General relativistic effects on the three-body problem are discussed.
One example is a general relativistic version of the figure-eight solution as a choreographic one. Second example is a general relativistic counterpart of the Euler's collinear solution and Lagrange's equilateral one.
Morse index of periodic solutions in the n-body problem
Using the variational method Chenciner and Montgomery proved the existence of a new periodic solution of figure-eight shape to the planar three-body problem.Since then a number of periodic and quasi-periodic solutions have been found as minimizers of variational formulation of the N-body problem in various different settings.
We prove the existence of periodic solutions which are not minimizers under symmetric constraints in the spatial n-body problem.
Colliding choreography in the prismatic 2n-body problem
正Ｎ角柱配置の2N体が自己重力下で回転する系において同時２体衝突を含む舞踏解を発見した。
その存在の解析的（変分法によらない）証明の概略を紹介する。
Choreographic solution with collision and quasi-periodic solution with collision in the rotating prismatic $2N$-body problem exist. The outline of the proof is given. Numerical observation for $N=2$ is illustrated.
Kuo-Chang Chen(the National Tsing Hua University)
The Kepler problem revisited
The Newtonian 2-body problem is also called the Kepler problem in honor of Johannes Kepler (1571-1630) for his discovery of three laws of planetary motion, based on which Newton deduced in 1687 the celebrated law of universal gravitation. It is widely considered a well-understood problem, as solving it with given initial data and proving Kepler's three laws require nothing more than tools from elementary calculus. In this talk I will present a nonclassical approach which gives us more insights into Keplerian orbits and which enables us to explore the n-body problem from a new perspective.
Efficient parallel computation of all-pairs n-body acceleration by do-loop folding
The computational load inside a do loop is equalized by folding the loop appropriately if the amount of load is a linear function of the loop index. Using this idea, we develop an efficient parallel computation scheme of Newtonian all-pairs N-body acceleration vectors with help from OpenMP architecture. Using a consumer PC with a quad-core eight-thread processor, the new parallel scheme runs 4.2.4.9 times faster than a serial computation when the number of particles exceeds a few hundred.
藤原俊朗(北里大学)
Motion in shape for planar three-body problem
Richard Moeckel and Richard Montgomery introduced a shape variable that is useful to describe the motion in shape for planar three-body problem. They write the Lagrangian by variables that describe the shape, size and rotation angle, and derived the equations of motion for these variables.
Using their formulation, we proved the Saari's homographic conjecture for the case of planar equal-mass three-body problem under the strong force potential $U=¥sum 1/r_{ij}^2$. Namely, we proved that the shape of the triangle whose vertices are the position of the three bodies is unchanged, while the size and the rotation angle can be changed, if and only if the product of the moment of inertia $I=¥sum r_{ij}^2$ and the potential $U$ is constant.
Families of symmetric relative periodic orbits originating from the circular Euler solution in the isosceles three-body problem
We study symmetric relative periodic orbits in the isosceles three-body problem using theoretical and numerical approaches. We first prove that another family of symmetric relative periodic orbits is born from the circular Euler solution besides the elliptic Euler solutions. Previous studies also showed that there exist infinitely many families of symmetric relative periodic orbits which are born from heteroclinic connections between triple collisions as well as planar periodic orbits with binary collisions. We carry out numerical continuation analyses of symmetric relative periodic orbits, and observe abundant families of symmetric relative periodic orbits bifurcating from the two families born from the circular Euler solution. As the angular momentum tends to zero, many of the numerically observed families converge to heteroclinic connections between triple collisions or planar periodic orbits with binary collisions described in the previous results, while some of them converge to “previously unknown” periodic orbits in the planar problem. This talk is based on a joint work with Mitsuru Shibayama.
山田慧生(弘前大学)
Collinear solution to the general relativistic three-body problem
I will give a poster presentation regarding a collinear solution to relativistic three-body problem,based on a collaboration with Hideki Asada.
The Newtonian three-body problem admits Euler's collinear solution,where three bodies move around the common center of mass with the same orbital period and always line up. The solution is unstable. Hence it is unlikely that such a simple configuration would exist owing to general relativistic forces dependent not only on the masses but also on the velocity of each body. However, we show that the collinear solution remains true with a correction to the spatial separation between masses at the post- Newtonian order in general relativity.
[Ref:Yamada and Asada, PRD82, 104019 (2010), PRD83, 024040 (2011)]
