arXiv subjects

Toshiaki Fujiwara

Publications and source records attributed to Toshiaki Fujiwara.

At least 19 recordsLinked to original sources

arXiv2025

N-body choreographies on a p-limacon curve

We consider an $N$--body problem under a harmonic potential of the form $\frac{1}{2}\sum κ_{jl} |q_j-q_l|^2$. A $p$-limaçon curve is a planar curve parametrized by $t$ given by $a(\cos t,\sin t)+b(\cos pt, \sin pt)$, where $a,b\in \mathbb{R}$, $p \in \mathbb{Z}$, and $t \in [0,2π]$. We study $N$-body choreographic motions constrained to a $p$-limaçon curve and establish necessary and sufficient conditions for their existence. Specifically, we prove that choreographic motions exist if and only if $p/N, (p \pm 1)/N \notin \mathbb{Z}$. Under an additional symmetry assumption on the force coefficients, we further refine these conditions. We also analyze the occurrence of collisions, showing that for given $p$ and $N$, at most $2(N-1)$ choices of $a/b$ lead to collisions. Furthermore, we find additional conserved quantities.

math.DS↗

arXiv2025

Equivalence of Rigid Motions and Relative Equilibria in the N-Body Problem on the Two-Sphere

We investigate the relationship between rigid motions and relative equilibria in the N-body problem on the two-dimensional sphere, S2. We prove that any rigid motion of the N-body system on S2 must be a relative equilibrium. Our approach extends the classical study of rigid body dynamics by Euler and utilizes a rotating frame attached to the particles to derive the corresponding equations of motion. We further show that our results can be extended to the N-body gravitational system in R3. The results are oriented to a broader understanding of the dynamics of N-body systems on curved surfaces.

math.DS↗

arXiv2024

Continuations and bifurcations of relative equilibria for the positive curved three body problem

The positive curved three body problem is a natural extension of the planar Newtonian three body problem to the sphere $\mathbb{S}^2$. In this paper we study the extensions of the Euler and Lagrange Relative equilibria ($RE$ in short) on the plane to the sphere. The $RE$ on $\mathbb{S}^2$ are not isolated in general. They usually have one-dimensional continuation in the three-dimensional shape space. We show that there are two types of bifurcations. One is the bifurcations between Lagrange $RE$ and Euler $RE$. Another one is between the different types of the shapes of Lagrange $RE$. We prove that bifurcations between equilateral and isosceles Lagrange $RE$ exist for equal masses case, and that bifurcations between isosceles and scalene Lagrange $RE$ exist for partial equal masses case.

math.CA↗

arXiv2024

Mass independent shapes for relative equilibria in the two dimensional constant positive curved three body problem

In the planar three-body problem under Newtonian potential, it is well known that any masses, located at the vertices of an equilateral triangle generates a relative equilibrium, known as the Lagrange relative equilibrium. In fact, the equilateral triangle is the unique mass independent shape for a relative equilibrium in this problem. The two dimensional positive curved three-body problem, is a natural extension of the Newtonian three-body problem to the sphere $\mathbb{S}^2$, where the masses are moving under the influence of the cotangent potential. S.~Zhu showed that in this problem, equilateral triangle on a rotating meridian can form a relative equilibria for any masses. This was the first report of mass independent shape on $\mathbb{S}^2$ which can form a relative equilibrium. % In this paper, we show that, in addition to the equilateral triangle, there exists one isosceles triangle on a rotating meridian, with two equal angles seen from the centre of $\mathbb{S}^2$ given by $2^{-1}\arccos((\sqrt{2}-1)/2)$, which always form a relative equilibrium for any choice of the masses. Additionally we prove that, the equilateral and the above isosceles relative equilibrium are unique with this characteristic. We also prove that each relative equilibrium generated by a mass independent shape is not isolated from the other relative equilibria.

