arXiv research

Paper proves equivalence of two Floer theories using pearly trees and Hamiltonian flows.

Develops Hamiltonian Score Matching and Generative Flows for machine learning.

The paper introduces a geometric flow for Lagrangian submanifolds that preserves Hamiltonian isotopy.

Higher KdV flows on spaces of closed equicentroaffine plane curves are studied and it is shown that the flows are described as certain multi-Hamiltonian systems on the spaces. Multi-Hamiltonian systems describing higher mKdV flows are also given on spaces of closed Euclidean plane curves via the geometric Miura transfo…

Paper extends port-Hamiltonian model to include internal energy for compressible and incompressible flow.

Let (Σ, ω) be a compact Riemann surface with constant curvature c. In this work, we proved that the mean curvature flow of a given Hamiltonian diffeomorphism on Σ provides a smooth path in Ham(Σ), the group of all Hamiltonian diffeomorphisms of Σ. This result gives a proof, in the case of graph of Hamiltonian diffeomor…

We prove a spectral flow formula for one-parameter families of Hamiltonian systems under homoclinic boundary conditions, which relates the spectral flow to the relative Maslov index of a pair of curves of Lagrangians induced by the stable and unstable subspaces, respectively. Finally, we deduce sufficient conditions fo…

Paper develops a new fluid flow model with energy exchange through boundaries.

New method reduces Bayesian inference time by optimizing Hamiltonian flows.

Study integrability of geodesic flow on specific Lie groups.

In this article we prove two formulas for the topological entropy of an F-optical Hamiltonian flow induced by a C^{\infty} Hamiltonian, where F is a Lagrangian distribution. In these formulas, we calculate the topological entropy as the exponential growth rate of the average of the determinant of the differential of th…

The article studies Hamiltonian flows on surface group representations induced by invariant multi-functions.

In [LW], we construct examples of two-dimensional Hamiltonian stationary self-shrinkers and self-expanders for Lagrangian mean curvature flows, which are asymptotic to the union of two Schoen-Wolfson cones. These self-shrinkers and self-expanders can be glued together to yield solutions of the Brakke flow - a weak form…

We classify all Hamiltonian stationary Lagrangian surfaces in complex Euclidean plane which are self-similar solutions of the mean curvature flow.

The curvature and the reduced curvature are basic differential invariants of the pair (Hamiltonian system, Lagrange distribution) on the symplectic manifold. It is shown that the negativity of the reduced curvature implies the hyperbolicity of any compact invariant set of the Hamiltonian flow restricted to a prescribed…

In this paper we generalize examples of Hamiltonian stationary Lagrangian submanifolds constructed by Lee and Wang in $\mathbb{C}^m$ to toric almost Calabi-Yau manifolds. We construct examples of weighted Hamiltonian stationary Lagrangian submanifolds in toric almost Calabi-Yau manifolds and solutions of generalized La…

The present paper is a review of counterexamples to the ``Hamiltonian Seifert conjecture'' or, more generally, of examples of Hamiltonian systems having no periodic orbits on a compact energy level. We begin with the discussion of the ``classical'' and volume--preserving Seifert conjectures. Then a construction of coun…

We study the determination of the second-order normal form for perturbed Hamiltonians $H_ε=H_0 +εH_1 +\frac{ε^2}{2} H_2$, relative to the periodic flow of the unperturbed Hamiltonian $H_0$. The formalism presented here is global, and can be easily implemented in any CAS. We illustrate it by means of two examples: the H…

The main result in this paper is the $C^{\infty}$ closing lemma for a large family of Hamiltonian flows on $4$-dimensional symplectic manifolds, which includes classical Hamiltonian systems. First we prove the $C^{\infty}$ closing lemma and the $C^r$ general density theorem for geodesic flows on closed Finsler surfaces…

For Hamiltonian flows we establish the existence of periodic orbits on a sequence of level sets approaching a Bott-nondegenerate symplectic extremum of the Hamiltonian. As a consequence, we show that a charge on a compact manifold with a nondegenerate (i.e. symplectic) magnetic field has periodic orbits on a sequence o…

It is known that the Schrödinger flow on a complex Grassmann manifold is equivalent to the matrix non-linear Schrödinger equation and the Ferapontov flow on a principal Adjoint U(n)-orbit is equivalent to the $n$-wave equation. In this paper, we give a systematic method to construct integrable geometric curve flows on …

Study of intersections in Hamiltonian orbits on cotangent bundles.

