Results on normal forms for FPU chains. (English) Zbl 1168.70005
Summary: We prove, among other results, that near the equilibirum position, any periodic Fermi-Pasta-Ulam (FPU) chain with an odd number \(N\) of particles admits a Birkhoff normal form up to order \(4\), whereas any periodic FPU chain with \(N\) even admits a resonant normal form up to order \(4\). This resonant normal form of order 4 turns out to be completely integrable. Further, for \(N\) odd, we obtain an explicit formula for Hessian of its Hamiltonian at the fixed point.
MSC:
| 70F99 | Dynamics of a system of particles, including celestial mechanics |
| 70K45 | Normal forms for nonlinear problems in mechanics |
| 70H06 | Completely integrable systems and methods of integration for problems in Hamiltonian and Lagrangian mechanics |
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