math.CA↗

arXiv2023

Three body relative equilibria on $\mathbb{S}^2$

We study relative equilibria ($RE$ in short) for three-body problem on $\mathbb{S}^2$, under the influence of a general potential which only depends on $\cosσ_{ij}$ where $σ_{ij}$ are the mutual angles among the masses. Explicit conditions for masses $m_k$ and $\cosσ_{ij}$ to form relative equilibrium are shown. Using the above conditions, we study the equal masses case under the cotangent potential. We show the existence of scalene and isosceles Euler $RE$, and isosceles and equilateral Lagrange $RE$.

math.CA↗

arXiv2023

A new method to study relative equilibria on $\mathbb{S}^2$

We develop a new geometrical technique to study relative equilibria for a system of $n$--positive masses, moving on the two dimensional sphere $\mathbb{S}^2$, under the influence of a general potential which only depends on the mutual distances among the masses. The big difficulty to study relative equilibria on $\mathbb{S}^2$, that we call $RE$ by short, is the absence of the center of mass as a first integral. We show that the two vanishing components of the angular momentum, for motions on $\mathbb{S}^2$, play the same role as the center of mass for motions on the Euclidean plane. From here we obtain that the rotation axis of a $RE$ is one of the principal axes of the inertia tensor. Conditions for have $RE$ and relations between the shape (given by the arc angles $σ_{ij}$ among the masses) and the configuration (given by the polar angles $θ_k$ and $ϕ_i - ϕ_j$ in spherical coordinates) are shown. For $n=3$, we show explicitly the conditions to have Euler and Lagrange $RE$ on $\mathbb{S}^2$. As an application of our method we study the the equal masses case for the positive curved three body problem where we show the existence of scalene and isosceles Euler $RE$ and isosceles Lagrange $RE$.

math.CA↗

arXiv2022

Equal masses Eulerian relative equilibria on a rotating meridian of S^2

Relative equilibria on a rotating meridian on $\mathbb{S}^2$ in equal-mass three-body problem under the cotangent potential are determined. We show the existence of scalene and isosceles relative equilibria. Almost all isosceles triangles, including equilateral, can form a relative equilibrium, except for the two equal arc angles $θ= π/2$. For $θ\in (0,2π/3)\setminus \{π/2\}$, the mid mass must be on the rotation axis, in our case, at the north or south pole of $\mathbb{S}^2$. For $θ\in (2π/3,π)$, the mid mass must be on the equator. For $θ=2π/3$, we obtain the equilateral triangle, where the position of the masses is arbitrary. When the largest arc angle $a_\ell$ is in $a_\ell\in (π/2,a_c)$, with $a_c=1.8124...$, two scalene configurations exist for given $a_\ell$.

math.CA↗

arXiv2022

Three-body relative equilibria on $\mathbb{S}^2$ I: Euler configurations

Using the properties of the angular momentum, we develop a new geometrical technique to study relative equilibria for a system of $3$--bodies with positive masses, moving on the two sphere under the influence of an attractive potential depending only on the mutual distances among the bodies. With the above techniques we do an analysis of the relative equilibria for the case of three bodies when they are moving on the same geodesic (Euler configurations).

math.CA↗

arXiv2022

Three-body relative equilibria on $S^2$ II: Extended Lagrangian configurations

This is a natural continuation of our first paper \cite{pre}, where we develop a new geometrical technique which allow us to study relative equilibria on the two sphere. We consider a system of three positive masses on $\mathbb{S}^2$ moving under the influence of an generic attractive potential which only depends on the mutual distances among the masses. We reduce the problem of finding extended Lagrangian relative equilibria to the analysis of the inertia tensor, then we obtain a more manageable equivalent inertia tensor which allow us to find new families of Lagrangian configurations.

math.CA↗

arXiv2020

Variational principle of action and group theory for bifurcation of figure-eight solutions

Figure-eight solutions are solutions to planar equal mass three-body problem under homogeneous or inhomogeneous potentials. They are known to be invariant under the transformation group $D_6$: the dihedral group of regular hexagons. Numerical investigation shows that each figure-eight solution has some bifurcation points. Six bifurcation patterns are known with respect to the symmetry of the bifurcated solution. In this paper we will show the followings. The variational principle of action and group theory show that the bifurcations of every figure-eight solution are determined by the irreducible representations of $D_6$. Each irreducible representation has one to one correspondence to each bifurcation. This explains numerically observed six bifurcation patterns. In general, in Lagrangian mechanics, bifurcations of a periodic solution is determined by irreducible representations of the transformation group that leaves this solution invariant.

