A solution for the Weinstein's Problem in the general framework of generalized Lie algebroids is the target of this paper. We present the mechanical systems called by use, mechanical (?; ?)-systems, Lagrange mechanical (?; ?)-systems or Finsler mechanical (?; ?)-systems and we develop their geometries. We obtain the ca…
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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.
This paper considers control systems defined on Lie algebroids. After deriving basic controllability tests for general control systems, we specialize our discussion to the class of mechanical control systems on Lie algebroids. This class of systems includes mechanical systems subject to holonomic and nonholonomic const…
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) …
In this paper, we consider a generalization of variational calculus which allows us to consider in the same framework different cases of mechanical systems, for instance, Lagrangian mechanics, Hamiltonian mechanics, systems subjected to constraints, optimal control theory and so on. This generalized variational calculu…
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 for discrete mechanical systems.
result The contact order of the integrator is the same as the contact order of the original systems.
The purpose of this paper is describe Lagrangian Mechanics for constrained systems on Lie algebroids, a natural framework which covers a wide range of situations (systems on Lie groups, quotients by the action of a Lie group, standard tangent bundles...). In particular, we are interested in two cases: singular Lagrangi…
Geometry of Mechanicsmath-ph
Studies geometric mechanics for autonomous and nonautonomous systems.
problem Understanding the geometric basis of mechanics.
method Geometric descriptions, Lagrangian, Hamiltonian, unified formalisms, symmetries, variational principles.
result Characterization of dynamical systems' properties and characteristics.
The paper simplifies complex mechanical systems with external forces.
problem Analyzing symmetric discrete mechanical systems with external forces.
method Lagrangian reduction and reconstruction for principal bundles.
result Evolution of momentum maps and Poisson structures under different conditions.
We design two mechanisms for the recommender system to collect user ratings. One is modified Laplace mechanism, and the other is randomized response mechanism. We prove that they are both differentially private and preserve the data utility.
The paper studies distributions and controllability in quantum mechanical systems.
problem Controlling quantum mechanical systems and their evolution.
method Analysis of distributions, controllability, and geodesics on sub-Finsler manifolds.
result Proves the Lie group decomposition and geodesics equivalence for quantum system steering.
Alternative discrete Dirac mechanics using Dirac structures.
problem Developing a new framework for discrete mechanics.
method Introducing 'continuous Dirac system' and proposing a definition of 'discrete Dirac system'.
result It is possible to recover discrete Lagrangian and Hamiltonian systems.
This work extends reduction processes for nonholonomic discrete mechanical systems.
problem Nonholonomic discrete mechanical systems and their reductions.
method Introduces a category of discrete-time dynamical systems and a two-stage reduction process.
result Two-stage reduction process produces systems isomorphic to one-stage reduction.
New integrators for mechanical systems on Lie groups simplify based on group properties.
problem Designing numerical integrators for mechanical systems on Lie groups.
method Leverage retraction maps and Lie group properties to design structure-preserving integrators.
result Simplified design of integrators for Euler-Poincare and Lie-Poisson equations.
In this paper we study a class of physical systems that combine a finite number of mechanical and thermodynamic observables. We call them finite dimensional thermo-mechanical systems. We introduce these systems by means of simple examples. The evolution equations of the involved observables are obtained in each example…
In this paper we propose a process of lagrangian reduction and reconstruction for nonholonomic discrete mechanical systems where the action of a continuous symmetry group makes the configuration space a principal bundle. The result of the reduction process is a discrete dynamical system that we call the discrete reduce…
Survey of Lagrangian reduction for discrete mechanical systems.
problem Understanding and reducing complex mechanical systems.
method Lagrangian reduction applied to discrete-time mechanical systems.
result Introduction to reduction techniques for various constraints and forces.
Study on relativistic nonholonomic mechanics with time-dependent constraints.
problem Formulating classical time-dependent nonholonomic mechanics.
method Invariant formulation using moving frames and Chaplygin systems.
result Hamiltonization of time-dependent constraints achieved.
Paper analyzes dynamics of nonholonomic systems with collisions using variational techniques.
problem Analyzing the dynamics of nonholonomic mechanical systems with impacts.
method Variational techniques extended to nonsmooth context for collisions.
result Variational formulation for implicit nonholonomic mechanical systems with energy-momentum preserving collisions.
Quantum mechanics models for financial Black-Scholes model.
problem Modeling financial derivatives using quantum mechanics.
method Noncommutative quantum mechanics applied to specific mechanical systems.
result Generalized noncommutative quantum mechanics of financial models.
Exact discrete mechanics for nonholonomic systems defined.
problem Discrete mechanics for nonholonomic systems.
method Constructing an exponential map and deriving exact discrete nonholonomic integrators.
result Reproduces continuous nonholonomic flow as discrete flow on constraint submanifold.
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.
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.
In order to obtain a framework in which both non-holonomic mechanical systems and non-holonomic mechanical systems with symmetry can be described, we introduce in this paper the notion of a Lagrangian system on a subbundle of a Lie algebroid.
