X(x)) \in T\mathbb{R}^{n}, }$$

(5.61)

$$\displaystyle{ \left \{\begin{array}{rcl} \frac{d} {\mathit{dt}}\tilde{x}(t,x)& =&X(\tilde{x}(t,x)),\\ \\ \tilde{x}(0,x)& =&x.\end{array} \right. }$$

(5.62)

Let consider the linearized differential equation of (5.62) around the curve solution (5.60):

$$\displaystyle{ \left \{\begin{array}{rcl} \frac{d} {\mathit{dt}}\tilde{h}(t,h_{0})& =&DX(\tilde{x}(t,x))\tilde{h}(t,h_{0}),\\ \\ \tilde{h}(0,h_{0})& =&h_{0}.\end{array} \right. }$$

(5.63)

Proposition 5.4

The flow of the linearized problem is the linearized of the flow of the non linear problem:

$$\displaystyle{ \tilde{h}(t,h_{0}) = D\varPhi _{X}^{t}(x)h_{ 0}\ \ \ \Big(= \frac{\partial \tilde{x}} {\partial x}(t,x)h_{0}\Big) }$$

(5.64)

Proof

$$\displaystyle\begin{array}{rcl} & \frac{d} {\mathit{dt}}\tilde{h}(t,h_{0}) = \frac{\partial ^{2}\tilde{x}} {\partial t\,\partial x}(t,x)h_{0} = \frac{\partial ^{2}\tilde{x}} {\partial x\,\partial t}(t,x)h_{0} = \frac{\partial } {\partial x}X(\tilde{x}(t,x))h_{0},& {}\\ & = DX(\tilde{x}(t,x))\frac{\partial \tilde{x}} {\partial x}(t,x)h_{0} = DX(\tilde{x}(t,x))\tilde{h}(t,h_{0}). & {}\\ \end{array}$$

 □