The study examines stability of Hamiltonian Poisson integrators on both integrable and non-integrable systems.
problem Investigating stability properties of Hamiltonian Poisson integrators.
method Examples of Lotka-Volterra dynamics and numerical investigations of a non-integrable system are used.
result The existence of a modified Hamiltonian is crucial for the stability of Hamiltonian Poisson integrators.
Kernel methods accurately predict Hamiltonian systems from data.
problem Data-driven simulation of Hamiltonian systems.
method Two-step and one-step kernel-based methods for identifying and forecasting Hamiltonian systems.
result Framework achieves accurate, data-efficient predictions across various benchmark systems.
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.
The paper discusses reducing Hamiltonian systems by scaling and standard symmetries, leading to Kirillov Hamiltonian systems.
problem Reduction of symplectic Hamiltonian systems by scaling and standard symmetries.
method Proof of Kirillov Hamiltonian systems and equivalence of reductions.
result Equivalent Kirillov Hamiltonian systems from different reduction orders.
Survey of recent developments in symmetric reductions and controls for Hamiltonian systems.
problem Understanding the internal relationships of geometric structures and controls in Hamiltonian systems with symmetry.
method Survey and introduction of recent developments in controlled Hamiltonian systems with symmetry.
result Reveals the relationships between geometric structures, nonholonomic constraints, dynamical vector fields, and controls.
New integrators preserve geometric structure in Hamiltonian systems.
problem Preserving geometric structure in Hamiltonian systems on Jacobi manifolds.
method Combining Poissonization and symplectic bi-realizations to construct structure-preserving integrators.
result Explicit construction and application of Jacobi Hamiltonian integrators.
This work generalizes Hamiltonian mechanics using closed differential forms.
problem Hidden invariants in classical Hamiltonian mechanics.
method Establishes a novel correspondence between generalized Hamiltonian mechanics and multisymplectic geometry.
result Key theorems linking classical and generalized Hamiltonian systems.
In this paper, for a variety of nonholonomic (reducible) Hamiltonian systems, we first give to various distributional Hamiltonian systems, by analyzing carefully the dynamics and structures of the nonholonomic Hamiltonian systems. Secondly, we derive precisely the geometric constraint conditions of the induced distribu…
Generalizes energy-momentum method for non-autonomous Hamiltonian systems.
problem Stability analysis of non-autonomous Hamiltonian systems with symmetries.
method Develops a new approach to relative equilibrium points and stability conditions for non-autonomous systems.
result Conditions ensuring stability of relative equilibrium points in non-autonomous Hamiltonian systems.
Arnold-Liouville systems cannot be bi-Hamiltonian generically.
problem The bi-Hamiltonian structure of Arnold-Liouville systems.
method Proving that a specific class of smooth functions is a meagre subset for the Fréchet topology, which implies Arnold-Liouville systems cannot be bi-Hamiltonian.
result Generically, Arnold-Liouville systems cannot be bi-Hamiltonian.
WSINDy identifies reduced Hamiltonian systems from particle interactions.
problem Coarse-graining Hamiltonian dynamics with approximate symmetries.
method WSINDy algorithm applied to Hamiltonian systems with timescale separation.
result WSINDy successfully identifies reduced Hamiltonian systems from noisy data.
Paper connects dynamics of mechanical systems to Reeb dynamics.
problem Understanding dynamics in mechanical systems with Poisson structures.
method Using Jacobi bundle metrics and linear Poisson structures.
result Extends classical results on Reeb dynamics to mechanical systems.
Develops integrators for contact Hamiltonian systems preserving geometric structure.
problem Creating integrators for dissipative systems with geometric structure.
method Structure-preserving splitting framework based on exact-contact subflows.
result Local universality of contact splitting integrators.
Develops control and observer methods for complex systems.
problem Controlling and observing infinite-dimensional systems with boundary actuation.
method Energy-Casimir method and port-Hamiltonian system representation.
result Control law and observer designed for Kirchhoff-Love plate example.
Study reveals new geometric structures for magnetic field Hamiltonian systems.
problem Understanding Hamiltonian systems in magnetic fields.
method Investigation of symplectic-Haantjes geometry.
result Non-trivial symplectic-Haantjes manifolds found.
Gaussian process model learns Hamiltonian systems from noisy data.
problem Learning Hamiltonian systems from long, noisy trajectories.
method Efficient decoupled parameterisation, energy-conserving shooting method.
result Robust inference from short and long trajectories.
