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.
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48 results for “Chaplygin system”
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.
Study of gyroscopic Chaplygin systems and magnetic flows on spheres.
problem Integrability and Hamiltonization of magnetic geodesic flows on spheres.
method Analysis of gyroscopic Chaplygin systems with magnetic forces, Hamiltonization, invariant measure existence.
result Integrable magnetic geodesic flows on spheres for .
We relate a Chaplygin type system to a Cartan decomposition of a real semi-simple Lie group. The resulting system is described in terms of the structure theory associated to the Cartan decomposition. It is shown to possess a preserved measure and when internal symmetries are present these are factored out via a process…
LR and L+R Systemsmath-ph
We consider coupled nonholonomic LR systems on the product of Lie groups. As examples, we study -dimensional variants of the spherical support system and the rubber Chaplygin sphere. For a special choice of the inertia operator, it is proved that the rubber Chaplygin sphere, after reduction and a time reparametrizat…
On the case of Goryachev-Chaplygin and new examples of integrable conservative systems on S^2math.DG
The aim of this paper is to describe a class of conservative systems on possessing an integral cubic in momenta. We prove that this class of systems consists off the case of Goryachev-Chaplygin, the one-parameter family of systems which has been found by the author in the previous paper (dg-ga/9711005) and a new …
The paper studies a natural -dimensional generalization of the classical nonholonomic Chaplygin sphere problem. We prove that for a specific choice of the inertia operator, the restriction of the generalized problem onto zero value of the SO(n-1)-momentum mapping becomes an integrable Hamiltonian system after an app…
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.
We study the rolling of the Chaplygin ball in over a fixed --dimensional sphere without slipping and without slipping and twisting. The problems can be naturally considered within a framework of appropriate modifications of the L+R and LR systems -- well known systems on Lie groups groups with an i…
Paper reduces nonholonomic systems with symmetries.
problem Nonholonomic systems with symmetries and conserved quantities.
method Two-step reduction procedure: first results in Chaplygin system, second in almost symplectic structure.
result Almost symplectic manifolds coincide with reduced nonholonomic brackets.
Geodesic extensions for systems with nonholonomic constraints.
problem Extending equations of motion for systems with nonholonomic constraints.
method Constructing extensions to second-order ODEs, investigating geodesic conditions.
result Conditions for nonholonomic trajectories to be geodesics of a Riemannian metric.
We study relations between vakonomically and nonholonomically constrained Lagrangian dynamics for the same set of linear constraints. The basic idea is to compare both situations at the level of variational principles, not equations of motion as has been done so far. The method seems to be quite powerful and effective.…
Projective geodesic extensions for nonholonomic systems are derived under conformal transformations.
problem Deriving conditions for projective geodesic extensions in nonholonomic mechanics.
method Analyzing necessary and sufficient conditions for existence under conformal modifications.
result Conditions for existence of projective geodesic extensions in nonholonomic systems under conformal transformations.
It has been proved that on 2-dimensional orientable compact manifolds of genus there is no integrable geodesic flow with an integral polynomial in momenta. There is a conjecture that all integrable geodesic flows on possess an integral quadratic in momenta. All geodesic flows on and possessing i…
Hidden symmetries of the Goryachev-Chaplygin and Kovalevskaya gyrostats spacetimes, as well as the Brdička-Eardley-Nappi-Witten pp-waves are studied. We find out that these spacetimes possess higher rank Stäckel-Killing tensors and that in the case of the pp-wave spacetimes the symmetry group of the Stäckel-Killing ten…
We study a time reparametrisation of the Newton type equations on Riemannian manifolds slightly modifying the Chaplygin multiplier method, allowing us to consider the Chaplygin method and the Maupertuis principle within a unified framework. As an example, the reduced nonholonomic problem of rolling without slipping and…
This paper studies the construction of geometric integrators for nonholonomic systems. We derive the nonholonomic discrete Euler-Lagrange equations in a setting which permits to deduce geometric integrators for continuous nonholonomic systems (reduced or not). The formalism is given in terms of Lie groupoids, specifyin…
The paper explores infinite-dimensional nonholonomic and vakonomic systems.
problem Understanding dynamics of infinite-dimensional systems with constraints.
method Visualizing and revisiting classical and new examples of nonholonomic and vakonomic systems.
result Infinite-dimensional systems exhibit both nonholonomic and vakonomic dynamics.
