Eur. Phys. J. B 38, 635-649 (2004)
https://doi.org/10.1140/epjb/e2004-00158-1

Real Hamiltonian forms of Hamiltonian systems

¹ Institute for Nuclear Research and Nuclear Energy, 72 Tzarigradsko chaussée, 1784 Sofia, Bulgaria
² Dipartimento di Fisica “E.R. Caianiello”, Universita di Salerno Istituto Nazionale di fisica Nucleare, Gruppo Collegato di Salerno, Salerno, Italy
³ Dipartimento di Scienze Fisiche, Università Federico II di Napoli and Istituto Nazionale di fisica Nucleare, Sezione di Napoli, Complesso Universitario di Monte Sant'Angelo, Via Cintia, 80126 Napoli, Italy

Corresponding author: ^a gerjikov@inrne.bas.bg

Received: 8 October 2003
Published online: 8 June 2004

Abstract

We introduce the notion of a real form of a Hamiltonian dynamical system in analogy with the notion of real forms for simple Lie algebras. This is done by restricting the complexified initial dynamical system to the fixed point set of a given involution. The resulting subspace is isomorphic (but not symplectomorphic) to the initial phase space. Thus to each real Hamiltonian system we are able to associate another nonequivalent (real) ones. A crucial role in this construction is played by the assumed analyticity and the invariance of the Hamiltonian under the involution. We show that if the initial system is Liouville integrable, then its complexification and its real forms will be integrable again and this provides a method of finding new integrable systems starting from known ones. We demonstrate our construction by finding real forms of dynamics for the Toda chain and a family of Calogero–Moser models. For these models we also show that the involution of the complexified phase space induces a Cartan-like involution of their Lax representations.