Completeness of Riemannian metrics: an application to the control of constrained mechanical systemseess.SY
We improve Riemannian metrics for constrained systems control.
problem Controlling mechanical systems with configuration constraints.
method Constructing complete Riemannian metrics by modifying incomplete ones.
result A controller can be found to satisfy a design criterion.
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 …
In this work we introduce a category of discrete Lagrange--Poincare systems LP_d and study some of its properties. In particular, we show that the discrete mechanical systems and the discrete mechanical systems obtained by the Lagrangian reduction of symmetric discrete mechanical systems are objects in LP_d. We introdu…
Proves a theorem for mechanical systems with reflections.
problem Action functionals on paths with reflections.
method Proves a Morse index theorem for action functionals on paths that can reflect.
result Action functionals on paths with reflections have a well-defined Morse index.
Study preserves symplectic structure in forced discrete mechanical systems.
problem Preserving symplectic structure in forced discrete mechanical systems.
method Analyzes a specific type of forced discrete mechanical system , preserving a symplectic structure on .
result The preserved symplectic structure can be seen as Marsden-Weinstein reduction of the canonical symplectic structure.
Defines hybrid systems on principal bundles and studies impact effects.
problem Understanding impact effects in hybrid mechanical systems.
method Defines hybrid systems on principal bundles, studies underlying geometry, and finds conditions for impact preservation.
result Conditions for preservation of both exterior and interior impacts by mechanical connections.
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.
Develops VAEs for learning complex physical systems from data.
problem Learning low-dimensional representations of nonlinear physical systems.
method Variational Autoencoders with manifold latent spaces.
result Effective in learning nonlinear Burgers equation and constrained mechanical systems.
In this paper, we explore dynamics of the nonholonomic system called vakonomic mechanics in the context of Lagrange-Dirac dynamical systems using a Dirac structure and its associated Hamilton-Pontryagin variational principle. We first show the link between vakonomic mechanics and nonholonomic mechanics from the viewpoi…
CICME estimates common and domain-specific causal mechanisms from multi-sensor data.
problem Inferring causal mechanisms from heterogeneous multi-sensor data across multiple domains.
method Three-step approach using Causal Transfer Learning (CTL).
result CICME reliably detects domain-invariant causal mechanisms and guides individual domain causal mechanism estimation.
We discuss the quantization of mechanical systems for which the Hamiltonian vector fields of observables form the deformation of -dimensional oscilator algebra. Because of this fact these systems can be considered as "deformations" of the harmonic oscillator. The set of abovementioned mechanical systems are realized…
We investigate local configuration controllability for mechanical control systems within the affine connection formalism. Extending the work by Lewis for the single-input case, we are able to characterize local configuration controllability for systems with degrees of freedom and input forces.
Quantum Econophysicsphysics.soc-ph
The relationships between game theory and quantum mechanics let us propose certain quantization relationships through which we could describe and understand not only quantum but also classical, evolutionary and the biological systems that were described before through the replicator dynamics. Quantum mechanics could be…
The geometry of a Lagrangian mechanical system is determined by its associated evolution semispray. We uniquely determine this semispray using the symplectic structure and the energy of the Lagrange space and the external force field. We study the variation of the energy and Lagrangian functions along the evolution and…
Global singularities propagate in magnetic mechanical systems on Riemannian manifolds.
problem Propagation of singularities in magnetic mechanical systems.
method Combines reduction from magnetic to Riemannian systems, analysis of reparameterized flows, and regularization techniques.
result Invariant singular set under generalized gradient flow dynamics.
A new Dirac algebroid approach for nonholonomic systems.
problem Nonholonomic constraints in mechanical systems.
method Developed a Dirac algebroid to generate phase equations for systems with linear nonholonomic constraints.
result Unified approach to describe systems with different potentials.
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…
Quantum systems learn like machine learning models, influenced by dissipation.
problem Understanding how quantum systems learn and evolve.
method Hydrodynamical formulation of quantum mechanics, gradient descent model, empirical demonstration.
result Quantum systems follow a disrupted gradient descent model, influenced by dissipation.
Rolling systems limit to billiard models with no-slip collisions.
problem Understanding how rolling systems behave as billiard models with no-slip collisions.
method Showed that no-slip billiards arise as limits of non-holonomic rolling systems.
result Rolling systems limit to billiard models with no-slip collisions.
In this paper, we investigate the existence of a subclass of quotients of affine connection control systems, which preserve the mechanical structures. Both local and global sufficient and necessary conditions are given for the geodesically accessible affine connection control systems such that they can admit this subcl…
Study symmetry breaking in quantum mechanics to understand many-body physics.
problem Understanding many-body physics from quantum mechanics.
method Analyzing potentials with unstable critical points and local minima.
result Emergence of many-body physics from spontaneous symmetry breaking.
In this paper, we carry a detailed study of mechanical systems with configuration space for which the base variables are being controlled. The overall system's motion is considered to be induced from the base one due to the presence of general non-holonomic constraints. It is shown that the…
Paper proves trajectories of Chaplygin systems are reparametrized geodesics.
problem Understanding trajectories of Chaplygin systems.
method Constructive proof using modified Riemannian metrics.
result Reparametrized geodesics of Chaplygin systems.
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.