On the Minimality of Keplerian Arcs with Fixed Negative Energy

Abstract

We revisit a classical result by Jacobi (J Reine Angew Math 17:68–82, 1837) on the local minimality, as critical points of the corresponding energy functional, of fixed-energy solutions of the Kepler equation joining two distinct points with the same distance from the origin. Our proof relies on the Morse index theorem, together with a characterization of the conjugate points as points of geodesic bifurcation.

Appendix: A Brief Recap on Geodesics and their Morse Index

In this section, we collect some known facts about geodesics and conjugate points. We will work in the simplified setting when the manifold is an open set of the Euclidean space, endowed with a Riemannian metric conformal to the standard one. For a more general treatment, we refer to [16].

1.1 Definition and First Properties

Let M be an open subset of $\mathbb {R}^N$; as such, M has a natural structure of differentiable manifold of dimension N. Given a smooth function $W: M \rightarrow \mathbb {R}$ satisfying $W(x) > 0$ for every $x \in M$, we can endow M with the Riemannian structure given by

$$\begin{aligned} g_W(x)[u,v] = W(x) \langle u,v \rangle , \qquad x \in M, \,u,v \in \mathbb {R}^N, \end{aligned}$$

where $\langle \cdot , \cdot \rangle$ denotes the Euclidean product on $\mathbb {R}^N$. Notice that smooth curves on M are nothing but curves with values in M which are smooth in the usual sense; however, the velocity and the length of a curve are influenced by the function W.

A geodesic on the Riemannian manifold $(M,g_W)$ is a smooth curve $\gamma : I \rightarrow M$ (with $I \subset \mathbb {R}$ an interval) solving the equation

$$\begin{aligned} \frac{d}{ds} \left( W(\gamma ) {{\dot{\gamma }}} \right) = \frac{1}{2} \vert {{\dot{\gamma }}} \vert ^2 \nabla W(\gamma ), \end{aligned}$$

(18)

where $\nabla W$ is the gradient of the function W with respect to the standard Euclidean structure. By the standard theory of ODEs, for any $p \in M$ and $v \in \mathbb {R}^N$, there exists a unique (maximal) geodesic $\gamma _{p,v}$ satisfying $\gamma _{p,v}(0) = p$ and ${{\dot{\gamma }}}_{p,v}(0) = v$.

We collect in the next proposition some simple properties which can be readily obtained from this definition.

Proposition 4.1

The following facts hold true:

• (i) Geodesics have constant velocity: namely, for any geodesic $\gamma$, there exists $c \in \mathbb {R}$ such that $\vert \dot{\gamma }(s) \vert \sqrt{W(\gamma (s))} \equiv c$;

• (ii) For every geodesic $\gamma$ and for every $a,b \in \mathbb {R}$ with $a \ne 0$, the curve ${{\tilde{\gamma }}}(s) = \gamma (as + b)$ is a geodesic, as well;

• (iii) (Rescaling Lemma) For any $\lambda \in \mathbb {R}$ with $\lambda \ne 0$, it holds that $\gamma _{p,\lambda v}(s) = \gamma _{p,v}(\lambda s)$ for any $s \in \mathbb {R}$ such that both sides of the equation are defined.

In view of this proposition, geodesics are often considered as geometric objects, rather than as parameterized curves. In some textbooks, curves which can be parameterized as geodesics are called pre-geodesics.

1.2 The Energy Functional and the Morse Index Theorem

Equation (18) can be meant as the Euler–Lagrange equation of the Lagrangian $L(\gamma ,{{\dot{\gamma }}}) = \vert {{\dot{\gamma }}} \vert ^2 W(\gamma )$; thus, geodesics can be regarded as critical points of the functional $E(\gamma ) = \int L(\gamma ,{{\dot{\gamma }}}) \,ds$. Below, we describe in more details this variational principle, by focusing on the case of geodesics joining two points $p,q \in M$ (a similar discussion can be made when periodic boundary conditions are taken into account, leading to the so-called closed geodesics).

Let us consider the functional

$$\begin{aligned} E(\gamma ) = \int _0^1 \vert {{\dot{\gamma }}}(s)\vert ^2 W(\gamma (s))\,ds, \end{aligned}$$

defined on the Hilbert manifold $\Omega _{p,q}$ of $H^1$-paths $\gamma : [0,1] \rightarrow M$ satisfying $\gamma (0) = p$ and $\gamma (1) = q$. It is standard to verify that E is a smooth functional on $\Omega _{p,q}$ with first Frechet differential given by

$$\begin{aligned} dE(\gamma )[\xi ] = \int _0^1 \left( 2 \langle {{\dot{\gamma }}}(s),{{\dot{\xi }}}(s)\rangle W(\gamma (s)) + \vert {{\dot{\gamma }}}(s) \vert ^2 \langle \nabla W(\gamma (s)),\xi (s)\rangle \right) \,ds, \end{aligned}$$

for any $\xi \in H^1_0([0,1];\mathbb {R}^2)$. A well-known argument based on integration by parts yields:

Proposition 4.2

A curve $\gamma : [0,1] \rightarrow M$ satisfying $\gamma (0) = p$ and $\gamma (1) = q$ is a geodesic if and only if it is a critical point of the functional E.

