A new method simplifies contact Hamiltonian mechanics.
problem Traditional contact Hamiltonian mechanics is complex.
method Introduces sections of line bundles over contact manifolds.
result Reduces contact Hamiltonian formalism to symplectic.
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48 results for “contact Hamiltonian mechanics”
The evolution operator connecting the Lagrangian and Hamiltonian formalisms for contact systemsmath-ph
This paper extends the evolution operator to contact mechanics, linking Lagrangian and Hamiltonian formulations.
problem Translating the evolution operator to contact mechanics for mechanical systems with dissipation.
method Using the evolution operator K to connect Lagrangian and Hamiltonian formalisms in contact mechanics.
result The evolution operator provides a geometric description of evolution equations and relates constraints.
Paper bridges quantum and classical mechanics for open systems.
problem Quantum open systems with bi-Lindblad structure.
method Develops a bridge between bi-Hamiltonian structures and GKSL formalism, introducing contact-compatible Lindblad generators.
result Provides a mathematical mechanism for semiclassical limit of quantum open systems.
We develop a new geometric framework suitable for dealing with Hamiltonian field theories with dissipation. To this end we define the notions of -contact structure and -contact Hamiltonian system. This is a generalization of both the contact Hamiltonian systems in mechanics and the -symplectic Hamiltonian syst…
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.
Generalizes Tulczyjew triples for contact manifolds in Hamiltonian and Lagrangian formalisms.
problem Tackles the need for a geometric tool in contact manifolds.
method Introduces a generalized Tulczyjew triple for contact manifolds.
result Contact Hamiltonians and Lagrangians as sections of line bundles determine dynamics on contact phase space.
Paper generalizes Hamiltonian mechanics using line bundles.
problem Mathematical foundations of measurand and units of measurement.
method Introduces line bundles over smooth manifolds as configuration spaces.
result Generalization successfully incorporates physical dimension and units.
On the role of geometry in statistical mechanics and thermodynamics I: Geometric perspectivecond-mat.stat-mech
The paper provides a geometric framework for understanding non-equilibrium thermodynamics.
problem Unclear geometric structure of GENERIC in non-equilibrium thermodynamics.
method Cotangent lifts of dynamics, splitting into holonomic and vertical representatives, and formulation within contact geometry.
result Physical meaning and explicit formulation of the second law of thermodynamics within evolution equations.
The paper adapts results for Reeb flows and Hamiltonian flows, showing all orbits are closed have identical periods.
problem Adapting results for Reeb flows and Hamiltonian flows with closed orbits.
method Adapting results from Geodesic circle foliations to Reeb and Hamiltonian flows.
result All orbits on connected contact manifolds with closed orbits have identical periods.
Reduces equations for contact mechanical systems on Lie groups by exploiting symmetries.
problem Contact mechanical systems on Lie groups with symmetries.
method Reduction process using Lie group actions and symmetries.
result Euler-Poincaré-Herglotz equations on the reduced phase space.
Paper introduces Eden bracket for nonholonomic systems.
problem Nonholonomic contact systems with dissipation.
method Contact Eden bracket for system evolution.
result Evolution of observables in constrained systems.
Study the commutativity of reduction and symplectification in contact Hamiltonian systems.
problem Understanding the commutativity of reduction and symplectification in contact Hamiltonian systems.
method Introduce symplectic and contact geometry, perform reduction via momentum map, analyze symplectification process.
result Commutativity relations between reduction and symplectification in contact Hamiltonian systems.
New framework models non-conservative stochastic processes without energy conservation constraints.
problem Existing Schrödinger Bridge methods are limited by energy-conservation assumptions.
method Introduces non-conservative generalized Schrödinger bridge (NCGSB) based on contact Hamiltonian mechanics.
result Contact Wasserstein geodesic (CWG) provides a broader class of real-world stochastic processes.
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.
Study on contact Hamiltonian functions for singular contact structures.
problem Understanding infinitesimal contact transformations on singular contact structures.
method Showed injectivity and provided an explicit local formula for the inverse map.
result Explicit local formula for the inverse map when contact structure has singularities of the first type.
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…
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.
A Hamilton-Jacobi theory for general dynamical systems, defined on fibered phase spaces, has been recently developed. In this paper we shall apply such a theory to contact Hamiltonian systems, as those appearing in thermodynamics and on geodesic flows in fluid mechanics. We first study the partial and complete solution…
Infinite-dimensional contact geometry explored.
problem Generalizing contact geometry to infinite dimensions.
method Generalization of cosymplectic, contact, and cocontact manifolds to infinite dimensions.
result Model examples of time-dependent and dissipative Hamiltonian systems calculated.
The usual formulations of time-dependent mechanics start from a given splitting of the coordinate bundle . From physical viewpoint, this splitting means that a reference frame has been chosen. Obviously, such a splitting is broken under reference frame transformations and time-dependent canonical …
We construct counterexamples to lifting properties of Hamiltonian and contact isotopies.
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.
Contact Lie systemsmath-ph
Contact Lie systems analyze integral curves of Hamiltonian vector fields.
problem Analyzing integral curves of Hamiltonian vector fields.
method Define and study contact Lie systems, including conservative systems.
result Develop Liouville theorems, contact reductions, and Gromov non-squeezing theorems.
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.
New Lie systems defined on -contact manifolds, with applications.
problem Understanding Lie systems on -contact manifolds.
method Distributional approach to -contact manifolds and Hamiltonian vector fields.
result Lie systems can be understood as Hamiltonian relative to a -contact manifold.
Unified geometric framework for integrability of conservative and dissipative systems.
problem Unified definition of integrability for both conservative and dissipative systems.
method Introducing Jacobi-Haantjes manifolds and contact-Haantjes manifolds to unify definitions.
result Equivalence of integrability in contact Hamiltonian systems and existence of Abelian extended Haantjes algebra.
