Related papers: Symplectic integrators for spin systems
We propose a variational symplectic numerical method for the time integration of dynamical systems issued from the least action principle. We assume a quadratic internal interpolation of the state between two time steps and we approximate…
We describe the symplectic structure and Hamiltonian dynamics for a class of Grassmannian manifolds. Using the two dimensional sphere ($S^2$) and disc ($D^2$) as illustrative cases, we write their path integral representations using…
Let M be a symplectic 4-manifold. A semitoric integrable system on M is a pair of real-valued smooth functions J, H on M for which J generates a Hamiltonian S^1-action and the Poisson brackets {J,H} vanish. We shall introduce new global…
The construction of modified equations is an important step in the backward error analysis of symplectic integrator for Hamiltonian systems. In the context of partial differential equations, the standard construction leads to modified…
This paper had no abstract originally. A second-order symplectic integration algorithm for guiding center motion is presented. The algorithm is based on the Poincar\'e (mid-point) generating function.
We introduce a recent symplectic integration scheme derived for solving physically motivated systems with non-separable Hamiltonians. We show its relevance to Riemannian manifold Hamiltonian Monte Carlo (RMHMC) and provide an alternative to…
Symplectic integrators are a foundation to the study of dynamical $N$-body phenomena, at scales ranging from from planetary to cosmological. These integrators preserve the Poincar\'e invariants of Hamiltonian dynamics. The $N$-body…
We propose a classical constrained Hamiltonian theory for the spin. After the Dirac treatment we show that due to the existence of second class constraints the Dirac brackets of the proposed theory represent the commutation relations for…
This paper is devoted to the study of symplectic manifolds and their connection with Hamiltonian dynamical systems. We review some properties and operations on these manifolds and see how they intervene when studying the complete…
The classical spin system consisting of three spins with Heisenberg interaction is an example of a completely integrable mechanical system. In this paper we explicitly calculate its time evolution and the corresponding action-angle…
Symplectic integrators that preserve the geometric structure of Hamiltonian flows and do not exhibit secular growth in energy errors are suitable for the long-term integration of N-body Hamiltonian systems in the solar system. However, the…
Symplectic integrators can be excellent for Hamiltonian initial value problems. Reasons for this include their preservation of invariant sets like tori, good energy behaviour, nonexistence of attractors, and good behaviour of statistical…
Symplectic integrator plays a pivotal role in the long-term tracking of charged particles within accelerators. To get symplectic maps in accurate simulation of single-particle trajectories, two key components are addressed: precise…
While symplectic integration methods based on operator splitting are well established in many branches of science, high order methods for Hamiltonian systems that split in more than two parts have not been studied in great detail. Here, we…
This contribution presents an integration method based on the Simpson quadrature. The integrator is designed for finite-dimensional nonlinear mechanical systems that derive from variational principles. The action is discretized using…
The kinetic term of the $N$-body Hamiltonian system defined on the surface of the sphere is non-separable. As a result, standard explicit symplectic integrators are inapplicable. We exploit an underlying hierarchy in the structure of the…
Most numerical integration algorithms are not designed specifically for Hamiltonian systems and do not respect their characteristic properties, which include the preservation of phase space volume with time. This can lead to spurious…
We introduce a novel numerical method to integrate partial differential equations representing the Hamiltonian dynamics of field theories. It is a multi-symplectic integrator that locally conserves the stress-energy tensor with an excellent…
This article is a survey of classical and quantum completely integrable systems from the viewpoint of local ``phase space'' analysis. It advocates the use of normal forms and shows how to get global information from glueing local pieces.…
We reconsider the variational derivation of symplectic partitioned Runge-Kutta schemes. Such type of variational integrators are of great importance since they integrate mechanical systems with high order accuracy while preserving the…