1.
Vetchanin E. V., Mokrushina L. N.
Cases of Explicit Integration in the $N$-Source Problem
Rus. J. Nonlin. Dyn.,
2026
Abstract
pdf (3.54 Mb)
The motion of $N$ point sources on a plane is considered. Equations of motion of this
system are represented in Hamiltonian form with a Hamiltonian that is a multivalued function
of coordinates. A detailed analysis is presented for the case of sources with zero total strength,
where a reduction by two degrees of freedom is possible. For three sources of which two have
identical strengths, an explicit solution to the reduced system (with one degree of freedom) is
constructed. The trajectories of the reduced system always remain on the same single-valued
branch of the Hamiltonian and arrive in finite time at a branching point of the Hamiltonian.
This point corresponds to collision of two sources with strengths of opposite signs. An exception
is the trajectories lying on an invariant manifold which contains a fixed point of the reduced
system. This fixed point corresponds to an equilateral triangular configuration. For four sources
of which two have identical positive strengths and the other two have identical negative strengths,
a reduced system with two degrees of freedom is presented. It is shown that the reduced system
admits three invariant manifolds, on which the motion is integrable. On the two manifolds the
sources form a collinear configuration, and on the third manifold the sources are located at the
vertices of a (convex or concave) deltoid. For the reduced system restricted to the invariant
manifold we have constructed first integrals, phase portraits, and have identified singularities
and fixed points.