We consider static spacetimes whose spatial part admits foliations with the extrinsic curvature tensor K_{ab}=0. There are two complementary cases when the gradient of the lapse function points 1) to the direction of foliation or 2) orthogonally to it. Case 1) gives generalization of metrics like Bertotti-Robinson or N…
Paper proves existence of isometric immersions for negatively curved surfaces with unbounded second fundamental form.
problem Existence of isometric immersions for surfaces with negative Gaussian curvature.
method Reformulated Gauss--Codazzi equations into hyperbolic conservation laws, applied theories of invariant regions and compensated compactness.
result Established existence of -isometric immersions for various families of metrics.
Overview of integrable systems with symmetries, focusing on toric and semitoric systems.
problem Classifying and understanding integrable systems with symmetries.
method Using decorated polygons and controlled bifurcations in one-parameter families of systems.
result Construction of explicit semitoric systems with prescribed invariants.
New method to derive integrable systems from existing Lax systems.
problem Deriving new integrable systems from existing ones.
method Systematic method of deriving new integrable systems from a given one.
result Examples of new integrable systems derived, including the dispersionless Hirota equation, the general heavenly equation, and the web equations.
Learning to control linear systems is statistically hard, especially for underactuated systems.
problem Statistical difficulty of learning to control linear systems, especially underactuated ones.
method Utilized minimax lower bounds and structural assumptions to prove learning complexity can be exponential.
result Learning complexity can be at most exponential with the controllability index of the system.
Discrete-time systems can be characterized by simple flat coordinates and their shifts.
problem Characterizing flatness of discrete-time systems.
method Developed a map from flat coordinates and their shifts to system state and input, fulfilling system equations identically.
result Derived necessary conditions for a system to be flat, without requiring differential geometry methods.
The paper explores when linear system identification is hard or easy, especially for under-actuated systems.
problem Statistical hardness of learning linear systems, especially under-actuated or under-excited systems.
method Using tools from minimax theory and recent statistical tools for finite sample analysis of system identification.
result The controllability index of linear systems affects the sample complexity of identification, making some systems hard to learn.
This paper improves system identification by reducing sample complexity for high-dimensional linear dynamical systems.
problem High sample complexity for learning partially observed linear dynamical systems in high dimensions.
method Introduces an -regularized estimation method that reduces sample complexity from linear to logarithmic with system dimension.
result Markov parameters can be learned with logarithmic number of samples relative to system dimension, improving sample complexity.
In integrable hydrodynamic systems, coordinates exist where generators and symmetries are simple.
problem Existence of Riemannian invariants for integrable systems of hydrodynamic type.
method Finding coordinates where the generator and all symmetries are diagonal.
result In integrable hydrodynamic systems, there exist coordinates where the generator and all symmetries are diagonal.
Nonholonomic Controlled Hamiltonian System: Symmetric Reduction and Hamilton-Jacobi Equationsmath.SG
This paper studies nonholonomic constraints in Hamiltonian systems, deriving equations and theorems.
problem Analyzing nonholonomic constraints in Hamiltonian systems.
method Deriving distributional RCH systems, geometric constraint conditions, and Hamilton-Jacobi theorems.
result Derives precise geometric constraint conditions and Hamilton-Jacobi theorems for nonholonomic systems.
New method models unknown systems with hidden parameters using neural networks.
problem Modeling unknown dynamical systems with hidden parameters.
method Training a deep neural network (DNN) model using trajectory data of the unknown system.
result DNN model accurately predicts unknown dynamical systems with new initial conditions.
Study absolute equivalence for Pfaffian systems, applying to control systems.
problem Absolute equivalence of Pfaffian systems with specific independence conditions.
method Structural results for Pfaffian systems of corank 3, applied to control systems.
result Dynamic feedback linearization of control systems with 2 inputs.
Solves selecting the best optimizing system problems.
problem Selecting the best system among contenders with unknown performance.
method Adaptive algorithms integrating stochastic gradient descent and sequential elimination.
result Exponential rates of convergence to zero for false selection probability.
