The paper defines and proves equivalence of nonholonomic brackets in contact mechanical systems.
problem Nonholonomic constraints in contact geometry.
method Construct a general framework for non-holonomic constraints, define and prove equivalence of different nonholonomic brackets.
result All nonholonomic brackets coincide and one is an almost Jacobi bracket.
We reexamine the relation between contact structures on supermanifolds and supersymmetric mechanics in the superspace formulation. This allows one to use the language of contact geometry when dealing with the d = 1, N = 2 super-Poincare algebra.
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.
The paper analyzes errors in mechanical systems with external forces.
problem Error analysis of mechanical systems with external forces.
method Analysis of variational integrators with contact order r for discrete mechanical systems. result The contact order of the integrator is the same as the contact order of the original systems.
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.
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 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.
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 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.
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…
The study of Reeb dynamics on contact manifolds without periodic orbits.
problem Existence of periodic Reeb orbits on bm-contact manifolds. method Generalization of the Weinstein conjecture, proof of periodic orbits, existence of traps.
result In dimension 3, there are infinitely many periodic orbits on the critical set.
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.
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…
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.
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.
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.
We present a generally covariant approach to quantum mechanics in which generalized positions, momenta and time variables are treated as coordinates on a fundamental "phase-spacetime." We show that this covariant starting point makes quantization into a purely geometric flatness condition. This makes quantum mechanics …
The paper proves a nonholonomic version of Maupertuis-Jacobi principle and shows that nonholonomic trajectories minimize length.
problem Nonholonomic dynamics and their length minimization.
method Contact bundle formulation and geometric equivalence between problems.
result Regular solutions of nonholonomic mechanical problems are reparametrizations of geodesics with minimized Riemannian length.
Polynomial invariants classify molecular chains based on their contact arrangements.
problem No established invariants for molecular chains with both hard and soft contacts.
method Developed polynomial invariants for circuit topology of molecular chains.
result Polynomial invariants efficiently classify chains with various contact types.
Extends coisotropic embedding theorem to various geometric settings.
problem Regularization problem of singular Lagrangian systems.
method Generic methodology applied to cosymplectic, contact, and multisymplectic manifolds.
result Fundamental basis for applying results to regularization problem.
Generative model creates frictional surfaces from friction laws.
problem Designing frictional interfaces with prescribed behavior is challenging.
method Uses Variational Autoencoders (VAEs) to infer surface topographies from friction laws.
result Efficiently generates candidate topographies without contact simulations.
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.
Study tackles causal effects of close contact on MRSA infections from entangled treatment data.
problem Estimating causal effects of close contact on MRSA infections from observational data with entangled treatments.
method NEAT method that models treatment assignment mechanism and mitigates confounding biases.
result NEAT method effectively estimates causal effects from entangled treatment data.
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.
Paper tackles dynamic behavior of variable topology mechanisms, presenting new transition conditions.
problem Dynamic behavior of mechanisms with changing kinematic topology.
method Presented new transition conditions for variable topology mechanisms using projected motion equations and Voronets equations.
result Results show the dynamic behavior of joint locking in 3R and 6DOF mechanisms.
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.
Geometric quantization scheme for contact 3-manifolds models gravity.
problem Quantizing contact structures for geometric understanding.
method Explicit embedding and Stein extension to define quantum Hilbert space.
result Reeb vector field geodesic and Einstein gravity model.
In the last two decades, significant effort has been put in understanding and designing so-called structure-preserving numerical methods for the simulation of mechanical systems. Geometric integrators attempt to preserve the geometry associated to the original system as much as possible, such as the structure of the co…
The usual formulations of time-dependent mechanics start from a given splitting Y=R×M of the coordinate bundle Y→R. 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 …
Starting from the observation that a flying saucer is a nonholonomic mechanical system whose 5-dimensional configuration space is a contact manifold, we show how to enrich this space with a number of geometric structures by imposing further nonlinear restrictions on the saucer's velocity. These restrictions define cert…
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.
