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We investigate the dynamics of chaotic trajectories in simple yet physically important Hamiltonian systems with non-hierarchical borders between regular and chaotic regions with positive measures. We show that the stickiness to the border…
Canonical transformation plays a fundamental role in simplifying and solving classical Hamiltonian systems. We construct flexible and powerful canonical transformations as generative models using symplectic neural networks. The model…
Modified Hamiltonians are used in the field of geometric numerical integration to show that symplectic schemes for Hamiltonian systems are accurate over long times. For nonlinear systems the series defining the modified Hamiltonian usually…
In systems with a mixed phase space, where regular and chaotic motion coexists, regular states are coupled to the chaotic region by dynamical tunneling. We give an overview on the determination of direct regular-to-chaotic tunneling rates…
Harmonic inversion has already been proven to be a powerful tool for the analysis of quantum spectra and the periodic orbit orbit quantization of chaotic systems. The harmonic inversion technique circumvents the convergence problems of the…
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…
Long-term stability studies of nonlinear Hamiltonian systems require symplectic integration algorithms which are both fast and accurate. In this paper, we study a symplectic integration method wherein the symplectic map representing the…
We present a method to construct high-order polynomial approximate invariants (AI) for non-integrable Hamiltonian dynamical systems, and apply it to modern ring-based particle accelerators. Taking advantage of a special property of one-turn…
Integrable systems have provided various insights into physical phenomena and mathematics. The way of constructing many-body integrable systems is limited to few ansatzes for the Lax pair, except for highly inventive findings of conserved…
We aim to completely formalize the rough topological analysis of integrable Hamiltonian systems admitting analytical solutions such that the initial phase variables along with the time derivatives of the auxiliary variables are expressed as…
A system of linearly coupled quantum harmonic oscillators can be diagonalized when the system is dynamically stable using a Bogoliubov canonical transformation. However, this is just a particular case of more general canonical…
Variation of coupling constants of integrable system can be considered as canonical transformation or, infinitesimally, a Hamiltonian flow in the space of such systems. Any function $T(\vec p, \vec q)$ generates a one-parametric family of…
In generic Hamiltonian systems tori of regular motion are dynamically separated from regions of chaotic motion in phase space. Quantum mechanically these phase-space regions are coupled by dynamical tunneling. We introduce a semiclassical…
The path integral formulation of constrained systems leads to obtain the equations of motion as total differential equations in many variables. If these equations are integrable then one can constuct a valid and a canonical phase space…
We present a coordinate-free approach for constructing approximate first integrals of generalized slow-fast Hamiltonian systems, based on the global averaging method on parameter-dependent phase spaces with $\mathbb{S}^1 -$symmetry.…
Semiclassical approximations often involve the use of stationary phase approximations. This method can be applied when $\hbar$ is small in comparison to relevant actions or action differences in the corresponding classical system. In many…
We introduce an iterative method to search for time-optimal Hamiltonians that drive a quantum system between two arbitrary, and in general mixed, quantum states. The method is based on the idea of progressively improving the efficiency of…
We develop a rigorous analytical Hamiltonian formalism adapted to the study of the motion of two planets in co-orbital resonance. By constructing a complex domain of holomorphy for the planetary Hamilto-nian, we estimate the size of the…
We introduce a new strategy for coupling the parallel in time (parareal) iterative methodology with multiscale integrators. Following the parareal framework, the algorithm computes a low-cost approximation of all slow variables in the…
We study dynamical systems which admit action-angle variables at leading order which are subject to nearly resonant perturbations. If the frequencies characterizing the unperturbed system are not in resonance, the long-term dynamical…