Symplectic forms can be preserved under small deformations on Calabi-Yau manifolds.
problem Preserving symplectic forms under deformations on Calabi-Yau manifolds.
method Dynamical stability of symplectic curvature flow.
result Any small symplectic deformation of a Kähler form remains Kähler on a compact Calabi-Yau manifold.
New neural net learns time-reversible symplectic dynamics.
problem Lack of time-reversibility in neural networks for symplectic systems.
method Proposes a new neural network architecture for time-reversible symplectic systems.
result Demonstrates learning of time-reversible symplectic dynamics from data.
The paper explores the dynamics of composite symplectic Dehn twists with nonuniform hyperbolicity.
problem Understanding the dynamics and properties of composite symplectic Dehn twists.
method Analyzing the form of nonuniform hyperbolicity, growth of Floer cohomology, and classification of symplectic mapping classes.
result Composite symplectic Dehn twists exhibit positive topological entropy and exponential growth in Floer cohomology.
Survey on strong closing lemmas in Hamiltonian dynamics.
problem Understanding dynamics in Hamiltonian systems.
method Use spectral invariants in symplectic geometry.
result Proofs of strong closing lemmas in various dimensions.
SRNNs learn dynamics of physical systems from data.
problem Learning dynamics of complex, noisy Hamiltonian systems.
method SRNNs model Hamiltonian functions with neural networks, using symplectic integration and optimization.
result SRNNs reliably learn dynamics of complex and noisy Hamiltonian systems.
Proposes a contact dynamics framework using generalized geometries.
problem Contact dynamics and related geometries.
method Generalizes symplectic and Morse families to contact framework.
result Establishes contact Hamiltonian and Lagrangian Dynamics as Legendrian submanifolds.
Invariant measures found for contact Hamiltonian systems split into Reeb and Liouville dynamics.
problem Finding invariant measures for contact Hamiltonian systems.
method Splitting the system into Reeb and Liouville dynamics; using invariant measures and symplectic sandwiches.
result Invariant measure found for Reeb dynamics; characterization of Liouville dynamics invariant measure.
This paper develops a general method for constructing Poisson integrators.
problem Lack of a general theory for Poisson integrators due to geometric challenges.
method Adapting structural results about symplectic realizations to create geometric approximations.
result Developed a general approach for constructing geometric integrators on Poisson manifolds.
Proves Arnold conjecture for singular symplectic manifolds using novel techniques.
problem Hamiltonian dynamics on singular symplectic manifolds.
method Introducing smooth symplectic forms to singular symplectic structures under mild conditions, using Floer homology.
result Proves a lower bound on the number of 1-periodic Hamiltonian orbits for b2m-symplectic manifolds. Book introduces Hofer's metric on symplectic diffeomorphisms.
problem Understanding dynamics and growth in symplectic geometry.
method Introduces Hofer's metric and analyzes its properties.
result Provides insights into the structure of symplectic diffeomorphisms.
SSINNs learn Hamiltonian systems from data with interpretable, low-memory models.
problem Learning Hamiltonian dynamical systems from data efficiently and accurately.
method Combines fourth-order symplectic integration with sparse regression for a learned Hamiltonian.
result Outperforms state-of-the-art techniques in system prediction and energy conservation.
New algebraic approach for approximating Hamiltonian dynamics.
problem Approximating Hamilton-Jacobi solutions on symplectic groupoids.
method Introducing a pre-Lie algebra and Butcher trees for symplectic groupoids.
result New class of Poisson integrators for Hamiltonian dynamics.
We consider trivializations of second iterated bundles of a Lie group that preserve lifted group structures. With such a trivialization, we elaborate Hamiltonian dynamics on cotangent, Lagrangian dynamics on tangent bundles and, both Hamiltonian and Lagrangian dynamics on Tulczyjew's symplectic space which is tangent o…
Stability of Type IIA flow ensures Kähler properties of Calabi-Yau 3-folds.
problem Ensuring the Kähler property of Calabi-Yau 3-folds under symplectic deformations.
method Established dynamical stability of Type IIA flow near stationary points.
result Stability of Type IIA flow ensures the stability of Kähler properties under symplectic deformations.
New Hamiltonian Monte Carlo method for non-canonical dynamics.
problem Incompatibility of canonical symplectic structure with non-canonical dynamics.
method Developed a framework for Hamiltonian Monte Carlo using non-canonical symplectic structures with implicit integration.
result Non-canonical Hamiltonian Monte Carlo provides sampling advantages.
