At least n brake orbits for classical-type Hamiltonians: the generalized Seifert conjecture for disk-shaped potential wells

Seifert conjectured in 1948 that a natural Hamiltonian system (Riemannian kinetic energy plus potential) confined to a compact potential well homeomorphic to the n-dimensional ball has at least n geometrically distinct brake orbits — periodic motions oscillating between two zero-momentum points on the boundary. Corona and Giannoni (Adv. Nonlinear Anal. 2022) reduced the same multiplicity question for classical-type Hamiltonians (even and strictly convex in the momenta, i.e. Finsler-kinetic systems) to counting orthogonal geodesic chords in a strongly concave, reversible Finsler disk built from the Jacobi metric, and explicitly left the multiplicity theorem as future work. The open problem: prove, or disprove by explicit counterexample, that every classical-type Hamiltonian with a compact disk-type potential well in dimension n possesses at least n geometrically distinct brake orbits.

Background

An autonomous Hamiltonian system on phase space $\mathbb{R}^{2n}$ with canonical coordinates $(q,p)$ is governed by $\dot q=\partial H/\partial p$, $\dot p=-\partial H/\partial q$, and conserves the energy $H$. Following the source, $H$ is of classical type if for each fixed $q$ the map $p\mapsto H(q,p)$ is even, and $\partial^2H/\partial p^2(q,p)$ is strictly positive definite for all $p$, namely there exists a continuous function $\nu:\mathbb{R}^n\to\mathbb{R}$ with $\nu(q)>0$ such that $\frac{\partial^2H}{\partial p^2}(q,p)[\xi,\xi]\ge\nu(q)\lVert\xi\rVert^2$ for all $p,\xi\in\mathbb{R}^n$. This class contains the natural Hamiltonians $H(q,p)=\tfrac12\langle A(q)p,p\rangle+V(q)$ with $A(q)$ positive definite — mechanical systems with Riemannian kinetic energy in the potential $V$ — and, more generally, systems with non-quadratic (Finsler-type) kinetic energies. Writing $V(q)=H(q,0)$, a potential well at energy $E$ is a bounded open set $D$ with $V<E$ on $D$ and $V=E$, $\nabla V\neq0$ on the $C^2$ boundary $\partial D$; following the source it is called disk-type when $\overline D$ is homeomorphic to the closed $n$-ball. A brake orbit of energy $E$ is a periodic solution of energy $E$ lying in $D$ except for two turning points on $\partial D$ where the momentum vanishes; evenness of $H$ in $p$ makes the reflected arc close up, so the orbit oscillates along a chord of the well. Two brake orbits are geometrically distinct if their traces in $D$ differ as point sets. In 1948 H. Seifert proved that a natural system in a disk-type well has at least one brake orbit and proposed, in a footnote, that such a well in dimension $n$ should carry at least $n$ geometrically distinct brake orbits — Seifert's conjecture. Existence of one brake orbit was extended to convex Hamiltonians by Weinstein (1978); a substantial literature also treats a different convex-hypersurface formulation (brake orbits on strictly convex energy hypersurfaces, e.g. Liu–Zhang 2014), which neither implies nor is implied by the potential-well disk problem. The most successful attack on the potential-well problem is geometric. By the Maupertuis principle, trajectories of energy $E$ are reparametrized geodesics of a Jacobi metric on $D$: for natural systems a Riemannian metric conformal to the kinetic-energy metric, degenerate at $\partial D$; for classical-type systems a Finsler metric, i.e. a norm $F(q,v)$ on each tangent space, smooth off the zero section with positive-definite fundamental tensor $g_v=\tfrac12\partial^2F^2/\partial v^2$. Because $H(q,\cdot)$ is even, this Jacobi–Finsler metric is reversible ($F(q,v)=F(q,-v)$). A curve in $\overline\Omega$ that is a geodesic with both endpoints on $\partial\Omega$ and with initial and final velocities orthogonal to the boundary is an orthogonal geodesic chord (OGC); a domain is strongly concave with respect to $F$ if every geodesic tangent to $\partial\Omega$ bends strictly into $\Omega$, the hypothesis that makes the action functional on curves with endpoints on $\partial\Omega$ variationally tractable. The school of Giambò–Giannoni–Piccione developed this program in the Riemannian case: existence of one OGC in concave $n$-disks (2010), at least two brake orbits in $m$-dimensional disk wells (2015), and — in the preprint arXiv:2002.09687 (v3 2022), using nonsmooth critical point theory (variational analysis of functionals that fail to be differentiable at some points) directly on the singular Jacobi metric — a claimed proof of the full Seifert conjecture. The source, Corona–Giannoni (2022), extends the reduction to classical-type Hamiltonians: its Theorem 1.9 constructs from any compact disk-type well $D$ a smaller domain $\Omega\subset D$ with $C^2$ boundary, homeomorphic to $D$ and strongly concave for an explicit Jacobi–Finsler metric $F$, such that brake orbits in $D$ correspond bijectively to OGCs in $(\Omega,F)$, each OGC extending uniquely to a brake orbit. The paper states these results will be exploited in the future to generalize Seifert's conjecture to classical-type Hamiltonians; the missing ingredient is a Finsler multiplicity theorem. The closest partial result is Corona (2021, Discrete Contin. Dyn. Syst.): in a compact Finsler $N$-disk, either some orthogonal-tangent Finsler geodesic chord exists (a chord leaving the boundary orthogonally and returning tangentially) or there are at least $N$ geometrically distinct OGCs when $F$ is reversible (at least two otherwise). Since the Jacobi–Finsler metric of a classical-type Hamiltonian is reversible, what remains open is precisely to exclude the tangent-chord alternative — or to bypass it, as the nonsmooth Riemannian argument attempts — and thereby obtain $n$ distinct chords in the strongly concave Finsler disks produced by the reduction. A tool toward this program — Lusternik–Schnirelmann relative category, a homotopy invariant of a pair (space, subspace) that lower-bounds the number of critical points of a functional — was developed by Corona–Giambò–Giannoni–Piccione (Topol. Methods Nonlinear Anal. 2023).

