arXiv research

Geometrically describes Jacobi equations for field theories with dissipation.

Defines Jacobi fields in nonholonomic mechanics.

We discuss some properties of Jacobi fields that do not involve assumptions on the curvature endomorphism. We compare indices of different spaces of Jacobi fields and give some applications to Riemannian geometry.

Survey reviews Hamilton-Jacobi theory in various geometric settings, focusing on Jacobi and Leibniz identities.

Characterizes affine vector fields on Finsler manifolds with rigidity results.

We show that an invariant surface allows to construct the Jacobi vector field along a geodesic and construct the formula for the normal component of the Jacobi field. If a geodesic is the transversal intersection of two invariant surfaces (such situation we have, for example, if the geodesic is hyperbolic), then we can…

Study magnetic Hamiltonian systems with constraints, deriving Hamilton-Jacobi equations.

The most general Jacobi brackets in $\mathbb{R}^3$ are constructed after solving the equations imposed by the Jacobi identity. Two classes of Jacobi brackets were identified, according to the rank of the Jacobi structures. The associated Hamiltonian vector fields are also constructed.

Develops Hamilton-Jacobi theory for non-conservative field theories in k-contact geometry.

This paper studies Hamilton-Jacobi equations for magnetic systems with constraints.

We study the geometric nature of the Jacobi equation. In particular we prove that Jacobi vector fields (JVFs) along a solution of the Euler-Lagrange (EL) equations are themselves solutions of the EL equations but considered on a non-standard algebroid (different from the tangent bundle Lie algebroid). As a consequence …

Solves specific mean-field game equations with ODEs.

Paper connects differential geometry with geometric calculus.

The paper defines conjugate points for systems of second-order ODEs and discusses their properties.

Diffieties formalize geometrically the concept of differential equations. We introduce and study Hamilton-Jacobi diffieties. They are finite dimensional subdiffieties of a given diffiety and appear to play a special role in the field theoretic version of the geometric Hamilton-Jacobi theory.

We extend the geometric Hamilton-Jacobi formalism for hamiltonian mechanics to higher order field theories with regular lagrangian density. We also investigate the dependence of the formalism on the lagrangian density in the class of those yelding the same Euler-Lagrange equations.

Introduces a new bracket for multicontact geometry and applies it to field theories.

This paper bridges Kahler geometry and quantum mechanics in lognormal statistical models.

Abstract: Study Hamiltonian systems on almost cosymplectic manifolds, extending contact Hamiltonian systems.

In this paper a method for the resolution of the differential equation of the Jacobi vector fields in the manifold V1 = Sp(2)/SU(2) is exposed. These results are applied to determine areas and volumes of geodesic spheres and balls.

In this paper, we give precisely the geometric constraint conditions of canonical symplectic form and regular reduced symplectic forms for the dynamical vector fields of a regular controlled Hamiltonian (RCH) system and its regular reduced systems, which are called the Type I and Type II of Hamilton-Jacobi equations. A…

This paper studies nonholonomic constraints in Hamiltonian systems, deriving equations and theorems.

It is proved that all invariant functions of a complex Finsler manifold can be totally recovered from the torsion and curvature of the connection introduced by Kobayashi for holomorphic vector bundles with complex Finsler structures. Equations of the geodesics and Jacobi fields of a generic complex Finsler manifold, ex…

The Jacobi curve of an extremal of optimal control problem is a curve in a Lagrangian Grassmannian defined up to a symplectic transformation and containing all information about the solutions of the Jacobi equations along this extremal. In our previous works we constructed the canonical bundle of moving frames and the …

In this paper, we first give the regular point reduction and the two types of Hamilton-Jacobi equation for a regular controlled Hamiltonian (RCH) system with symmetry and momentum map on the generalization of a semidirect product Lie group. Next, as an application of the theoretical results, we consider the underwater …

The paper studies conjugate loci on ellipsoids and Liouville manifolds.

In sub-Riemannian geometry the coefficients of the Jacobi equation define curvature-like invariants. We show that these coefficients can be interpreted as the curvature of a canonical Ehresmann connection associated to the metric, first introduced in [Zelenko-Li]. We show why this connection is naturally nonlinear, and…

Study on LOB dynamics using mean-field game theory.

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…

A close relationship between the classical Hamilton-Jacobi theory and the kinematic reduction of control systems by decoupling vector fields is shown in this paper. The geometric interpretation of this relationship relies on new mathematical techniques for mechanics defined on a skew-symmetric algebroid. This geometric…

First-order ODEs linked to flat surfaces, leading to integrability.

Deep Galerkin Method estimates value function for mean-field control problem.

A new definition for vector fields extends the Jacobi set concept.

Model analyzes competitive pricing strategies in large markets of perishable products.

Deep learning method proves convergence for high-dimensional PDEs.

A Jacobi field on a Riemannian manifold M is defined along a geodesic. We generalize this notion to an arbitrary smooth curve, and call it an infinitesimal isometry along the curve. We give two approaches to this: 1) compute the complete prolongation of the Killing equation and then restrict to the curve, and 2) comput…

New vector fields integrate first-order ODEs.

The connection between Jacobi fields and odular structures of affine manifold is established. It is shown that the Jacobi fields generate the natural geoodular structure of affinely connected manifolds.

Unified geometric framework for integrability of conservative and dissipative systems.

Abstract: From complex financial models to simpler Hamilton-Jacobi equations.

By resorting to Noether's Second Theorem, we relate the generalized Bianchi identities for Lagrangian field theories on gauge-natural bundles with the kernel of the associated gauge-natural Jacobi morphism. A suitable definition of the curvature of gauge-natural variational principles can be consequently formulated in …

New method uses TT approximations to solve HJB equations for efficient sampling.

Equations of motion for linear Hamiltonians in the real Jacobi group

A variational principle is proposed for obtaining the Jacobi equations in systems admitting a Lagrangian description. The variational principle gives simultaneously the Lagrange equations of motion and the Jacobi variational equations for the system. The approach can be of help in finding constants of motion in the Jac…

A framework solves parametric families of MFGs efficiently.

Minimal surface doublings have specific index and nullity values.

Symmetric Poisson structures linked to geodesic foliations and Jordan algebras.

The geodesic flow on the tangent bundle is the flow of a certain vector field which is called the spray $S:TM\to TTM$. The flow lines of the vector field $\ka_{TM}øTS:TTM\to TTTM$ project to the Jacobi fields on $TM$. This could be called the Jacobi flow.