We classify the periodic Hamiltonian flows on compact four dimensional symplectic manifolds up to isomorphism of Hamiltonian S^1 spaces. Additionally, we show that all these spaces are Kaehler, that every such space is obtained from a simple model by a sequence of symplectic blowups, and that if the fixed points are is…

The paper adapts results for Reeb flows and Hamiltonian flows, showing all orbits are closed have identical periods.

The chaotic geodesic flow on a jet space is non-integrable.

The paper explores the geometric properties of fluid flows and their symmetries.

Hamiltonian dynamical systems tend to have infinitely many periodic orbits. For example, for a broad class of symplectic manifolds almost all levels of a proper smooth Hamiltonian carry periodic orbits. The Hamiltonian Seifert conjecture is the existence problem for regular compact energy levels without periodic orbits…

Study shows no periodic geodesics in jet space.

HF-opt uses Hamiltonian dynamics to optimize functions, achieving accelerated rates with randomized integration time.

The paper studies geometric Airy curve flows on R^n and their properties.

We prove that a topological contact isotopy uniquely defines a topological contact Hamiltonian. Combined with previous results from [MS11], this generalizes the classical one-to-one correspondence between smooth contact isotopies and their generating smooth contact Hamiltonians and conformal factors to the group of top…

For integrable Hamiltonian systems with two degrees of freedom whose Hamiltonian vector fields have incomplete flows, an analogue of the Liouville theorem is established. A canonical Liouville fibration is defined by means of an "exact" 2-parameter family of flat polygons equipped with certain pairing of sides. For the…

Minimal Lagrangians in certain curved spaces are stable under specific flows.

Quantum computers can simulate flow models efficiently.

We develop the differential geometric and geometric analytic studies of Hamiltonian systems. Key ingredients are the curvature operator, the weighted Laplacian, and the associated Riccati equation. We prove the appropriate generalizations of Bochner--Weitzenböck formula and Laplacian comparison theorem, and study the h…

Classifies geodesic flows on projective plane with potential field.

This paper shows how pseudo-Anosov flows represent stable Hamiltonian classes and limits the ways 3-manifolds can be obtained from knots.

Study of Hamiltonian flows on character varieties for self-intersecting curves.

Paper presents a new port-Hamiltonian model for vehicle manipulators.

We study a family of fermionic extensions of the Camassa-Holm equation. Within this family we identify three interesting classes: (a) equations, which are inherently hamiltonian, describing geodesic flow with respect to an H^1 metric on the group of superconformal transformations in two dimensions, (b) equations which …

In this paper, we generalize several results for the Hamiltonian stability and the mean curvature flow of Lagrangian submanifolds in a Kähler-Einstein manifold to more general Kähler manifolds including a Fano manifold equipped with a Kähler form $ω\in 2πc_1(M)$ by using the methodology proposed by T. Behrndt. Namely, …

We present a generalization of the Nambu mechanics on the base of Liouville's theorem. We prove that the Poisson structure of an n-dimensional multisymplectic phase space is induced by (n-1)-Hamiltonian k-vector field seach of which requires introduction of k-Hamiltonians.

On the one hand, we prove that the Clifford torus in $\mathbb{C}^2$ is unstable for Lagrangian mean curvature flow under arbitrarily small Hamiltonian perturbations, even though it is Hamiltonian $F$-stable and locally area minimising under Hamiltonian variations. On the other hand, we show that the Clifford torus is r…

The theory of differential forms began with a discovery of Poincare who found conservation laws of a new type for Hamiltonian systems - The Integral Invariants. Even in the absence of non-trivial integrals of motion, there exist invariant differential forms: a symplectic two-form, or a contact one-form for geodesic flo…

The paper extends Vlasov kinetic theory to time-dependent dynamics using cosymplectic and cocontact manifolds.

New methods improve efficiency of sampling algorithms for complex systems.

This paper introduces equivariant hamiltonian flows, a method for learning expressive densities that are invariant with respect to a known Lie-algebra of local symmetry transformations while providing an equivariant representation of the data. We provide proof of principle demonstrations of how such flows can be learnt…

The Hamiltonian formalism plays a central role in classical and quantum physics. Hamiltonians are the main tool for modelling the continuous time evolution of systems with conserved quantities, and they come equipped with many useful properties, like time reversibility and smooth interpolation in time. These properties…