scholarly journals Integrability of Hamiltonian systems and differential Galois groups of higher variational equations

2007 ◽  
Vol 40 (6) ◽  
pp. 845-884 ◽  
Author(s):  
J MORALESRUIZ ◽  
J RAMIS ◽  
C SIMO
Keyword(s):  
Hamiltonian Systems ◽  
Galois Groups ◽  
Variational Equations

scholarly journals EFFICIENT INTEGRATION OF THE VARIATIONAL EQUATIONS OF MULTIDIMENSIONAL HAMILTONIAN SYSTEMS: APPLICATION TO THE FERMI–PASTA–ULAM LATTICE

2012 ◽  
Vol 22 (09) ◽  
pp. 1250216 ◽  
Author(s):  
ENRICO GERLACH ◽  
SIEGFRIED EGGL ◽  
CHARALAMPOS SKOKOS
Keyword(s):  
Hamiltonian Systems ◽  
Coupled Oscillators ◽  
Taylor Series Expansion ◽  
Expansion Method ◽  
Variational Equations ◽  
Integration Schemes ◽  
Chaos Detection ◽  
Efficient Integration ◽  
Low Dimensional ◽  
Series Expansion Method

We study the problem of efficient integration of variational equations in multidimensional Hamiltonian systems. For this purpose, we consider a Runge–Kutta-type integrator, a Taylor series expansion method and the so-called "Tangent Map" (TM) technique based on symplectic integration schemes, and apply them to the Fermi–Pasta–Ulam β (FPU-β) lattice of N nonlinearly coupled oscillators, with N ranging from 4 to 20. The fast and accurate reproduction of well-known behaviors of the Generalized Alignment Index (GALI) chaos detection technique is used as an indicator for the efficiency of the tested integration schemes. Implementing the TM technique — which shows the best performance among the tested algorithms — and exploiting the advantages of the GALI method, we successfully trace the location of low-dimensional tori.

scholarly journals DIFFERENTIAL GALOIS THEORY AND INTEGRABILITY

2009 ◽  
Vol 06 (08) ◽  
pp. 1357-1390 ◽  
Author(s):  
ANDRZEJ J. MACIEJEWSKI ◽  
MARIA PRZYBYLSKA
Keyword(s):  
Galois Group ◽  
Hamiltonian Systems ◽  
Integrable Systems ◽  
Degrees Of Freedom ◽  
Galois Theory ◽  
Differential Galois Theory ◽  
Differential Galois Group ◽  
Variational Equations ◽  
Homogeneous Potentials ◽  
Particular Solution

This paper is an overview of our works that are related to investigations of the integrability of natural Hamiltonian systems with homogeneous potentials and Newton's equations with homogeneous velocity independent forces. The two types of integrability obstructions for these systems are presented. The first, local ones, are related to the analysis of the differential Galois group of variational equations along a non-equilibrium particular solution. The second, global ones, are obtained from the simultaneous analysis of variational equations related to all particular solutions belonging to a certain class. The marriage of these two types of the integrability obstructions enables to realize the classification programme of all integrable homogeneous systems. The main steps of the integrability analysis for systems with two and more degrees of freedom as well as new integrable systems are shown.

Non-integrability by discrete quadratures

2014 ◽  
Vol 2014 (687) ◽  
Author(s):  
Guy Casale ◽  
Julien Roques
Keyword(s):  
Difference Equations ◽  
Galois Groups ◽  
Necessary Condition ◽  
Variational Equations ◽  
Algebraic Solutions

Abstract.We give a necessary condition for integrability by discrete quadratures of systems of difference equations: the discrete variational equations along algebraic solutions must have virtually solvable Galois groups. This necessary condition à la Morales and Ramis is used in order to prove that

scholarly journals Complex Dynamics of Some Hamiltonian Systems: Nonintegrability of Equations of Motion

2019 ◽  
Vol 2019 ◽  
pp. 1-10
Author(s):  
Jingjia Qu
Keyword(s):  
Galois Group ◽  
Dynamic Behavior ◽  
Hamiltonian Systems ◽  
Complex Dynamics ◽  
Large Range ◽  
Equations Of Motion ◽  
Differential Galois Group ◽  
Variational Equations ◽  
Particular Solutions ◽  
Double Well Potential

The main purpose of this paper is to study the complexity of some Hamiltonian systems from the view of nonintegrability, including the planar Hamiltonian with Nelson potential, double-well potential, and the perturbed elliptic oscillators Hamiltonian. Some numerical analyses show that the dynamic behavior of these systems is very complex and in fact chaotic in a large range of their parameter. I prove that these Hamiltonian systems are nonintegrable in the sense of Liouville. My proof is based on the analysis of normal variational equations along some particular solutions and the investigation of their differential Galois group.

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