Related papers: Geometry of discrete-time spin systems
We develop a class of C1-continuous time integration methods that are applicable to conservative problems in elastodynamics. These methods are based on Hamilton's law of varying action. From the action of the continuous system we derive a…
An explicit high-order noncanonical symplectic algorithm for ideal two-fluid systems is developed. The fluid is discretized as particles in the Lagrangian description, while the electromagnetic fields and internal energy are treated as…
The solutions of Hamiltonian equations are known to describe the underlying phase space of a mechanical system. In this article, we propose a novel spatio-temporal model using a strategic modification of the Hamiltonian equations,…
Splines and subdivision curves are flexible tools in the design and manipulation of curves in Euclidean space. In this paper we study generalizations of interpolating splines and subdivision schemes to the Riemannian manifold of shell…
We implement and investigate the numerical properties of a new family of integrators connecting both variants of the symplectic Euler schemes, and including an alternative to the classical symplectic mid-point scheme, with some additional…
We propose the difference discrete variational principle in discrete mechanics and symplectic algorithm with variable step-length of time in finite duration based upon a noncommutative differential calculus established in this paper. This…
In the first part of the paper we introduce some geometric tools needed to describe slow-fast Hamiltonian systems on smooth manifolds. We start with a smooth Poisson bundle $p: M\to B$ of a regular (i.e. of constant rank) Poisson manifold…
We propose a topological characterization of Hamiltonians describing classical waves. Applying it to the magnetostatic surface spin waves that are important in spintronics applications, we settle the speculation over their topological…
Bifibrations, in symplectic geometry called also dual pairs, play a relevant role in the theory of superintegrable Hamiltonian systems. We prove the existence of an analogous bifibrated geometry in dynamical systems with a symmetry group…
Numerical algorithms based on variational and symplectic integrators exhibit special features that make them promising candidates for application to general relativity and other constrained Hamiltonian systems. This paper lays part of the…
This work develops a symplectic framework for quantum computing to be applied to classical Hamiltonian systems, exploiting the intrinsic geometric compatibility between unitary quantum evolution and symplectic phase-space dynamics in a…
In this work, we design and analyze semi/fully-discrete virtual element approximations for the time-dependent Navier--Stokes-Cahn--Hilliard equations, modeling the dynamics of two-phase incompressible fluid flows with diffuse interfaces. A…
We develop a dynamical symmetry approach to path integrals for general interacting quantum spin systems. The time-ordered exponential obtained after the Hubbard-Stratonovich transformation can be disentangled into the product of a finite…
In this paper, we investigate the asymptotic error distributions of symplectic methods for stochastic Hamiltonian systems and further provide Hamiltonian-specific analysis that clarifies the superiority of symplectic methods. Our…
For oriented surfaces $\Sigma$ with boundary, we consider the infinite-dimensional deformation space of projective structures on $\Sigma$ with nondegenerate boundary, up to isotopies fixing the boundary. We show that this space carries a…
The exceptional point, known as the non-Hermitian degeneracy, has special topological structure, leading to various counterintuitive phenomena and novel applications, which are refreshing our cognition of quantum physics. One particularly…
The main purpose of this paper is to study the length minimizing property of Hamiltonian paths on closed symplectic manifolds $(M,\omega)$ such that there are no spherical homology class $A \in H_2(M)$ with $$ \omega(A) > 0 \quad \text{and}…
This paper presents a "historical" formalism for dynamical systems, in its Hamiltonian version (Lagrangian version was presented in a previous paper). It is universal, in the sense that it applies equally well to time dynamics and to field…
We recently derived a spin-mapping approach for treating the nonadiabatic dynamics of a two-level system in a classical environment [J. Chem. Phys. 151, 044119 (2019)] based on the well-known quantum equivalence between a two-level system…
Studying single-particle dynamics over many periods of oscillations is a well-understood problem solved using symplectic integration. Such integration schemes derive their update sequence from an approximate Hamiltonian, guaranteeing that…