Symmetric and regularized coordinates on the plane triple collision manifold

The force field described by a potential function of the form U = ?n k=1 ak/rk (r = distance between particles, ak = real parameters) models various concrete situations belonging to astronomy, physics, mechanics, astrodynamics, etc. The two-body problem is being tackled in such a field. The motion equations and the first integrals of energy and angular momentum are established. The McGehee-type coordinates are used to blow up the collision singularity and to paste the resulting manifold on the phase space. The flow on the collision manifold is depicted. Then, using the rotational symmetry of the problem and the angular momentum integral, the local flow near collision is described and interpreted in terms of physical motion.

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Exact solution of the planar motion of three arbitrary point vortices

Modern Physics Letters B ◽

10.1142/s0217984915300173 ◽

2015 ◽

Vol 29 (35n36) ◽

pp. 1530017

Author(s):

Robert Conte ◽

Laurent de Seze

Keyword(s):

Exact Solution ◽

Equilibrium Configuration ◽

Relative Equilibrium ◽

Arbitrary Point ◽

Triple Collision ◽

Stability Conditions ◽

Adiabatic Invariants ◽

The Third ◽

Elastic Diffusion ◽

Equilateral Triangles

We give an exact quantitative solution for the motion of three vortices of any strength, which Poincaré showed to be integrable. The absolute motion of one vortex is generally biperiodic: in uniformly rotating axes, the motion is periodic. There are two kinds of relative equilibrium configuration: two equilateral triangles and one or three colinear configurations, their stability conditions split the strengths space into three domains in which the sets of trajectories are topologically distinct. According to the values of the strengths and the initial positions, all the possible motions are classified. Two sets of strengths lead to generic motions other than biperiodic. First, when the angular momentum vanishes, besides the biperiodic regime there exists an expansion spiral motion and even a triple collision in a finite time, but the latter motion is nongeneric. Second, when two strengths are opposite, the system also exhibits the elastic diffusion of a vortex doublet by the third vortex. For given values of the invariants, the volume of the phase space of this Hamiltonian system is proportional to the period of the reduced motion, a well known result of the theory of adiabatic invariants. We then formally examine the behaviour of the quantities that Onsager defined only for a large number of interacting vortices.

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