Problem Statement

Prove, or disprove by an explicit counterexample, the generalized Seifert conjecture in the form set up by Corona–Giannoni (2022). Let $n\ge2$ and let $H$ be a $C^2$ Hamiltonian of classical type on the phase space $\mathbb{R}^{2n}$ with canonical coordinates $(q,p)\in\mathbb{R}^n\times\mathbb{R}^n$, i.e. for each $q$ the map $p\mapsto H(q,p)$ is even and there exists a continuous function $\nu:\mathbb{R}^n\to\mathbb{R}$ with $\nu(q)>0$ such that $\frac{\partial^2H}{\partial p^2}(q,p)[\xi,\xi]\ge\nu(q)\lVert\xi\rVert^2$ for all $p,\xi\in\mathbb{R}^n$ (Definition 1.1 of the source). Let $D\subset\mathbb{R}^n$ be a compact disk-type potential well of $H$ at energy $E$: writing $V(q)=H(q,0)$, $D$ is a bounded open set with $V<E$ on $D$, with $V=E$ and $\nabla V\neq0$ on the boundary $\partial D$ of class $C^2$, and with closure $\overline D$ homeomorphic to the closed $n$-ball. Must the Hamiltonian flow of $H$ at energy $E$ possess at least $n$ geometrically distinct brake orbits in $D$ (brake orbits whose traces in $D$ are different point sets)? By Theorem 1.9 of the source, an equivalent geometric formulation is: every strongly concave compact Finsler disk $(\Omega,F)$ obtained as the Jacobi–Finsler reduction of such data — with $F$ reversible — carries at least $n$ geometrically distinct orthogonal geodesic chords. An affirmative answer must hold for all classical-type Hamiltonians and disk-type wells in all dimensions $n\ge2$, without symmetry, nondegeneracy, or analyticity assumptions beyond those in the definitions above; a disproof consists of an explicit Hamiltonian and well of this class together with a rigorous proof that fewer than $n$ geometrically distinct brake orbits of energy $E$ exist. The equivalence between the Hamiltonian formulation and the orthogonal-geodesic-chord formulation is already established by the source and need not be reproved.

References

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