Researchers present and compare different representations of dissipative Hamiltonian DAE systems.
problem Understanding and transforming dissipative Hamiltonian DAE systems.
method Global geometric and algebraic points of view, translations between representations, characterizations, and numerical methods for computing structural information.
result A general DAE system can be transformed into a dissipative Hamiltonian or port-Hamiltonian DAE system.
Study discretizes Dirac and port-Hamiltonian systems using manifolds.
problem Discretization of Dirac and port-Hamiltonian systems.
method Retraction and discretization maps on manifolds for Dirac structures, applied to port-Hamiltonian systems.
result Numerical integrators for port-Hamiltonian systems derived from discretization techniques.
New product structures encode superintegrable Hamiltonian systems in Euclidean spaces.
problem Encoding superintegrable Hamiltonian systems using product structures.
method Introducing commutative and associative product structures on Euclidean spaces of dimension at least three, satisfying specific conditions.
result All abundant superintegrable Hamiltonian systems on Euclidean space of dimension at least three arise from these product structures.
In this paper we introduce the concept of Hamiltonian system in the canonical and Poisson settings. We will discuss the quantization of the Hamiltonian systems in the Poisson context, using formal deformation quantization and quantum group theories.
Proves local bi-integrability of bi-Hamiltonian systems via bi-Poisson reduction.
problem Local bi-integrability of bi-Hamiltonian systems.
method Bi-Poisson reduction to prove local bi-integrability.
result Constructs a complete set of functions in bi-involution for bi-Hamiltonian systems.
Reduces symplectic Hamiltonian systems to contact systems, realizing Poincaré's dream.
problem Scaling symmetries in Hamiltonian systems.
method Contact reduction of symplectic Hamiltonian systems.
result Generically possible reduction to contact Hamiltonian systems, reducing inputs needed.
Abstract: Study Hamiltonian systems on almost cosymplectic manifolds, extending contact Hamiltonian systems.
problem Extend Hamiltonian systems to almost cosymplectic manifolds.
method Determine Hamiltonian vector field on odd-dimensional almost cosymplectic manifolds.
result Extend equations of motion to generalized transitive almost cosymplectic structures.
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.
The study analyzes stochastic Lie systems and their applications in various models.
problem Analyzing stochastic differential equations on manifolds.
method Coalgebra method for Hamiltonian stochastic Lie systems.
result New examples of stochastic Lie systems and Hamiltonian stochastic Lie systems are analyzed.
Develops integrators for Hamiltonian systems in Jacobi manifolds.
problem Modeling conservative systems with dissipative and thermodynamic phenomena.
method Constructs structure-preserving integrators for Hamiltonian systems in Jacobi manifolds.
result Proposes a numerical integration technique compatible with Jacobi dynamics.
The Eisenhart lift connects Hamiltonian systems to geodesics in pp-wave spacetimes.
problem Studying the stability and dynamics of Hamiltonian systems.
method Eisenhart lift to pp-wave spacetimes and conformal classes of ODEs.
result Existence of a constant of the motion generalizing conservation of energy.
In this paper, we elucidate the key role played by the cosymplectic geometry in the theory of time dependent Hamiltonian systems. In particular, we generalize the cosymplectic structures to time-dependent Nambu-Poisson Hamiltonian systems and corresponding Jacobi's last multiplier for 3D systems. We illustrate our cons…
We develop variational integrators from discrete Hamiltonian systems with external forces.
problem Creating accurate discrete models of continuous Hamiltonian systems.
method Constructing discrete Hamiltonian systems with external forces, analyzing symplectic structure, and combining methods to build variational integrators.
result We derive variational integrators that approximate continuous Hamiltonian systems with high accuracy.
In this paper, from the viewpoint of completeness of Marsden-Weinstein reduction, we illustrate how to give the definitions of a controlled Hamiltonian (CH) system and a reducible controlled Hamiltonian system with symmetry; and how to describe the dynamics of a CH system and the controlled Hamiltonian equivalence; as …
Study investigates Hamiltonian systems in fibered almost-symplectic manifolds.
problem Existence and properties of non-vertical fully-Hamiltonian vector fields in almost symplectic manifolds with Lagrangian fibrations.
method Investigates vector fields in 2n-dimensional almost symplectic manifolds with Lagrangian fibrations, focusing on partially-Hamiltonian and fully-Hamiltonian vector fields.
result Non-vertical fully-Hamiltonian vector fields exist under certain genericity conditions and can be reduced to families of symplectic-Hamiltonian vector fields.
