Homothetic Solution Research Articles

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.

Deforming a hypersurface by principal radii of curvature and support function

We study the motion of smooth, closed, strictly convex hypersurfaces in $$\mathbb {R}^{n+1}$$ expanding in the direction of their normal vector field with speed depending on the kth elementary symmetric polynomial of the principal radii of curvature $$\sigma _k$$ and support function h. A homothetic self-similar solution to the flow that we will consider in this paper, if exists, is a solution of the well-known $$L_p$$ -Christoffel–Minkowski problem $$\varphi h^{1-p}\sigma _k=c$$ . Here $$\varphi $$ is a preassigned positive smooth function defined on the unit sphere, and c is a positive constant. For $$1\le k\le n-1, p\ge k+1$$ , assuming the spherical hessian of $$\varphi ^{\frac{1}{p+k-1}}$$ is positive definite, we prove the $$C^{\infty }$$ convergence of the normalized flow to a homothetic self-similar solution. One of the highlights of our arguments is that we do not need the constant rank theorem/deformation lemma of Guan and Ma (Invent Math 151:553–577, 2003) and thus we give a partial answer to a question raised in Guan and Xia (Calc Var Part Differ Equ 57:69, 2018. https://doi.org/10.1007/s00526-018-1341-y . Moreover, for $$k=n, p\ge n+1$$ , we prove the $$C^{\infty }$$ convergence of the normalized flow to a homothetic self-similar solution without imposing any condition on $$\varphi .$$ In the final section of the paper, for $$1\le k<n$$ , we will give an example that spherical hessian of $$\varphi ^{\frac{1}{p+k-1}}$$ is negative definite at some point and the solution to the flow loses its smoothness.

Free time minimizers for the three-body problem

Free time minimizers of the action (called"semi-static" solutions by Ma\~ne) play a central role in the theory of weak KAM solutions to the Hamilton-Jacobi equation (see Fathi). We prove that any solution to Newton's three-body problem which is asymptotic to Lagrange's parabolic homothetic solution is eventually a free time minimizer. Conversely, we prove that every free time minimizer tends to Lagrange's solution, provided the mass ratios lie in a certain large open set of mass ratios. We were inspired by the work of Da Luz-Maderna who had shown that every free time minimizer for the N-body problem is parabolic, and therefore must be asymptotic to the set of central configurations. We exclude being asymptotic to Euler's central configurations by a second variation argument. Central configurations correspond to rest points for the McGehee blown-up dynamics. The large open set of mass ratios are those for which the linearized dynamics at each Euler rest point has a complex eigenvalue.

Contracting convex immersed closed plane curves with slow speed of curvature

The authors study the contraction of a convex immersed plane curve with speed 1 α k α \frac {1}{\alpha }k^{\alpha } , where α ∈ ( 0 , 1 ] \alpha \in (0,1] is a constant, and show that, if the blow-up rate of the curvature is of type one, it will converge to a homothetic self-similar solution. They also discuss a special symmetric case of type two blow-up and show that it converges to a translational self-similar solution. In the case of curve shortening flow (i.e., when α = 1 \alpha =1 ), this translational self-similar solution is the familiar “Grim Reaper” (a terminology due to M. Grayson).

Convex curves moving translationally in the plane

We show that a nontrivial homothetic self-similar solution can happen only when F ( k ) = k α or F ( k ) = − k − α . We also derive a parametric representation of a translational self-similar solution. A translational self-similar solution may have self-intersections but cannot be a simple closed curve for any F ( k ) .

On the Symmetric Collinear Four-Body Problem

Abstract The global geometry of the phase structure in a special case of the general Newtonian four-body problem was studied both analytically and numerically in the case of negative energy. Our method consists of blow-up of total collision by McGehee’s coordinates. and representation of orbits by symbol sequences. The analytical study for arbitrary masses clarifies the macroscopic behavior in phase space: escape motion, vortical motion around vertical lines along which the escape motion occurs, and vertical convective flow. We numerically examined a distribution of the symbol sequences on a surface of section in the case of equal masses. The result has clarified that there never exist orbits whose symbol sequences contain special words: un-realizable words. On the other hand, the existence of oscillatory motions are shown under a reasonable assumption. We semi-analytically obtained the initial conditions leading to escape using escape criteria established in the present study. Additionally, we established a way to find the fastest capture-escape orbits, the ejection-collision orbits besides the homothetic solution, the capture-collision orbits, and the ejection-escape orbits. Moreover, quasi-periodic orbits containing a Schubart-like orbit, and unstable periodic orbits were found. The result displays a similarity between the symmetric collinear four-body problem and the collinear three-body problem.

