Jacobi-Maupertuis Metric Research Articles

Existence of partially hyperbolic motions in the 𝑁-body problem

In the context of the Newtonian N N -body problem, we prove the existence of a partially hyperbolic motion with prescribed positive energy and any initial collisionless configuration. Moreover, it is a free time minimizer of the respective supercritical Newtonian action or equivalently a geodesic ray for the respective Jacobi-Maupertuis metric.

Geodesic Rays of the N-Body Problem

For the Newtonian N-body problem, we study the Jacobi-Maupertuis metric of the nonnegative energy levels. We show that the geodesic rays are expansive, that is to say, all the distances between the bodies must be divergent functions. More precisely, we prove that the evolution of such motions asymptotically decomposes into free particles and subsystems in completely parabolic expansion. The theorem applies in particular to the maximal characteristic curves of any given global viscosity solution of the stationary Hamilton-Jacobi equation $H(x,d_xu) = h$.

The three-body problem as a geodesic billiard map with singularities

This paper deals with the planar three-body problem with equal masses and moving under 1/r2 potential. The complete reduction of this problem is done by using its finite group of symmetries and the Jacobi-Maupertuis metric. We show that it is the unfolding of a flow which is topologically conjugate to a geodesic billiard problem with the Jacobi-Maupertuis metric. Finally there is a numerical exploration of the corresponding billiard map.

Minimal Geodesics of the Isosceles Three Body Problem

The isosceles three-body problem with nonnegative energy is studied from a variational point of view based on the Jacobi–Maupertuis metric. The solutions are represented by geodesics in the two-dimensional configuration space. Since the metric is singular at collisions, an approach based on the theory of length spaces is used. This provides an alternative to the more familiar approach based on the principle of least action. The emphasis is on the existence and properties of minimal geodesics, that is, shortest curves connecting two points in configuration space. For any two points, even singular points, a minimal geodesic exists and is nonsingular away from the endpoints. For the zero energy case, it is possible to use knowledge of the behavior of the flow on the collision manifold to see that certain solutions must be minimal geodesics. In particular, the geodesic corresponding to the collinear homothetic solution turns out to be a minimal for certain mass ratios.

Minimizers for the Kepler Problem

We characterize the minimizing geodesics for the Kepler problem endowed with the Jacobi-Maupertuis metric. We focus on the positive energy case, but do all energies. The more complicated negative energy case was solved in Jacobi (Crelles J 17:68–82, 1837. https://doi.org/10.1515/crll.1837.17.68), with his work translated and completed by Todhunter (Researches in the Calculus of Variations, Principally on the Theory of Discontinuous Solutions. Macmillan and Co., Cambridge, 1871), and later summarized in Wintner’s book. Our discussion of these old results includes a new proof for the positive energy case and perspectives coming from metric and differential geometry. For the negative energy result we need Lambert’s theorem which we discuss.

Classical three rotor problem: Periodic solutions, stability and chaos.

This paper concerns the classical dynamics of three coupled rotors: equal masses moving on a circle subject to attractive cosine interparticle potentials. This system arises as the classical limit of a model of coupled Josephson junctions. In appropriate units, the non-negative energy E of the relative motion is the only free parameter. We find families of periodic solutions: pendulum and isosceles solutions at all energies and choreographies up to moderate energies. The model displays order-chaos-order behavior: it is integrable at zero and infinitely high energies but displays a fairly sharp transition from regular to chaotic behavior as E is increased beyond Ec≈4 and a more gradual return to regularity. The transition to chaos is manifested in a dramatic rise of the fraction of the area of the Hill region of Poincaré surfaces occupied by chaotic sections and also in the spontaneous breaking of discrete symmetries of Poincaré sections present at lower energies. Interestingly, the above pendulum solutions alternate between being stable and unstable, with the transition energies cascading geometrically from either sides at E=4. The transition to chaos is also reflected in the curvature of the Jacobi-Maupertuis metric that ceases to be everywhere positive when E exceeds four. Examination of Poincaré sections also indicates global chaos in a band of energies (5.33≲E≲5.6) slightly above this transition.

On the Sectional Curvature Along Central Configurations

In this paper we characterize planar central configurations in terms of a sectional curvature value of the Jacobi-Maupertuis metric. This characterization works for the $N$-body problem with general masses and any $1/r^{\alpha}$ potential with $\alpha> 0$. We also observe dynamical consequences of these curvature values for relative equilibrium solutions. These curvature methods work well for strong forces ($\alpha \ge 2$).

