##
**Existence and numerical approximation of periodic motions of an infinite lattice of particles.**
*(English)*
Zbl 0838.34046

The lattice is autonomous. Its particles interact with their nearest neighbours by a force belonging to a certain class characterized by the properties of the potential. Purely attractive forces are considered. Assumptions are given which admit a nonzero periodic non constant solution with finite energy. First, a finite system is considered, then an appropriate imbedding of this case in the infinite system is given and the limit process is taken. The application of a known algorithm is shown for finite lattices in the case of repulsive-attractive forces; investigating the infinite lattices a process of approximation in the proof is based on these results. Some numerical experiments for finite lattices are shown.

Reviewer: Á.Bosznay (Budapest)

### MSC:

34C25 | Periodic solutions to ordinary differential equations |

70F99 | Dynamics of a system of particles, including celestial mechanics |

PDF
BibTeX
XML
Cite

\textit{G. Arioli} and \textit{F. Gazzola}, Z. Angew. Math. Phys. 46, No. 6, 898--912 (1995; Zbl 0838.34046)

Full Text:
DOI

### References:

[1] | A. Ambrosetti and P. H. Rabinowitz,Dual variational methods in critical point theory and applications, J. Funct. Anal14, 349-381 (1973). · Zbl 0273.49063 |

[2] | G. Arioli and F. Gazzola,Periodic motions of an infinite lattice of particles with nearest neighbor interaction, to appear in Nonlinear Analysis TMA. · Zbl 0867.70004 |

[3] | G. Arioli, F. Gazzola and S. Terracini,Multibump periodic motions of an infinite lattice of particles, to appear in Mathematische Zeitschrift. · Zbl 0871.34028 |

[4] | V. Benci, G. F. Dell’Antonio and B. D’Onofrio,Index theory and stability of periodic solutions of Lagrangian systems, C. R. Acad. Sci. Paris Ser. I-Math315 (5), 583-588 (1992). |

[5] | V. Benci and P. H. Rabinowitz,Critical points theorems for indefinite functional, Inv. Math.52, 241-273 (1979). · Zbl 0465.49006 |

[6] | Y. S. Choi and J. McKenna,A mountain pass method for the numerical solution of semilinear elliptic problems, Nonlinear Analysis TMA20 (4), 417-437 (1993). · Zbl 0779.35032 |

[7] | V. Coti Zelati and P. H. Rabinowitz,Homoclinic orbits for second order Hamiltonian systems possessing superquadratic potentials, J. AMS4, 693-727 (1991). · Zbl 0744.34045 |

[8] | P. L. Lions,The concentration-compactness principle in the calculus of variations. The locally compact case, Part 1, Ann. Inst. H. Poincaré (A.N.L.)1 (2), 109-145 (1984). · Zbl 0541.49009 |

[9] | P. H. Rabinowitz,Some critical point theorems and applications to semilinear elliptic partial differential equations, Ann. Sc. Norm. Sup. Pisa4 (5), 215-223 (1978). · Zbl 0375.35026 |

[10] | B. Ruf and P. N. Srikanth,On periodic motions of lattices of Toda type via critical point theory, Arch. Rat. Mech. Anal.126, 369-385 (1994). · Zbl 0809.34056 |

[11] | K. Tanaka,Homoclinic orbits in a first order superquadratic Hamiltonian system: convergence of subharmonic orbits, J. Diff. Eq.94, 315-339 (1991). · Zbl 0787.34041 |

[12] | M. Toda,Theory of Nonlinear Lattices, Springer-Verlag, Berlin 1989. · Zbl 0694.70001 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.