Physica D 152–153 (2001) 110–123 Mirror transformations of Hamiltonian systems Jishan Hu ∗ , MinYan, Tat-Leung Yee Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong, PR China Abstract We demonstrate, through various examples of Hamiltonian systems, that symplectic structures have been encoded into the Painlevé test. Each principal balance in the Painlevé test induces a mirror transformation that regularizes movable singularities. Moreover, for finite-dimensional Hamiltonian systems, the mirror transformations are canonical. © 2001 Elsevier Science B.V. All rights reserved. Keywords: Hamiltonian systems; Painlevé test; Symptectic structures 1. Introduction The dynamics of a system of differential equations may be studied by considering the geometry of the corre- sponding flow in the phase space. However, it may not be easy to get the global information because the flow stops at singularities. On the other hand, if the system is integrable, it is often possible to complete its phase space by “adding singularities” and extend the flow over the singularities. As a result, we get a global flow (i.e., defined for all time) on the completion of the phase space, and the study of the geometry of this global flow provides a complete and global picture on the dynamics of the system of differential equations. The key technical step in the above approach is the completion process. Moreover, if the given system is Hamil- tonian, we would further like to preserve the symplectic structure in the completion process. In the late 1980s, the idea was carried out for autonomous finite-dimensional Hamiltonian systems by Adler and van Moerbeke [1], and Ercolani and Siggia [2,3]. It was assumed that only movable singularities are poles and the system passes certain hi- erarchical type Painlevé test (i.e., principal and lower balances forming a “coherent” tree structure). The completion of the phase space was constructed by introducing changes of variables at the pole singularities so that the Laurent series solutions are regularized. For principal balances, Adler and van Moerbeke used the Laurent series solutions directly as the change of variables. The change of variables constructed by Ercolani and Siggia is more refined and is often canonical (so the symplectic structure on the complete phase space is easily understood). However, their constructions are ad hoc and involve some guess works. In [4,5], we demonstrated a systematic way of regularizing the principal balances for general ODE systems passing the Painlevé test. In [6], we further showed that the method is equivalent to the Painlevé test. The resulting change of variable, which we call the mirror transformation, is very similar to Ercolani and Siggia and can be used to complete the phase space. In this paper, we demonstrate, through several examples, that he mirror transformation for ∗ Corresponding author. 0167-2789/01/$ – see front matter © 2001 Elsevier Science B.V. All rights reserved. PII:S0167-2789(01)00164-6