On the roles of impatiencein homothetic growth paths

This article reexamines the role of long-run endogenous impatience in homothetic growth paths. Using a class of intertemporal preferences assuming a continuous time generator representation, the existence, uniqueness and determinacy properties of a homothetic growth path with endogenous rates of time preference are checked. The argument is based on extensive use of prospective utility. The associated utilities are shown to exhibit a rate of time preference function that becomes invariant to prospective utility along the homothetic growth solution. They also enter in a normal way in the generating function. Finally and at the opposite of earlier conclusions for steady states, the nil rate of time preference case associated with multiplicatively separable generators cannot anymore be discarded.

Courbures de Riemann dans le problème plan des trois corps — Stabilité

We develop a method for studying the curvature of the configuration space of a dynamical system equipped with the Maupertuis metric, for a given value of the energy constant, using the symbolic manipulation program Macsyma. We apply this method in the planar three-body problem. Discussion of the sign of the Riemann curvatures enables one to study the stability of particular configurations. For Lagrange's homothetic triangular solution, we found conditions when all the Riemann curvatures are negative. This means that under these conditions, the homothetic Lagrange solution is exponentially unstable.

Anisotropic fluid spherically symmetric space–times admitting a kinematic self-similarity

Anisotropic fluid spherically symmetric space–times admitting a kinematic self-similar vector are investigated. The geodesic case is considered, and some special subcases in which the anisotropic fluid satisfies additional physical conditions are investigated in detail. A number of other special cases are studied. Particular attention is focused on the possible asymptotic behavior of the models, and it is shown that the models considered always asymptote towards an exact homothetic solution, which is in general either a perfect fluid model or a static solution.

Comment on the Roberts Solution for the Spherically-Symmetric Einstein-Scalar Field Equations

We critically examine the Roberts homothetic solution for the spherically symmetric Einstein-scalar field equations in double null coordinates, and show that the Roberts solution indeed solves the field equations only for one non-trivial case. We generalize this solution and discuss its relations with other known exact solutions.

Kinematic self-similarity

Self-similarity in general relativity is briefly reviewed and the differences between self-similarity of the first kind (which can be obtained from dimensional considerations and is invariantly characterized by the existence of a homothetic vector in perfect-fluid spacetimes) and generalized self-similarity are discussed. The covariant notion of a kinematic self-similarity in the context of relativistic fluid mechanics is defined. It has been argued that kinematic self-similarity is an appropriate generalization of homothety and is the natural relativistic counterpart of self-similarity of the more general second (and zeroth) kind. Various mathematical and physical properties of spacetimes admitting a kinematic self-similarity are discussed. The governing equations for perfect-fluid cosmological models are introduced and a set of integrability conditions for the existence of a proper kinematic self-similarity in these models is derived. Exact solutions of the irrotational perfect-fluid Einstein field equations admitting a kinematic self-similarity are then sought in a number of special cases, and it is found that: (i) in the geodesic case the 3-spaces orthogonal to the fluid velocity vector are necessarily Ricci-flat; (ii) in the further specialization to dust (i.e. zero pressure) the differential equation governing the expansion can be completely integrated and the asymptotic properties of these solutions can be determined; (iii) the solutions in the case of zero expansion consist of a class of shear-free and static models and a class of stiff perfect-fluid (and non-static) models; and (iv) solutions in which the kinematic self-similar vector is parallel to the fluid velocity vector are necessarily Friedmann - Robertson - Walker (FRW) models. Solutions in which the kinematic self-similarity is orthogonal to the velocity vector are also considered. In addition, the existence of kinematic self-similarities in FRW spacetimes is comprehensively studied. It is known that there are a variety of circumstances in general relativity in which self-similar models act as asymptotic states of more general models. Finally, the questions of under what conditions are models which admit a proper kinematic self-similarity asymptotic to an exact homothetic solution and under what conditions are the asymptotic states of cosmological models represented by exact solutions of Einstein's field equations which admit a generalized self-similarity are addressed.

