arXiv research

New action-angle coordinates found for singular symplectic manifolds.

The $k$-symplectic structures appear in the geometric study of the partial differential equations of classical field theories. Meanwhile, we present a new application of the $k$-symplectic structures to investigate a type of systems of first-order ordinary differential equations, the $k$-symplectic Lie systems. In part…

We prove that an integrable system over a symplectic manifold, whose symplectic form is covariantly constant w.r.t. the Gauss-Manin connection, carries a natural hyper-symplectic structure. Moreover, a special Kaehler structure is induced on the base manifold.

New neural net learns time-reversible symplectic dynamics.

We study the symplectic Bott-Chern cohomology by L.-S. Tseng and S.-T. Yau for solvmanifolds endowed with left-invariant symplectic structures. Our results are applicable to cohomology with values in local systems. Studying symplectic Bott-Chern cohomology of solvmanifolds with values in local systems, we give some rem…

Study reveals new geometric structures for magnetic field Hamiltonian systems.

A Lie system is a 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 vector fields: a so-called Vessiot-Guldberg Lie algebra. We suggest the definition of a particular class of Lie systems, the $k…

Study preserves symplectic structure in forced discrete mechanical systems.

Symplectic classification for a specific type of singularity in integrable systems.

SSINNs learn Hamiltonian systems from data with interpretable, low-memory models.

The paper classifies symplectic invariants of specific singularities in integrable Hamiltonian systems.

Study rigidity of Hamiltonians near a minimum in symplectic and magnetic settings.

Proposes NSSNNs to predict nonseparable Hamiltonian systems.

We propose Symplectic Recurrent Neural Networks (SRNNs) as learning algorithms that capture the dynamics of physical systems from observed trajectories. An SRNN models the Hamiltonian function of the system by a neural network and furthermore leverages symplectic integration, multiple-step training and initial state op…

Two reduction schemes for symplectic manifolds are shown equivalent.

Study investigates Hamiltonian systems in fibered almost-symplectic manifolds.

In this paper we will present Lagrangian and Hamiltonian $k$-symplectic formalisms, we will recall the notions of symmetry and conservation law and we will define the notion of pseudosymmetry as a natural extension of symmetry. Using symmetries and pseudosymmetries, without the help of a Noether type theorem, we will o…

Let G be a split semi-simple algebraic group over Q. Let S be a decorated surface, that is a topological oriented surface with a finite set of marked points on the boundary, considered modulo isotopy. We introduce a moduli space D(G,S) and define a collection of special rational coordinate systems on it. The moduli spa…

We discuss normal forms and symplectic invariants of parabolic orbits and cuspidal tori in integrable Hamiltonian systems with two degrees of freedom. Such singularities appear in many integrable systems in geometry and mathematical physics and can be considered as the simplest example of degenerate singularities. We a…

The main purpose of this paper is to give a topological and symplectic classification of completely integrable Hamiltonian systems in terms of characteristic classes and other local and global invariants.

For any Lie group $G$, we construct a $G$-equivariant analogue of symplectic capacities and give examples when $G = \mathbb{T}^k\times\mathbb{R}^{d-k}$, in which case the capacity is an invariant of integrable systems. Then we study the continuity of these capacities, using the natural topologies on the symplectic $G$-…

Paper reduces nonholonomic systems with symmetries.

We show that from an even degree symplectic NQ-manifold, whose homological vector field Q preserves the symplectic form, one can construct a weight system for tri-valent graphs with values in the Q-cohomology ring, satisfying the IHX relation. Likewise, given a representation of the homological vector field, one can co…

For any k<2n we construct a complete system of invariants in the problem of classifying singularities of immersed k-dimensional submanifolds of a symplectic 2n-manifold at a generic double point.

Locally symplectic structure found on Kerr space-time.

In recent years, we have established the iteration theory of the index for symplectic matrix paths and applied it to periodic solution problems of nonlinear Hamiltonian systems. This paper is a survey on these results.

Unified geometric framework for integrability of conservative and dissipative systems.

Classical mechanical systems are modeled by a symplectic manifold $(M,ω)$, and their symmetries, encoded in the action of a Lie group $G$ on $M$ by diffeomorphisms that preserves $ω$. These actions, which are called "symplectic", have been studied in the past forty years, following the works of Atiyah, Delzant, Duister…

New method solves optimization problems on manifolds using symplectic integrators.

Invariant measures found for contact Hamiltonian systems split into Reeb and Liouville dynamics.

New method preserves convergence rates in gradient-based optimization.

We introduce symplectic structures on "Lie pairs" of (real or complex) algebroids as studied by Chen, Stienon and the second author (From Atiyah classes to homotopy Leibniz algebras, arXiv:1204.1075), encompassing homogeneous symplectic spaces, symplectic manifolds with a $\mathfrak g$-action and holomorphic symplectic…

The paper constructs a symplectic groupoid for a specific Poisson structure.

Study the commutativity of reduction and symplectification in contact Hamiltonian systems.

For each simple symplectic triple system over the real numbers, the standard enveloping Lie algebra and the algebra of inner derivations of the triple provide a reductive pair related to a semi-Riemannian homogeneous manifold. It is proved that this is an Einstein manifold.

Symplectic GP regression models Hamiltonian systems for particle tracing.

Study magnetic Hamiltonian systems with constraints, deriving Hamilton-Jacobi equations.

The paper explores scaling symmetries in symplectic geometry and their applications to central configurations.

Several new mutation-periodic quivers of period higher than 1 are introduced as well as the associated discrete dynamical systems. The reduction of these systems is developed using either a presymplectic or a Poisson approach. The presymplectic approach leads to a reduced system whose iteration map is symplectic with r…

The paper discusses reducing Hamiltonian systems by scaling and standard symmetries, leading to Kirillov Hamiltonian systems.

We provide an alternative method for obtaining of compatible Poisson structures on Lie groups by means of the adjoint representations of Lie algebras. In this way, we calculate some compatible Poisson structures on four dimensional and nilpotent six dimensional symplectic real Lie groups. Then using Magri-Morosi's theo…

This paper studies Hamilton-Jacobi equations for magnetic systems with constraints.

SVD-based methods reduce computational cost for stochastic systems.

We describe a reduction process for symplectic principal $\mathbb{R}$-bundles in the presence of a momentum map. This type of structures plays an important role in the geometric formulation of non-autonomous Hamiltonian systems. We apply this procedure to the standard symplectic principal $\mathbb{R}$-bundle associated…

Classifies almost-toric systems in four dimensions.

A novel symplectic integrator for Hamiltonian equations on $S_2^n \times T^{\ast} \RR^m$ is developed and studied. Partitioned Runge--Kutta methods for Hamiltonian systems on products of Hamiltionian manifolds are studied, specifically, algebraic conditions for their symplecticity are derived.

This article is the second of a pair of articles about the Goldman symplectic form on the PGL(V)-Hitchin component of a closed, connected, oriented, hyperbolic surface S. We show that any ideal triangulation on S and any compatible bridge system determine a symplectic trivialization of the tangent bundle to the PGL(V)-…

Study of symplectic Stiefel and Grassmann manifolds with geodesics and applications.