scholarly journals Singularities of solutions of time dependent Hamilton-Jacobi equations. Applications to Riemannian geometry

2021 ◽  
Author(s):  
Piermarco Cannarsa ◽  
Wei Cheng ◽  
Albert Fathi
Keyword(s):  
Riemannian Manifold ◽  
Distance Function ◽  
Riemannian Geometry ◽  
Jacobi Equation ◽  
Compact Manifold ◽  
Complete Riemannian Manifold ◽  
Time Dependent ◽  
Jacobi Equations ◽  
Uniformly Continuous ◽  
Hamilton Jacobi Equation

AbstractIf $U:[0,+\infty [\times M$ U : [ 0 , + ∞ [ × M is a uniformly continuous viscosity solution of the evolution Hamilton-Jacobi equation $$ \partial _{t}U+ H(x,\partial _{x}U)=0, $$ ∂ t U + H ( x , ∂ x U ) = 0 , where $M$ M is a not necessarily compact manifold, and $H$ H is a Tonelli Hamiltonian, we prove the set $\Sigma (U)$ Σ ( U ) , of points in $]0,+\infty [\times M$ ] 0 , + ∞ [ × M where $U$ U is not differentiable, is locally contractible. Moreover, we study the homotopy type of $\Sigma (U)$ Σ ( U ) . We also give an application to the singularities of the distance function to a closed subset of a complete Riemannian manifold.

scholarly journals Particle dynamics inside shocks in Hamilton–Jacobi equations

2010 ◽  
Vol 368 (1916) ◽  
pp. 1579-1593 ◽  
Author(s):  
Konstantin Khanin ◽  
Andrei Sobolevski
Keyword(s):  
Velocity Field ◽  
Jacobi Equation ◽  
Particle Dynamics ◽  
Velocity Fields ◽  
Vanishing Viscosity ◽  
Effective Velocity ◽  
Lagrangian Action ◽  
The Moment ◽  
Jacobi Equations ◽  
Hamilton Jacobi Equation

The characteristic curves of a Hamilton–Jacobi equation can be seen as action-minimizing trajectories of fluid particles. For non-smooth ‘viscosity’ solutions, which give rise to discontinuous velocity fields, this description is usually pursued only up to the moment when trajectories hit a shock and cease to minimize the Lagrangian action. In this paper we show that, for any convex Hamiltonian, there exists a uniquely defined canonical global non-smooth coalescing flow that extends particle trajectories and determines the dynamics inside shocks. We also provide a variational description of the corresponding effective velocity field inside shocks, and discuss the relation to the ‘dissipative anomaly’ in the limit of vanishing viscosity.

The Hamilton-Jacobi Equations for a Relativistic Charged Particle

1963 ◽  
Vol 6 (3) ◽  
pp. 341-350 ◽  
Author(s):  
J. R. Vanstone
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Jacobi Equation ◽  
Classical Particle ◽  
Quantum Mechanical ◽  
Order Partial Differential Equation ◽  
First Order ◽  
Relativistic Charged Particle ◽  
Jacobi Equations ◽  
Hamilton Jacobi Equation

In the problem of finding the motion of a classical particle one has the choice of dealing with a system of second order ordinary differential equations (Lagrange's equations) or a single first order partial differential equation (the Hamilton-Jacobi equation, henceforth referred to as the H-J equation). In practice the latter method is less frequently used because of the difficulty in finding complete integrals. When these are obtainable, however, the method leads rapidly to the equations of the trajectories. Furthermore it is of fundamental theoretical importance and it provides a basis for quantum mechanical analogues.

scholarly journals Instability of the Dirichlet problem for Hamilton-Jacobi equation

1997 ◽  
Vol 55 (2) ◽  
pp. 311-319
Author(s):  
Kewei Zhang
Keyword(s):  
Dirichlet Problem ◽  
Jacobi Equation ◽  
Jacobi Equations ◽  
Hamilton Jacobi Equation ◽  
General Conditions

We show the instability of solutions of the Dirichlet problem for Hamilton-Jacobi equations under quite general conditions.

scholarly journals Canonical transformations and the Hamilton-Jacobi theory in quantum mechanics

