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Periodic motion of a system of two or three charged particles. (English) Zbl 0970.34036

This paper presents a necessary and sufficient condition for the existence of periodic solutions to the system \[ \begin{aligned} x_1 '' &= -Ag(x_2 -x_1)-Bg(x_3 -x_1)+h(t),\\ x_2'' &= Ag(x_2 -x_1)-Cg(x_3 -x_2)+k(t),\\ x_3'' &= Bg(x_3 -x_1)+Cg(x_3-x_2)+p(t),\end{aligned} \] where \(A\), \(B\) and \(C\) are positive numbers and \(h\), \(k\) and \(p\) are periodic functions. The conditions imposed on \(g\) include the example \(g(x)=x^{-\alpha}\) with \(\alpha \geq 1 \). For \(\alpha =2\) this system governs the forced motion of three particles placed on a line and having charges of the same sign. The authors also discuss systems of two particles. In this situation they can also analyze particles with charges of opposite sign.

MSC:

34C25 Periodic solutions to ordinary differential equations
78A35 Motion of charged particles
34B15 Nonlinear boundary value problems for ordinary differential equations
37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010)
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