Symmetric Liapunov center theorem for minimal orbit. (English) Zbl 1385.37069
Summary: Using the techniques of equivariant bifurcation theory we prove the existence of non-stationary periodic solutions of \(\Gamma\)-symmetric systems \(\ddot{q}(t) = - \nabla U(q(t))\) in any neighborhood of an isolated orbit of minima \(\Gamma(q_0)\) of the potential \(U\). We show the strength of our result by proving the existence of new families of periodic orbits in the Lennard-Jones two- and three-body problems and in the Schwarzschild three-body problem.
MSC:
| 37G40 | Dynamical aspects of symmetries, equivariant bifurcation theory |
| 37G15 | Bifurcations of limit cycles and periodic orbits in dynamical systems |
| 37J20 | Bifurcation problems for finite-dimensional Hamiltonian and Lagrangian systems |
| 37J45 | Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010) |
| 70H33 | Symmetries and conservation laws, reverse symmetries, invariant manifolds and their bifurcations, reduction for problems in Hamiltonian and Lagrangian mechanics |
| 70F07 | Three-body problems |
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