1999 ◽  
Vol 77 (6) ◽  
pp. 411-425 ◽  
Author(s):  
J -H Kim ◽  
H -W Lee
Keyword(s):  
Quantum Mechanics ◽  
Phase Space ◽  
Generating Functions ◽  
Jacobi Equation ◽  
Classical Case ◽  
Time Dependent ◽  
Canonical Transformations ◽  
Hamilton Jacobi Equation ◽  
Number Form

Canonical transformations using the idea of quantum generating functions are applied to construct a quantum Hamilton-Jacobi theory, based on the analogy with the classical case. An operator and a c-number form of the time-dependent quantum Hamilton-Jacobi equation are derived and used to find dynamical solutions of quantum problems. The phase-space picture of quantum mechanics is discussed in connection with the present theory.PACS Nos.: 03.65-w, 03.65Ca, 03.65Ge

A METRIC APPROACH TO PLASTICITY VIA HAMILTON–JACOBI EQUATION

2010 ◽  
Vol 20 (09) ◽  
pp. 1617-1647
Author(s):  
FERDINANDO AURICCHIO ◽  
ELENA BONETTI ◽  
ANTONIO MARIGONDA
Keyword(s):  
Jacobi Equation ◽  
Constitutive Relations ◽  
Yield Function ◽  
Generalized Gradients ◽  
Jacobi Equations ◽  
Thermodynamical Consistency ◽  
Hamilton Jacobi Equation ◽  
New Framework

Thermodynamical consistency of plasticity models is usually written in terms of the so-called "maximum dissipation principle". In this paper, we discuss constitutive relations for dissipative materials written through suitable generalized gradients of a (possibly non-convex) metric. This new framework allows us to generalize the classical results providing an interpretation of the yield function in terms of Hamilton–Jacobi equations theory.

scholarly journals Local singular characteristics on $$\mathbb {R}^2$$

2021 ◽  
Author(s):  
Piermarco Cannarsa ◽  
Wei Cheng
Keyword(s):  
Jacobi Equation ◽  
Differential Inclusions ◽  
Space Dimension ◽  
Uniqueness Result ◽  
Singular Set ◽  
Uniqueness Criterion ◽  
The Moment ◽  
Jacobi Equations ◽  
Hamilton Jacobi Equation ◽  
Singular Characteristics

AbstractThe singular set of a viscosity solution to a Hamilton–Jacobi equation is known to propagate, from any noncritical singular point, along singular characteristics which are curves satisfying certain differential inclusions. In the literature, different notions of singular characteristics were introduced. However, a general uniqueness criterion for singular characteristics, not restricted to mechanical systems or problems in one space dimension, is missing at the moment. In this paper, we prove that, for a Tonelli Hamiltonian on $$\mathbb {R}^2$$ R 2 , two different notions of singular characteristics coincide up to a bi-Lipschitz reparameterization. As a significant consequence, we obtain a uniqueness result for the class of singular characteristics that was introduced by Khanin and Sobolevski in the paper [On dynamics of Lagrangian trajectories for Hamilton-Jacobi equations. Arch. Ration. Mech. Anal., 219(2):861–885, 2016].

scholarly journals Hamilton-Jacobi Equation of Time Dependent Hamiltonians

2020 ◽  
Vol 5 (1-2) ◽  
pp. 09-15
Author(s):  
Anoud K. Fuqara ◽  
Amer D. Al-Oqali ◽  
Khaled I. Nawafleh
Keyword(s):  
Hamiltonian Systems ◽  
Jacobi Equation ◽  
Classical Mechanics ◽  
Time Dependent ◽  
Calculus Of Variation ◽  
Jacobi Formalism ◽  
Equation Of Time ◽  
Hamilton Jacobi Equation

In this work, we apply the geometric Hamilton-Jacobi theory to obtain solution of Hamiltonian systems in classical mechanics that are either compatible with two structures: the first structure plays a central role in the theory of time- dependent Hamiltonians, whilst the second is used to treat classical Hamiltonians including dissipation terms. It is proved that the generalization of problems from the calculus of variation methods in the nonstationary case can be obtained naturally in Hamilton-Jacobi formalism.

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