We are now in a position to define the crucial concept of this section. By Morse index$\mu (\gamma )$ of a geodesic $\gamma$ we mean its Morse index as a critical point of the functional E, that is, the dimension of the maximal subspace $V \subset H^1_0([0,1];\mathbb {R}^2)$ such that $d^2 E(\gamma )[\xi ,\xi ] < 0$ for any $\xi \in V$. Since

$$\begin{aligned} d^2 E(\gamma )[\xi ,\xi ]&= 2\int _0^1 \vert {{\dot{\xi }}}(s)\vert ^2 W(\gamma (s))\,ds + 4 \int _0^1 \langle {{\dot{\gamma }}}(s),{{\dot{\xi }}}(s) \rangle \langle \nabla W(\gamma (s),\xi (s)\rangle \,ds \\&+ \int _0^1 \vert {{\dot{\gamma }}}(s) \vert ^2 \langle \nabla ^2 W(\gamma (s))\xi (s),\xi (s)\rangle \,ds \end{aligned}$$

for any $\xi \in H^1_0([0,1];\mathbb {R}^2)$, functions $\xi$ lying in the kernel of $d^2 E(\gamma )$ are nothing but solutions of the linear equation

$$\begin{aligned}&\frac{d}{ds}\Big ( W(\gamma (s)) {{\dot{\xi }}} + \langle \nabla W(\gamma (s),\xi \rangle {{\dot{\gamma }}}(s) \Big ) \nonumber \\&\quad = \frac{1}{2} \vert \dot{\gamma }(s) \vert ^2 \nabla ^2 W(\gamma (s))\xi + \langle {{\dot{\gamma }}}(s),{{\dot{\xi }}} \rangle \nabla W(\gamma (s)) \end{aligned}$$

(19)

satisfying the boundary condition $\xi (0) = \xi (1) = 0$.

The celebrated Morse index theorem asserts that the Morse index $\mu (\gamma )$ equals the number of some special points along $\gamma$, the so-called conjugate points. Precisely, we say that the point $\gamma (s^*) = p_{s^*}$, with $s^* \in \mathopen {]}0,1\mathclose {]}$, is conjugate to $\gamma (0) = p$ along $\gamma$ if the linear equation (19) admits nontrivial solutions $\xi$ satisfying $\xi (0) = \xi (s^*) = 0$. The multiplicity $m(p_{s^*})$ of the conjugate point is, by definition, the dimension of the space of such solutions. Incidentally, notice that $\gamma (1) = q$ is conjugate to $\gamma (0)$ along $\gamma$ if and only if the kernel of $d^2 E(\gamma )$ is not trivial (that is, if and only if $\gamma$ is a degenerate critical point of E).

With this in mind, the Morse index theorem reads as follows (see for instance [21] and the references therein).

Theorem 4.3

(Morse Index Theorem) Let $\gamma : [0,1] \rightarrow M$ be a non-constant geodesic. Then the set of conjugate points along $\gamma$ is finite and

$$\begin{aligned} \mu (\gamma ) = \sum _{s^* \in \,\mathopen {]}0,1\mathclose {[}} m(p_{s^*}). \end{aligned}$$

Remark 4.4

The choice of parameterizing the geodesic $\gamma$ on the interval [0, 1] is completely conventional: any other interval could be used, since geodesics are preserved by affine reparameterizations (recall Proposition 4.1). Of course, conjugate points and the Morse index do not depend on this choice.

1.3 Bifurcation of Geodesics

A key role in our arguments will be played by the notion of geodesic bifurcation, as introduced in the paper [20].

Definition 4.5

Let $\gamma : [0,1] \rightarrow M$ be a non-constant geodesic. A point $\gamma (s^*)$, with $s^* \in \mathopen {]}0,1\mathclose {[}$, is a bifurcation point along $\gamma$ if there exists a sequence $\{s_n\}_n \subset [0,1]$ with $s_n \rightarrow s^*$ and a sequence of geodesics $\{\gamma _n\}_n$ (defined on [0, 1]), with $\gamma _n \ne \gamma$, such that

• (i) $\gamma _n(0) = \gamma (0)$, for every n,

• (ii) $\gamma _n(s_n) = \gamma (s_n)$, for every n,

• (iii) ${{\dot{\gamma }}}_n(0) \rightarrow {{\dot{\gamma }}}(0)$, for $n \rightarrow +\infty$.

According to the discussion in [20, p. 122] together with [20, Corollary 5.6], the relationship between conjugate points and bifurcation points can be stated as follows.

Theorem 4.6

Let $\gamma : [0,1] \rightarrow M$ be a non-constant geodesic. Then, a point $\gamma (s^*)$, with $s^* \in \mathopen {]}0,1\mathclose {[}$, is a bifurcation point if and only if it is conjugate to $\gamma (0)$ along $\gamma$.