math-ph↗

arXiv2019

Variational principle for bifurcation in Lagrangian mechanics

An application of variational principle to bifurcation of periodic solution in Lagrangian mechanics is shown. A few higher derivatives of the action integral at a periodic solution reveals the behaviour of the action in function space near the solution. Then the variational principle gives a method to find bifurcations from the solution. The second derivative (Hessian) of the action has an important role. At a bifurcation point, an eigenvalue of Hessian tends to zero. Inversely, if an eigenvalue tends to zero, the zero point is a bifurcation point. The third and higher derivatives of the action determine the properties of the bifurcation and bifurcated solution.

physics.class-ph↗

arXiv2019

Morse index and bifurcation for figure-eight choreographies of the equal mass three-body problem

We report on the Morse index and periodic solutions bifurcating from the figure-eight choreography for the equal mass three-body problem under homogeneous potential $-1/r^a$ for $a \ge 0$, and under Lennard-Jones (LJ) type potential $1/r^{12}-1/r^6$, where $r$ is a distance between bodies. It is shown that the Morse index changes at a bifurcation point and all solutions bifurcating are approximated by variational functions responsible for the change of the Morse index. Inversely we observed %numerically bifurcation occurs at every point where the Morse index changes for the figure-eight choreography under $-1/r^a$, and for $α$ solution under LJ type potential, where $α$ solution is a figure-eight choreography tending to that under $-1/r^6$ for infinitely large period. Thus, to our numerical studies, change of the Morse index is not only necessary but also sufficient condition for bifurcation for these choreographies. Further we observed that the change of the Morse index is equal to the number of bifurcated solutions regarding solutions with congruent orbits as the same solution.

math-ph↗

arXiv2018

Decomposition of the Hessian matrix for action at choreographic three-body solutions with figure-eight symmetry

We developed a method to calculate the eigenvalues and eigenfunctions of the second derivative (Hessian) of action at choreographic three-body solutions that have the same symmetries as the figure-eight solution. A choreographic three-body solution is a periodic solution to equal mass planar three-body problem under potential function $\sum_{i<j} U(r_{ij})$, in which three masses chase each other on a single closed loop with equal time delay. We treat choreographic solutions that have the same symmetries as the figure-eight, namely, symmetry for choreography, for time reversal, and for time shift of half period. The function space of periodic functions are decomposed into five subspaces by these symmetries. Namely, one subspace of trivial oscillators with eigenvalue $4π^2/T^2\times k^2$, $k=0,1,2,\dots$, four subspaces of choreographic functions, and four subspaces of "zero-choreographic" functions. Therefore, the matrix representation of the Hessian is also decomposed into nine corresponding blocks. Explicit expressions of base functions and the matrix representation of the Hessian for each subspaces are given. The trivial eigenvalues with $k\ne 0$ are quadruply degenerated, while with $k=0$ are doubly degenerated that correspond to the conservation of linear momentum in $x$ and $y$ direction. The eigenvalues in choreographic subspace have no degeneracy in general. In "zero-choreographic" subspace, every eigenvalues are doubly degenerated.

math-ph↗

arXiv2018

Morse index for figure-eight choreographies of the planar equal mass three-body problem

