Lie–Hamilton systems on the plane: applications and superposition rules

Abstract

A Lie-Hamilton system is a nonautonomous system of first-order ordinary differential equations describing the integral curves of a t-dependent vector field taking values in a finitedimensional real Lie algebra of Hamiltonian vector fields with respect to a Poisson structure. We provide new algebraic/geometric techniques to easily determine the properties of such Lie algebras on the plane, e.g., their associated Poisson bivectors. We study new and known Lie-Hamilton systems on R 2 with physical, biological and mathematical applications. New results cover Cayley-Klein Riccati equations, the here defined planar diffusion Riccati systems, complex Bernoulli differential equations and projective Schrödinger equations. Constants of motion for planar Lie-Hamilton systems are explicitly obtained which, in turn, allow us to derive superposition rules through a coalgebra approach. MSC: 34A26 (primary) 70G45, 70H99 (secondary)Hence, system (1.4) with a R 1 (t) = 0 is a LH system as it is related to a t-dependent vector field taking values in a Vessiot-Guldberg Lie algebra V of Hamiltonian vector fields relative to Λ. Since V ≃ R ⋉ R 2 ≃ X 1 ⋉ X 2 , X 3 , X 1 ∧ X 2 = 0 and ad X 1 : X i ∈ X 2 , X 3 → [X 1 , X i ] ∈ X 2 , X 3 is not diagonalizable over R, we see in view of table 1 that the Lie algebra V belongs to class P 1 and V ≃ iso(2). Meanwhile, the LH algebra spanned by h 1 , h 2 , h 3 , h 0 is isomorphic to the centrally extended Euclidean algebra iso(2) (see also [13] for further details).On the other hand, we recall that superposition rules for LH systems can be obtained in an algebraic way by applying a Poisson coalgebra approach [19]. In contrast, other methods to derive superposition rules require to integrate a Vessiot-Guldberg Lie algebra, e.g., the group theoretical method [2], or to solve a family of PDEs [9]. Winternitz and coworkers have also derived superposition rules for Lie systems in particular forms [2,5]. The application of this latter result for general Lie systems requires to map them into the canonical form for which the superposition rule was obtained. The coalgebra procedure makes these transformations unnecessary in many cases.The structure of the paper is as follows. In section 2 we summarize the local classification of Vessiot-Guldberg Lie algebras as of Hamiltonian vector fields on the plane performed in [13], where the corresponding symplectic structures were derived by solving a system of PDEs. As a first new achievement, we show in section 3 that such symplectic structures can be determined through algebraic and geometric methods. Although our techniques are heavily based upon the Lie algebra structure of Vessiot-Guldberg Lie algebras, they also depend on their geometric properties as Lie algebras of vector fields. In particular, a new method to construct symplectic structures for LH systems on the plane related to non-simple Vessiot-Guldberg Lie algebras is described.Next we remark that a Vessiot-Guldberg Lie algebra on the plane isomorphic to sl(2) can be diffeomorphic to either P 2 , I 3 ...

A Unified Approach to Poisson–Hopf Deformations of Lie–Hamilton Systems Based on $$\mathfrak {sl}$$(2)

“…4 The non-standard Poisson-Hopf algebra deformation of sl (2) Amongst the LH systems in the plane (see [2,5,10] for details and applications), those with a Vessiot-Guldberg Lie algebra isomorphic to sl(2) are of both mathematical and physical interest; they cover complex Riccati, Milne-Pinney and Kummer-Schwarz equations as well as the harmonic oscillator, all of them with t-dependent coefficients. Furthermore, sl(2)-LH systems are related to three non-diffeomorphic Vessiot-Guldberg Lie algebras on the plane [2,10]. This gives rise to different nonequivalent Poisson-Hopf deformations.…”

“…Our systematization permits us to give directly the Poisson-Hopf deformed system from the classification of LH systems [2,10], further suggesting a notion of Poisson-Hopf Lie systems based on a z-parameterized family of Poisson algebra morphisms. Our methods seem to be extensible to study also LH systems and their deformations on other more general manifolds.…”

Geometric Models for Lie–Hamilton Systems on ℝ2

“…Every Lie system on R 2 can be endowed with a VG Lie algebra that is, around a generic point, locally diffeomorphic to one of the twenty-eight possible classes of VG Lie algebras on R 2 described in [12,15,17]. In particular, only twelve classes can be considered, again locally around a generic point, as VG Lie algebras of Hamiltonian vector fields relative to a symplectic form (see [6,17] and Table 1). Lie systems admitting a VG Lie algebra of Hamiltonian vector fields relative to a Poisson bivector are called Lie-Hamilton systems [6,10,17].…”

“…We use two types of geometric models for studying Lie-Hamilton systems on R 2 , and we propose methods to derive geometrically their associated symplectic structures. [6,12,14]). In particular, every Lie algebra consists of Hamiltonian vector fields on the submanifold of R 2 where the given symplectic form is well defined.…”

Multisymplectic structures and invariant tensors for Lie systems

“…MSC 2010: 34A26 (primary); 34A05, 34C14, 53C15, 16T15 (secondary) the above-mentioned articles in an ad-hoc manner or by solving systems of PDEs. Then, our work simplifies their derivation.Remarkably, we show that the coalgebra method to derive superposition rules for Lie-Hamilton systems developed in [6,8] can be retrieved as a particular case of our techniques when it concerns Lie-Hamilton systems related to symplectic forms. Moreover, our methods give rise to tensor field invariants for multisymplectic Lie systems from invariants of tensor algebras, which are more general than the invariant structures appearing in the standard coalgebra method, e.g.…”