Initial-value constraints and generation of isometries and homothetic motions.

We discuss the generation of spacetime symmetries from an initial-value point of view. After reviewing the effective set of initial-value constraints (a combination of the familiar gravitational constraints and an additional set arising from projections of ${\mathcal{L}}_{\ensuremath{\xi}}g=\ensuremath{\alpha}g$ on the initial surface), we focus our attention on the construction of initial data generating spacetimes admitting homothetic motion. From the analysis of the constrained equations we show that there are only two classes of homothetic, spherically symmetric spacetimes ($M, g$) where (a) $M$ admits a flat Cauchy hypersurface $\ensuremath{\Sigma}$, (b) $g$ is a perfect-fluid solution of Einstein's equation with $P=k\ensuremath{\rho}$, $0\ensuremath{\le}k\ensuremath{\le}1$ as the equation of state and with an initial fourvelocity parallel to the normal of $\ensuremath{\Sigma}$, and (c) the projection of the infinitesimal homothetic generator $\ensuremath{\xi}$ along the normal of $\ensuremath{\Sigma}$ is everywhere a nonvanishing constant. The first class is the spatially flat Friedmann-Robertson-Walker (FRW) class of metrics. The second is a particular class of the marginally bound Tolman class of metrics. In combination with recent work by Eardley et al. [which shows the absence of spacetimes ($M, g$) with $g$ a nonvacuum homothetic solution of Einstein's equations and $M$ admitting a compact Cauchy hypersurface of constant extrinsic curvature], these results appear to suggest that only a "few" metrics (a) are compatible with Einstein's (nonvacuum) equations, and (b) allow the existence of a homothetic symmetry.

The variational equation of the three-body problem

Letx 0 (t),x 0 ɛℝ4 be a homothetic solution of the planar three-body problem with total energyh, described in relative coordinates with respect to one body. It is shown that the variational equation of the problem atx 0 (t) can be solved explicitly in terms of hypergeometric functions. This is done by using the scaled true anomaly of the one-dimensional Kepler motion as the independent variable. The classical theorems about hypergeometric functions allow a simple calculation of all the values needed in applications. By means of this theory the past of a homothetic triple close encounter may be described in a linearized approximation.

Triple Collision as an Unstable Equilibrium

AbstractThe family of trajectories near a triple collision solution in the planar problem of three bodies is investigated by means of linearization in the neighborhood of the parabolic free fall. The local topological structure of this family is found to be that of a saddle point in the R8. The corresponding stable manifold is the set of all triple collision solutions, whereas the instable manifold is formed by the parabolic solutions.In important cases of nonzero total energy the family of close encounters is quantitatively described in terms of hypergeometric functions. By means of homothetic transformations the close triple encounters are then related to three-body motion with zero energy and zero angular momentum. In this way almost all close encounters near a homothetic solution can be treated by using a small number of particular solutions of the three-body problem that may be calculated once for all.In practical examples the first order theory presented here predicts the escape velocity after a close triple encounter with a relative accuracy comparable to the closeness of the encounter.

Triple collision as un unstable equilibrium

The family of trajectories near a triple collision solution in the planar problem of three bodies is investigated by means of linearization in the neighbourhood of the parabolic free fall. The local topological structure of this family is found to be that of a saddle point in the R8. The corresponding stable manifold is the set of all triple collision solutions, whereas the instable manifold is formed by the parabolic solutions. In important cases of nonzero total energy the family of close encounters is quantitatively described in terms of hypergeometric functions. By means of homothetic transformations the close triple encounters are then related to three-body motion with zero energy and zero angular momentum. In this way almost all close encounters near a homothetic solution can be treated by using a small number of particular solutions of the three-body problem that may be calculated once for all. In practical examples the first order theory presented here predicts the escape velocity after a close triple encounter with a relative accuracy comparable to the closeness of the encounter.