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…
Superintegrable systems on curved manifolds found to have Hessian structures.
problem Characterizing superintegrable systems on curved manifolds.
method Identifying and computing Hessian coordinates for superintegrable systems.
result Examples of superintegrable systems in 2D and 3D have natural Hessian coordinates.
Measures of Systemic Riskq-fin.RM
Systemic risk refers to the risk that the financial system is susceptible to failures due to the characteristics of the system itself. The tremendous cost of systemic risk requires the design and implementation of tools for the efficient macroprudential regulation of financial institutions. The current paper proposes a…
Researchers solve boundary and scattering rigidity problems for magnetic systems.
problem Recovering magnetic systems from boundary or scattering data.
method Reduced to magnetic systems and applied results from [DPSU07].
result Recovering -system up to a gauge.
The paper develops methods to derive mixed superposition rules for Lie systems and applies them to various physical systems.
problem Finding general solutions for Lie systems.
method Develops mixed superposition rules for Lie systems with imprimitive Lie algebras and semidirect sums.
result Extends coalgebra method to Lie systems of partial differential equations.
Paper develops reduction theory for controlled Lagrangian systems with symmetry and momentum map.
problem Reduction of controlled Lagrangian systems with symmetry and momentum map.
method Using Legendre transformation and Euler-Lagrange vector field, the paper extends symmetric reduction theory.
result Established regular reduction theory for RCL systems with symmetry and momentum map.
The paper studies connections in superintegrable systems, revealing geometric insights.
problem Understanding non- and semi-degenerate superintegrable systems.
method Analyzes two torsion-free connections associated with superintegrable systems.
result Semi-degenerate secondary structure tensor is the Ricci curvature of a natural torsion-free connection.
This paper proposes a system-agnostic policy for dynamic scheduling.
problem Dynamic scheduling in changing systems is challenging due to system-specific optimal policies.
method Descriptive policy that learns a system-agnostic scheduling principle.
result System-agnostic meta-learning enables adaptation to unseen system characteristics.
Learn dynamics of a system using auxiliary data from similar systems.
problem Learning dynamics of a linear system with limited data.
method Weighted least squares approach, incorporating auxiliary data.
result Auxiliary data can help reduce intrinsic error due to noise.
The inability to see and quantify systemic financial risk comes at an immense social cost. Systemic risk in the financial system arises to a large extent as a consequence of the interconnectedness of its institutions, which are linked through networks of different types of financial contracts, such as credit, derivativ…
New Lie systems derived from Goursat distributions with applications to differential equations.
problem Analyzing Lie systems associated with Goursat distributions and their applications.
method Analyzing bracket-generating distributions and their relation to Lie systems, focusing on reductions and reconstructions.
result Lie systems associated with Goursat distributions can be reduced and solutions reconstructed from reduced systems.
Elliptic systems are characterized by Darboux integrability.
problem Characterizing elliptic differential systems with holomorphic solutions.
method Using a complex manifold and associated holomorphic Pfaffian system.
result Elliptic systems are Darboux integrable under generic conditions.
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.
Reduces multisymplectic Lie systems through symmetry analysis.
problem Solving multisymplectic Lie systems using symmetry reduction.
method Using momentum maps for reduction and reconstruction of multisymplectic Lie systems.
result Solves the original problem by analyzing simpler multisymplectic Lie systems.
Study integrable discretizations of cyclic systems with circular coordinate lines.
problem Integrable discretizations of 3D cyclic systems with circular coordinate lines.
method Investigate circle congruences and flat connections in the context of discrete cyclic systems.
result Characterization of circle congruences and existence of certain flat connections.
A new chaotic financial system is proposed by considering ethics involvement in a four-dimensional financial system with market confidence. A five-dimensional conformable derivative financial system is presented by introducing conformable fractional calculus to the integer-order system. A discretization scheme is propo…
A distributed system identification method for LTI systems using reverse experience replay.
problem Online system identification of LTI systems over multi-agent networks.
method DSGD-RER, a distributed variant of SGD-RER with backward updates.
result The estimation error decreases as the network size grows.