Normal Form of Periodic FPU Chain with Four Degrees of Freedom and Alternating Masses
Stephanus Ardyanto, Johan Matheus Tuwankotta (*)

Analysis and Geometry Group, Faculty of Mathematics and
Natural Sciences, Institut Teknologi Bandung, Bandung,
Indonesia
*jmtuwankotta[at]itb.ac.id

Abstract

In this paper we study the periodic Fermi-Pasta-Ulam (FPU) chain. It is a one dimensional chain of oscillators which endpoints are connected and has nearest-neighbor interaction only. We specify our research by considering the chain with four degrees of freedom and has alternating masses \(1,m,1,m\). Moreover, we also consider a more general potential function in the Hamiltonian function of the system.

The analysis is done by using the near identity transformation in phase space. The transformation is defined by using the flow of a linear Hamiltonian system, which is clearly symplectic so that the Hamiltonian structure can be preserved. The transformed Hamiltonian is then called in a so-called Birkhoff-Gustavson normal form. The structure as to the remaining terms in the normal form, depends on the choice of \(a=1/m\). Due to the nature of the problem, there are some discrete symmetries in phase space which simplify the normal form further. Our main focus is to analyze the case when \(a = 1\) (homogeneous chain) and \(a = 3\). Depending on the value of the parameter, the system has topologically nonequivalent phase space which will be classified. The two cases which are considered express two different class of resonances. This is one of the reason why FPU chain is an interesting model to study.

Keywords: Periodic FPU chain, alternating masses, Hamiltonian system, normal form

Topic: MATHEMATICS AND STATISTICS