The application of the Legendre transformation to a hyperregular Lagrangian system results in a Hamiltonian vector field generated by a Hamiltonian defined on the phase space of the mechanical system. The Legendre transformation in its usual interpretation can not be applied to homogeneous Lagrangians found in relativi…
We study bi-Hamiltonian systems of hydrodynamic type with non-singular (semisimple) non-local bi-Hamiltonian structures and prove that such systems of hydrodynamic type are diagonalizable. Moreover, we prove that for an arbitrary non-singular (semisimple) non-locally bi-Hamiltonian system of hydrodynamic type, there ex…
Paper presents a new port-Hamiltonian model for vehicle manipulators.
problem Complex mechanical systems' energy flow and conservation.
method Derives port-Hamiltonian dynamics from Hamiltonian reduction theory.
result Establishes mathematical equivalence with existing formulations.
Given a Poisson structure (or, equivalently, a Hamiltonian operator) P, we show that its Lie derivative Lτ(P) along a vector field τ defines another Poisson structure, which is automatically compatible with P, if and only if [Lτ2(P),P]=0, where [⋅,⋅] is the Schouten bracket. We further prove that…
Differentiable simulations control molecular Hamiltonians for desired outcomes.
problem Control and learning of molecular Hamiltonians for desired outcomes.
method Differentiable simulations to differentiate Hamiltonians with respect to target observables.
result Control and learning of molecular Hamiltonians for desired outcomes.
The paper extends Marsden-Weinstein reduction to mechanical presymplectic structures for time-dependent Hamiltonian systems.
problem Limitations of Marsden-Weinstein reduction for cosymplectic structures in time-dependent Hamiltonian systems.
method Developed Marsden-Weinstein reduction for mechanical presymplectic structures.
result Mechanical presymplectic structures provide a more suitable framework for time-dependent Hamiltonian systems than cosymplectic structures.
In this paper an approach is proposed to represent a class of dissipative mechanical systems by corresponding infinite-dimensional Hamiltonian systems. This approach is based upon the following structure: for any non-conservative classical mechanical system and arbitrary initial conditions, there exists a conservative …
We develop a new geometric framework suitable for dealing with Hamiltonian field theories with dissipation. To this end we define the notions of k-contact structure and k-contact Hamiltonian system. This is a generalization of both the contact Hamiltonian systems in mechanics and the k-symplectic Hamiltonian syst…
Proves local bi-integrability of bi-Hamiltonian systems on real smooth manifolds.
problem Proving local bi-integrability of bi-Hamiltonian systems on real smooth manifolds.
method Proves bi-integrability by constructing a complete set of functions in bi-involution and showing differentials can realize any bi-Lagrangian subspace.
result Bi-Hamiltonian systems are locally bi-integrable on real smooth manifolds.
We investigate multi-dimensional Hamiltonian systems associated with constant Poisson brackets of hydrodynamic type. A complete list of two- and three-component integrable Hamiltonians is obtained. All our examples possess dispersionless Lax pairs and an infinity of hydrodynamic reductions.
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.
In this paper, we consider nonlinear PDEs in a port-Hamiltonian setting based on an underlying jet-bundle structure. We restrict ourselves to systems with 1-dimensional spatial domain and 2nd-order Hamiltonian including certain dissipation models that can be incorporated in the port- Hamiltonian framework by means of a…
An efficient method to construct Hamiltonian structures for nonlinear evolution equations is described. It is based on the notions of variational Schouten bracket and l*-covering. The latter serves the role of the cotangent bundle in the category of nonlinear evolution PDEs. We first consider two illustrative examples …
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.
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…
A new description, different by the classical theory of Hamiltonian Mechanics, in the general framework of generalized Lie algebroids is presented. In the particular case of Lie algebroids, new and important results are obtained. We present the \emph{dual mechanical systems} called by use, \emph{dual mechanical}$(ρ,η) …
Meta learning enables cross-domain Hamiltonian dynamics.
problem Adapting to new physical systems with different laws.
method Graph Neural Network (GNN) with meta learning.
result Unified Hamiltonian representation across multiple system domains.