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A class of C∞-integrable Hamiltonian systems

Published online by Cambridge University Press:  14 November 2011

W. M. Oliva  and
M. S. A. C. Castilla
W. M. Oliva
Affiliation:
Department of Applied Mathematics and Department of Mathematics, Institute of Mathematicsand Statistics, University of Sao Paulo, Caixa Postal 20.570, Sao Paulo (SP), Brazil
M. S. A. C. Castilla
Affiliation:
Department of Applied Mathematics and Department of Mathematics, Institute of Mathematicsand Statistics, University of Sao Paulo, Caixa Postal 20.570, Sao Paulo (SP), Brazil
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Synopsis

We discuss the C∞ complete integrability of Hamiltonian systems of type q = —grad V(q) = F(q), in which the closure of the cone generated (with nonnegative coefficients) by the vectors F(q), q ϵ ℝn, does not contain a line. The components of the asymptotic velocities are first integrals and the main aim is to prove their smoothness as functions of the initial conditions. The Toda-like system with potential V(q)=ΣNi=1 exp(fi∣ q) is a special case of the considered systems ifthe cone C(f1,…,fN)={ΣNi=1cifi,ci≧0} does notcontain a line. In any number of degrees of freedom, if C(f1,…,fN) has amplitude not too large (ang (fi, fj ≦π/2i,j=1,2,…, N), the first integrals are C∞ functions. In two degrees of freedom, without restriction on the amplitude of the cone, C∞-integrability is proved even in a case in which it is known that there is no other meromorphic integral of motion independent of energy. In three degrees of freedom the C∞-integrability of a deformation of the classic nonperiodic Toda system is proved. Some other examples are also discussed.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 113 , Issue 3-4 , 1989 , pp. 293 - 314
Copyright
Copyright © Royal Society of Edinburgh 1989

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