Embedding the Kepler Problem as a Surface of Revolution

Solutions of the planar Kepler problem with fixed energy h determine geodesics of the corresponding Jacobi–Maupertuis metric. This is a Riemannian metric on ℝ2 if h ⩾ 0 or on a disk D ⊂ ℝ2 if h < 0. The metric is singular at the origin (the collision singularity) and also on the boundary of the disk when h < 0. The Kepler problem and the corresponding metric are invariant under rotations of the plane and it is natural to wonder whether the metric can be realized as a surface of revolution in ℝ3 or some other simple space. In this note, we use elementary methods to study the geometry of the Kepler metric and the embedding problem. Embeddings of the metrics with h ⩾ 0 as surfaces of revolution in ℝ3 are constructed explicitly but no such embedding exists for h < 0 due to a problem near the boundary of the disk. We prove a theorem showing that the same problem occurs for every analytic central force potential. Returning to the Kepler metric, we rule out embeddings in the three-sphere or hyperbolic space, but succeed in constructing an embedding in four-dimensional Minkowski spacetime. Indeed, there are many such embeddings.

The three-body problem and equivariant Riemannian geometry

We study the planar three-body problem with 1/r2 potential using the Jacobi-Maupertuis metric, making appropriate reductions by Riemannian submersions. We give a different proof of the Gaussian curvature’s sign and the completeness of the space reported by Montgomery [Ergodic Theory Dyn. Syst. 25, 921–947 (2005)]. Moreover, we characterize the geodesics contained in great circles.

Jacobi-Maupertuis-Eisenhart metric and geodesic flows

The Jacobi metric derived from the line element by one of the authors is shown to reduce to the standard formulation in the non-relativistic approximation. We obtain the Jacobi metric for various stationary metrics. Finally, the Jacobi-Maupertuis metric is formulated for time-dependent metrics by including the Eisenhart-Duval lift, known as the Jacobi-Eisenhart metric.

Curvature and geodesic instabilities in a geometrical approach to the planar three-body problem

The Maupertuis principle allows us to regard classical trajectories as reparametrized geodesics of the Jacobi-Maupertuis (JM) metric on configuration space. We study this geodesic reformulation of the planar three-body problem with both Newtonian and attractive inverse-square potentials. The associated JM metrics possess translation and rotation isometries in addition to scaling isometries for the inverse-square potential with zero energy E. The geodesic flow on the full configuration space ℂ3 (with collision points excluded) leads to corresponding flows on its Riemannian quotients: the center of mass configuration space ℂ2 and shape space ℝ3 (as well as 𝕊3 and the shape sphere 𝕊2 for the inverse-square potential when E = 0). The corresponding Riemannian submersions are described explicitly in “Hopf” coordinates which are particularly adapted to the isometries. For equal masses subject to inverse-square potentials, Montgomery shows that the zero-energy “pair of pants” JM metric on the shape sphere is geodesically complete and has negative gaussian curvature except at Lagrange points. We extend this to a proof of boundedness and strict negativity of scalar curvatures everywhere on ℂ2, ℝ3, and 𝕊3 with collision points removed. Sectional curvatures are also found to be largely negative, indicating widespread geodesic instabilities. We obtain asymptotic metrics near collisions, show that scalar curvatures have finite limits, and observe that the geodesic reformulation “regularizes” pairwise and triple collisions on ℂ2 and its quotients for arbitrary masses and allowed energies. For the Newtonian potential with equal masses and zero energy, we find that the scalar curvature on ℂ2 is strictly negative though it could have either sign on ℝ3. However, unlike for the inverse-square potential, geodesics can encounter curvature singularities at collisions in finite geodesic time.

Fitting hyperbolic pants to a three-body problem

Consider the three-body problem with an attractive 1/r2 potential. Modulo symmetries, the dynamics of the bounded zero-angular-momentum solutions is equivalent to a geodesic flow on the thrice-punctured sphere, or ‘pair of pants’. The sphere is the shape sphere. The punctures are the binary collisions. The metric generating the geodesics is the Jacobi–Maupertuis metric. It is complete, has infinite area, and its ends, the neighborhoods of the punctures, are asymptotically cylindrical. Our main result is that when the three masses are equal then the metric has negative curvature everywhere except at two points (the Lagrange points). A corollary of this negativity is the uniqueness of the 1/r2 figure-eight, a complete symbolic dynamics for encoding the collision-free solutions, and the fact that collision solutions are dense within the bound solutions.