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The classical Liouville Theorem on conformal transformations determines local conformal transformations on the Euclidean space of dimension $\geq 3$. Its natural adaptation to the general framework of Riemannian structures is the 2-rigidity…
In this paper we systematically consider various ways of generating integrable and separable Hamiltonian systems in canonical and in non-canonical representations from algebraic curves on the plane. In particular, we consider St\"ackel…
In a recent paper, Hassoul et al.[1], the authors proposed an analysis of the quantum dynamics for general time-dependent three coupled oscillators through an approach based on their decouplement using the unitary transformation method.…
A Hamiltonian formulation of generic many-particle systems with space-dependent balanced loss and gain coefficients is presented. It is shown that the balancing of loss and gain necessarily occurs in a pair-wise fashion. Further, using a…
Given a first order dynamical system possessing a commutative algebra of dynamical symmetries, we show that, under certain conditions, there exists a Poisson structure on an open neighbourhood of its regular (not necessarily compact)…
We prove that, under some natural conditions, Hamiltonian systems on a contact manifold $C$ can be split into a Reeb dynamics on an open subset of $C$ and a Liouville dynamics on a submanifold of $C$ of codimension 1. For the Reeb dynamics…
We examine the reduction process of a system of second-order ordinary differential equations which is invariant under a Lie group action. With the aid of connection theory, we explain why the associated vector field decomposes in three…
For the 1+1 dimensional Lax pair with a symplectic symmetry and cyclic symmetries, it is shown that there is a natural finite dimensional Hamiltonian system related to it by presenting a unified Lax matrix. The Liouville integrability of…
We describe natural Hamiltonian systems using projective geometry. The null lift procedure endows the tangent bundle with a projective structure where the null Hamiltonian is identified with a projective conic and induces a Weyl geometry.…
We investigate the energy transfer dynamics in a donor-acceptor model by developing a time-local master equation technique based on a variational transformation of the underlying Hamiltonian. The variational transformation allows a…
Constrained Hamiltonian systems are investigated by using the Hamilton-Jacobi method. Integration of a set of equations of motion and the action function is discussed. It is shown that we have two types of integrable systems: a) ${\it…
In many stochastic models, the observables of interest are naturally encoded in double transforms (e.g., Laplace transforms) that couple spatial and temporal variables. Notably, the double transform often provides the only analytically…
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 finite-dimensional real Lie algebra of Hamiltonian vector…
We present developments of the Hamiltonian approach to problems of the freely decay of isotropic turbulence, and also consider specific applications of the modified Prelle-Singer procedure to isotropic turbulence. It demonstrates that a…
A general program to show quantum-classical correspondence for bound conservative integrable and chaotic systems is described. The method is applied to integrable systems and the nature of the approach to the classical limit, the…
In this paper, we study singular systems with complete sets of involutive constraints. The aim is to establish, within the Hamilton-Jacobi theory, the relationship between the Frobenius' theorem, the infinitesimal canonical transformations…
The density-matrix and Heisenberg formulations of quantum mechanics follow--for unitary evolution--directy from the Schr"odinger equation. Nevertheless, the symmetries of the corresponding evolution operator, the Liouvillian L=i[.,H], need…
We investigate an interacting Pais-Uhlenbeck oscillator with a Landau-Ginzburg type interaction term and analyse its classical dynamics from a geometric and numerical point of view. We show that the resulting fourth-order equation of motion…
We study a second order differential equation corresponding to rotationally symmetric $F$-harmonic maps between certain noncompact manifolds. We show unique continuation and Liouville's type theorems for positive solutions. Asymptotic…
In this paper we revisit the classification of the gauge transformations in the Euler top system using the generalized classical Hamiltonian dynamics of Nambu. In this framework the Euler equations of motion are bi-Hamiltonian and $SL(2,…