Geometrically characterizes virtual nonlinear nonholonomic constraints using symplectic methods.
problem Characterizing virtual nonlinear nonholonomic constraints geometrically.
method Geometric characterization using symplectic structures and Chetaev equations.
result A unique control law exists to satisfy virtual constraints, and closed-loop dynamics are projections of uncontrolled dynamics.
While symplectic manifolds have no local invariants, they do admit many global numerical invariants. Prominent among them are the so-called symplectic capacities. Different capacities are defined in different ways, and so relations between capacities often lead to surprising relations between different aspects of sympl…
On a symplectic manifold a family of generalized Poisson brackets associated with powers of the symplectic form is studied. The extreme cases are related to the Hamiltonian and Liouville dynamics. It is shown that the Dirac brackets can be obtained in a similar way.
The study analyzes momentum-based optimization algorithms from dynamical systems perspective.
problem Understanding convergence rates of momentum-based optimization algorithms.
method Exploits dynamical systems, control theory, and symplectic perspectives to analyze convergence rates.
result Provides closed-form expressions relating algorithm parameters to convergence rates.
Abstract: Connections between Lie algebras and symplectic nilmanifolds explored.
problem Understanding the relationship between Lie algebras and symplectic nilmanifolds.
method Central extensions of Lie algebras, dynamical systems, and symplectic nilmanifolds constructed.
result Covering Lie groups for symplectic nilmanifolds can have any rank as solvable Lie groups.
Taking configuration space as a Lie group, the trivialized Euler-Lagrange and Hamilton's equations are obtained and presented as Lagrangian submanifolds of the trivialized Tulczyjew's symplectic space. Euler-Poincaré and Lie-Poisson equations are presented as Lagrangian submanifolds of the reduced Tulczyjew's symplecti…
Develops computational methods for simulating rigid body dynamics on SO(3).
problem Simulating rotational dynamics of rigid bodies on SO(3).
method Discrete Mechanics, Variational Integrators, Newton-Raphson algorithm.
result Preserves symplectic structure of SO(3) manifold dynamics.
Characterizes Anosov flows in 3D using symplectic and contact geometry.
problem Understanding Anosov flows in 3D.
method Purely contact and symplectic geometric methods.
result Characterization of Anosov flows based on Reeb flows and underlying (bi)-contact structures.
Study shows knots in 3-manifolds imply 3-sphere if certain conditions met.
problem Characterizing 3-manifolds based on knot properties.
method Analyzing contact forms and their behavior near boundaries.
result Knots in 3-manifolds imply the manifold is a 3-sphere under specific conditions.
In this paper, we will see that the symplectic creed by Weinstein "everything is a Lagrangian submanifold" also holds for Hamilton-Poincaré and Lagrange-Poincaré reduction. In fact, we show that solutions of the Hamilton-Poincaré equations and of the Lagrange-Poincaré equations are in one-to-one correspondence with dis…
The paper extends Gromov's non-squeezing theorem to deformed symplectic forms.
problem Extending Gromov's non-squeezing theorem to deformed symplectic forms.
method Trap idea for holomorphic curves analogous to dynamical systems.
result The classical Gromov argument breaks down for deformed forms.
Investigates the rotating Kepler problem for energy values ≤ -3/2.
problem Understanding periodic orbits and symplectic structures in rotating celestial mechanics.
method Ligon-Schaaf and Levi-Civita symplectic regularizations, special concave toric domain construction.
result Identification of a special concave toric domain (SCTD) for the RKP phase space.
We describe the reduction procedure for a symplectic Lie algebroid by a Lie subalgebroid and a symmetry Lie group. Moreover, given an invariant Hamiltonian function we obtain the corresponding reduced Hamiltonian dynamics. Several examples illustrate the generality of the theory.
Study magnetic Hamiltonian systems with constraints, deriving Hamilton-Jacobi equations.
problem Understanding dynamics of controlled magnetic Hamiltonian systems with constraints.
method Defined CMH system, derived Hamilton-Jacobi equations for different constraints.
result Invariant solutions of Hamilton-Jacobi equations under CMH-equivalence.
Proposes NSSNNs to predict nonseparable Hamiltonian systems.
problem Predicting nonseparable Hamiltonian systems with coupled dynamics.
method Augmented symplectic time integrator to decouple position and momentum.
result Long-term, accurate, and robust predictions for large-scale Hamiltonian systems.
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…
Floer homology theories solve complex dynamics problems.
problem Proving conjectures in symplectic and contact dynamics.
method Construction and application of various Floer homologies.
result Floer homologies have proven to be powerful tools in dynamics and topology.