I begin by giving a general discussion of completely integrable Hamiltonian systems in the setting of contact geometry. We then pass to the particular case of toric contact structures on the manifold . In particular we give a complete solution to the contact equivalence problem for a class of toric conta…
Invariant reduction preserves Poisson structures in PDEs.
problem Preserving Poisson structures in invariant solutions of PDEs.
method Invariant reduction applied to PDEs through Hamiltonian operators and Poisson bivectors.
result Inherited Poisson brackets match original systems up to sign.
Analyzes perturbed contact instantons with Legendrian boundary conditions using geometric analysis.
problem Analyzing nonlinear elliptic systems associated with contact Hamiltonian trajectories.
method Identifies correct action and energy functionals, develops elliptic regularity theory, and proves asymptotic convergence.
result Established convergence of perturbed contact instantons under finite energy hypothesis.
Lecture notes introduce contact complete integrability for odd-dimensional manifolds.
problem Integrability on odd-dimensional manifolds using contact geometry.
method Introduce contact geometry concepts, discuss contact Hamiltonian vector field, Jacobi bracket, and contact complete integrability.
result Two different notions of contact complete integrability coincide.
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.
Paper uses second-order differential geometry to study stochastic mechanics.
problem Stochastic differential equations and their symmetries.
method Develops second-order differential geometry to study symmetries of SDEs and constructs stochastic mechanics.
result Establishes stochastic Lagrangian and Hamiltonian mechanics and their relations with HJB equations.
Mechanical Hamiltonian systems with respect to linear Poisson structures and Jacobi-Reeb dynamicsmath.DG
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.
The paper connects a second order ODE to Sasakian structures and bi-Hamiltonian systems.
problem Defining and analyzing Sasakian structures associated with second order ODEs.
method Defining contact metric structures and Poisson structures, showing compatibility with bi-Hamiltonian systems.
result A compatible bi-Hamiltonian structure for the Reeb vector field is found, and conditions for the vanishing of the first Chern class are derived.
A new metriplectic system on contact manifolds is introduced for thermodynamic consistency.
problem Developing a thermodynamically consistent dynamical system on contact manifolds.
method Introducing a metriplectic dynamical system on the one-jet bundle .
result The metriplectic system is thermodynamically consistent, with and .
The study constructs K-contact manifolds with minimal closed Reeb orbits and provides conditions for their homeomorphism to spheres.
problem Understanding K-contact manifolds with minimal closed Reeb orbits and their homeomorphism properties.
method Using Boothby-Wang fibration and Hamiltonian torus actions, the study constructs and analyzes K-contact manifolds.
result The existence of K-contact manifolds with minimal closed Reeb orbits that are not homeomorphic to spheres and have unique cohomology rings.
This paper is concerned with the rational symplectic field theory in the Floer case. For this observe that in the general geometric setup for symplectic field theory the contact manifolds can be replaced by mapping tori of symplectic manifolds with symplectomorphisms. While the cylindrical contact homology is given by …
New geometric framework for non-conservative field theories with time-dependent terms.
problem Describing non-conservative field theories with explicit space-time dependence.
method Combining k-cosymplectic and k-contact formulations to develop Hamiltonian and Lagrangian formalisms.
result Illustrated with the nonlinear damped wave equation, demonstrating the new formalism's applicability.
Paper presents a new approach to continuum mechanics using port-Hamiltonian framework.
problem Geometric formulation of solid and fluid mechanics.
method Port-Hamiltonian framework, Dirac structures, Hamiltonian reduction theory.
result Systematic derivation of port-Hamiltonian models for solid and fluid mechanics.
Decomposition of symmetric tensor fields in the presence of a flat contact projective structuremath.DG
Let be an odd-dimensional Euclidean space endowed with a contact 1-form . We investigate the space of symmetric contravariant tensor fields on as a module over the Lie algebra of contact vector fields, i.e. over the Lie subalgebra made up by those vector fields that preserve the contact structure. If we cons…
Contact reductions explained through symplectic reductions.
problem Explaining contact reductions using symplectic methods.
method Lifting contact structures to symplectic covers and Hamiltonian actions.
result Contact structures are equivalent to certain symplectic structures.
In some previous papers, a Legendre duality between Lagrangian and Hamiltonian Mechanics has been developed. The (ρ,η)-tangent application of the Legendre bundle morphism associated to a Lagrangian L or Hamiltonian H is presented. Using that, a Legendre description of Lagrangian Mechanics and Hamiltonian Mechanics is d…
Study contact instantons and Legendrian links, proving energy inequalities.
problem Estimating Reeb-untangling energy of Legendrian submanifolds.
method Develop contact Hamiltonian geometry, introduce tame contact manifolds, construct moduli spaces, prove convergence results.
result Self Reeb-untangling energy of compact Legendrian submanifolds is greater than period gap.
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) …
Proves Weinstein's and Arnold's conjectures using contact instantons.
problem Proving Weinstein's and Arnold's conjectures in contact geometry.
method Existence of fundamental class in Legendrian contact instanton cohomology, evaluation transversality, and geometric construction of contactomorphisms.
result Proves Weinstein's and Arnold's conjectures in full generality.
We prove that the number of Reeb chords between a Legendrian submanifold and its contact Hamiltonian push-off is at least the sum of the -Betti numbers of the submanifold, provided that the contact isotopy is sufficiently small when compared to the smallest Reeb chord on the Legendrian. Moreover, the esta…
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.
In this article we develop a theory of contact systems with nonholonomic constraints. We obtain the dynamics from Herglotz's variational principle, by restricting the variations so that they satisfy the nonholonomic constraints. We prove that the nonholonomic dynamics can be obtained as a projection of the unconstraine…