This work tackles force control for contact-rich manipulation tasks with rigid robots using RL.
problem Challenges in working with real robotic hardware, especially position-controlled robots.
method Combines RL with traditional force control techniques, implementing parallel position/force control and admittance control.
result Validated methods on both simulation and real robot (UR3 e-series) for force control.
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.
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.
We give an elegant formulation of the structure equations (of Cartan) and the Bianchi identities in terms of exterior calculus without reference to a particular basis and without the exterior covariant derivative. This approach allows both structure equations and the Bianchi identities to be expressed in terms of forms…
A theory of double affine and special double affine bundles, i.e. differential manifolds with two compatible (special) affine bundle structures, is developed as an affine counterpart of the theory of double vector bundles. The motivation and basic examples come from Analytical Mechanics, where double affine bundles hav…
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.
The purpose of this article is to study co-dimension 2 iso-contact embeddings of closed contact manifolds. We first show that a closed contact manifold (M2n−1,ξM) iso-contact embeds in a contact manifold (N2n+1,ξN), provided M contact embeds in (N,ξN) with a trivial normal bundle and the contact s…
The paper defines and studies contact surgery numbers for contact 3-manifolds.
problem Understanding the minimal number of components of a surgery link describing a contact 3-manifold.
method Defined and studied various versions of contact surgery numbers, relating them to other invariants and computing specific cases.
result There exist infinitely many non-isotopic contact structures on certain manifolds that cannot be obtained by a single rational contact surgery from the standard tight contact 3-sphere.
Study weak quasi contact metric manifolds to generalize K-contact and Sasakian manifolds criteria.
problem Generalize K-contact and Sasakian manifolds criteria using weak quasi contact metric manifolds.
method Study weak quasi contact metric manifolds and generalize theorems for K-contact and Sasakian manifolds.
result Provide new criterions for K-contact and Sasakian manifolds in terms of curvature tensor and geometric objects.
Simplified construction of contact triad connection for analysis.
problem Analyzing contact instanton equation on contact triads.
method Characterization via almost contact structure and construction using almost contact moving frame.
result Simpler and more canonical construction of contact triad connection.
The paper examines conditions for contact surgeries on rational homology 3-spheres.
problem Conditions for contact surgeries on rational homology 3-spheres.
method Analyzes sufficient conditions for contact surgeries using Legendrian knots and links.
result Provides sufficient conditions for surgeries to have vanishing contact invariants or to be overtwisted.
We study the contact equivalence problem for toric contact structures on S3-bundles over S2. That is, given two toric contact structures, one can ask the question: when are they equivalent as contact structures while inequivalent as toric contact structures? In general this appears to be a difficult problem. To f…
Abstract: Survey on contact submanifolds.
problem Understanding contact submanifolds.
method None specified, survey of existing knowledge.
result Discussion of various contact submanifolds.
New contact structures on folded sums of contact mapping tori are tight under certain conditions.
problem Understanding tight contact structures on folded sums of contact mapping tori.
method Alternative bundle-theoretical construction and gluing process near the fold.
result Folded contact structures on folded sums of contact mapping tori are tight under specific conditions.
Study explores weak generalized K-contact structures in contact metric spaces.
problem Exploring weak generalized K-contact structures in contact metric spaces.
method Introducing a weak (κ,μ) condition and proving existence of K-contact and (κ,μ=2)-structures. result Existence of K-contact and (κ,μ=2)-structures under certain conditions on the Boeckx invariant. New contact structures detected by contact homology.
problem Detecting pseudo-Anosov flows in contact structures.
method Introducing pseudo-Anosov contact structures and using contact homology.
result Contact homology detects pseudo-Anosov flows and contact structures properties.
Local flatness theorem for paraquaternionic contact structures.
problem Local flatness of paraquaternionic contact manifolds.
method Defined paraquaternionic contact conformal curvature tensor and showed local flatness condition.
result Paraquaternionic contact conformal curvature vanishing implies local flatness.