The Weinstein conjecture is extended to a new class of manifolds.
problem Existence of Reeb 2-curves in locally conformally symplectic manifolds.
method Extended Gromov-Witten theory and elliptic curve counts.
result Partial verification of the conjecture in higher dimensions.
A new ML method predicts long-time-step molecular dynamics, preserving symplectic and time-reversible properties.
problem Limited computational efficiency in long-time-step molecular dynamics simulations.
method Learning data-driven structure-preserving maps to generate long time-step classical dynamics.
result The method eliminates artifacts like lack of energy conservation and loss of equipartition.
We introduce the notion of a symplectic Lie affgebroid and their Lagrangian submanifolds in order to describe the Lagrangian (Hamiltonian) dynamics on a Lie affgebroid in terms of this type of structures. Several examples are discussed.
We address the question of duality for the dynamical Poisson groupoids of Etingof and Varchenko over a contractible base. We also give an explicit description for the coboundary case associated with the solutions of the classical dynamical Yang-Baxter equation on simple Lie algebras as classified by the same authors. O…
Shearing deformations in Hitchin representations are computed for a symplectic form.
problem Computing symplectic form pairings for Hitchin representations.
method Shearing deformations of Hitchin representations.
result Pairings of shearing deformations computed for the Atiyah-Bott-Goldman symplectic form.
A new MCMC method on Lie groups using irreversible dynamics.
problem Sampling from distributions on Lie groups efficiently.
method Constructing an irreversible HMC-like algorithm on Lie groups using dissipation fields and symplectic integrators.
result The method can recover HMC as a special case and is faster than reversible MCMC.
In this paper we develope, in a geometric framework, a Hamilton-Jacobi Theory for general dynamical systems. Such a theory contains the classical theory for Hamiltonian systems on a cotangent bundle and recent developments in the framework of general symplectic, Poisson and almost-Poisson manifolds (including some appr…
This paper studies Hamilton-Jacobi equations for magnetic systems with constraints.
problem Understanding dynamics of magnetic systems with geometric constraints.
method Developed Hamilton-Jacobi equations for magnetic systems with nonholonomic constraints.
result Revealed relationships between magnetic structures, constraints, and dynamics.
Investigates metric degeneracies on symplectic leaves using a generalized gradient flow.
problem Degeneracies in metrics on symplectic leaves of Poisson manifolds.
method Introduces the generalized double bracket (GDB) vector field to generalize gradient dynamics.
result Identifies admissible regions where the double bracket metric remains non-degenerate on symplectic leaves, enabling GDB as a gradient flow.
This paper simplifies complex nonholonomic systems using momentum map reduction.
problem Reducing complex nonholonomic systems with symmetries.
method Using nonholonomic momentum bundle map and gauge transformation.
result Reduced manifolds are Chaplygin-type leaves with an almost symplectic form.
Invites study of contact structures and Reeb flows dynamics.
problem Relating dynamics of two Reeb flows of the same contact structure.
method Gathers results and poses many questions and conjectures.
result Many new questions and conjectures posed.
We introduce and study the basic notion of polarized Poisson manifolds generalizing the classical case of Poisson manifolds and extend this last notion for the k−% symplectic stuctures. And also, we show that for any polarized Hamiltonian map, the associated Nambu's dynamical system and polarized Hamiltonian system…
SymODEN learns physical systems dynamics from data.
problem Learning dynamics of physical systems from limited data.
method Physics-informed deep learning with Hamiltonian dynamics and control.
result SymODEN generalizes well with fewer samples and interpretable models.
In some previous papers, a geometric description of Lagrangian Mechanics on Lie algebroids has been developed. In the present paper, we give a Hamiltonian description of Mechanics on Lie algebroids. In addition, we introduce the notion of a Lagrangian submanifold of a symplectic Lie algebroid and we prove that the Lagr…
Proposes a new Langevin flow approach for VAEs.
problem Difficulty in constructing low variance ELBO for VAEs with large datasets.
method Integrates Langevin dynamic with quasi-symplectic integrator to improve posterior estimation.
result Shows theoretical and practical effectiveness compared to gradient flow-based methods.
The aim of this paper is to study the relationship between Hamiltonian dynamics and constrained variational calculus. We describe both using the notion of Lagrangian submanifolds of convenient symplectic manifolds and using the so-called Tulczyjew's triples. The results are also extended to the case of discrete dynamic…