We report on numerical calculations of Morse index for figure-eight choreographic solutions to a system of three identical bodies in a plane interacting through homogeneous potential, $-1/r^a$, or through Lennard-Jones-type (LJ) potential, $1/r^{12} - 1/r^6$, where $r$ is a distance between the bodies. The Morse index is a number of independent variational functions giving negative second variation $S^{(2)}$ of action functional $S$. We calculated three kinds of Morse indices, $N$, $N_c$ and $N_e$, in the domain of the periodic, the choreographic and the figure-eight choreographic function, respectively. For homogeneous system, we obtain $N=4$ for $0 \le a < a_0$, $N=2$ for $a_0 < a < a_1$, $N=0$ for $a_1 < a$, and $N_c=N_e=0$ for $0 \le a$, where $a_0=0.9970$ and $a_1=1.3424$. For $a=1$, we show a strong relationship between the figure-eight choreography and the periodic solution found by Simó through the $S^{(2)}$. For LJ system, we calculated the index for the solution tending to the figure-eight solution of $a=6$ homogeneous system for the period $T \to \infty$. We obtain $N$, $N_c$ and $N_e$ as monotonically increasing functions of the gradual change in $T$ from $T \to \infty$, which start with $N=N_c=N_e=0$, jump at the smallest $T$ by $1$, and reach $N=12$, $N_c=4$, and $N_e=1$ for $T \to \infty$ in the other branch.

math-ph↗

arXiv2017

Figure-eight choreographies of the equal mass three-body problem with Lennard-Jones-type potentials

We report on figure-eight choreographic solutions to a system of three identical particles interacting through a potential of Lennard-Jones type, $1/r^{12}-1/r^6$ where $r$ is a distance between the particles. By numerical search, we found there are a multitude of such solutions. A series of them are close to a figure-eight solutions to a homogeneous system with no $1/r^{12}$ term in the potential. The rest are very different from them and have several points with large curvatures in their figure-eight orbits, at which particles are repelled. Here figure-eight choreographies are the periodic motion whose shape is symmetric in both horizontal and vertical axis, starting with an isosceles triangle configuration and going back to an isosceles triangle configuration with opposite direction through Euler configuration. Thus the lobe of this figure-eight may be complex shape and needs not to be convex.

math.DS↗

arXiv2015

Saari's homographic conjecture for general masses in planar three-body problem under Newton potential and a strong force potential

Saari's homographic conjecture claims that, in the N-body problem under the homogeneous potential, $U=α^{-1}\sum m_i m_j/r_{ij}^α$ for $α\ne 0$, a motion having constant configurational measure $μ=I^{α/2}U$ is homographic, where $I$ represents the moment of inertia defined by $I=\sum m_i m_j r_{ij}^2/\sum m_k$, $m_i$ the mass, and $r_{ij}$ the distance between particles. We prove this conjecture for general masses $m_k>0$ in the planar three-body problem under Newton potential ($α=1$) and a strong force potential ($α=2$).

math-ph↗

arXiv2012

Saari's homographic conjecture for planar equal-mass three-body problem in Newton gravity

Saari's homographic conjecture in N-body problem under the Newton gravity is the following; configurational measure μ=\sqrt{I}U, which is the product of square root of the moment of inertia I=(\sum m_k)^{-1}\sum m_i m_j r_{ij}^2 and the potential function U=\sum m_i m_j/r_{ij}, is constant if and only if the motion is homographic. Where m_k represents mass of body k and r_{ij} represents distance between bodies i and j. We prove this conjecture for planar equal-mass three-body problem. In this work, we use three sets of shape variables. In the first step, we use ζ=3q_3/(2(q_2-q_1)) where q_k \in \mathbb{C} represents position of body k. Using r_1=r_{23}/r_{12} and r_2=r_{31}/r_{12} in intermediate step, we finally use μitself and ρ=I^{3/2}/(r_{12}r_{23}r_{31}). The shape variables μand ρmake our proof simple.

math-ph↗

arXiv2011

Saari's homographic conjecture for planar equal-mass three-body problem under a strong force potential

Donald Saari conjectured that the $N$-body motion with constant configurational measure is a motion with fixed shape. Here, the configurational measure $μ$ is a scale invariant product of the moment of inertia $I=\sum_k m_k |q_k|^2$ and the potential function $U=\sum_{i 0$. Namely, $μ= I^{α/2}U$. We will show that this conjecture is true for planar equal-mass three-body problem under the strong force potential $\sum_{i<j} 1/|q_i-q_j|^2$.

math-ph↗