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Questions tagged [hamilton-jacobi-equation]

Use this tag for questions related to the Hamilton-Jacobi equation, which in mathematics is a necessary condition describing extremal geometry in generalizations of problems from the calculus of variations and in physics is an alternative formulation of classical mechanics.

In mathematics, the Hamilton-Jacobi equation (HJE) is a necessary condition describing extremal geometry in generalizations of problems from the calculus of variations and is a special case of the Hamilton-Jacobi-Bellman equation.

In physics, the HJE is an alternative formulation of classical mechanics such as Newton's laws of motion, Lagrangian mechanics, or Hamiltonian mechanics. The HJE is particularly useful in identifying conserved quantities for mechanical systems, which may be possible even when the mechanical problem itself cannot be solved completely.

The HJE is the only formulation of mechanics in which the motion of a particle can be represented as a wave.

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Level sets of solution to Nonlinear PDE

I work on Stochastic Control theory and BSDE's for my research. In my research, I characterized the set I am interested in as the level set of a function which is a viscosity solution to nonlinear PDE (HJB Equation). I was able to prove that this…
chandu1729
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Optimality — Hamilton-Jacobi-Bellman (HJB) versus Riccati

Most of the literature on optimal control discuss Hamilton-Jacobi-Bellman (HJB) equations for optimality. In dynamics however, Riccati equations are used instead. Jacobi Bellman equations are also used in Reinforcement learning. Are there any…
GENIVI-LEARNER
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On the Liouville-Arnold theorem

A system is completely integrable (in the Liouville sense) if there exist $n$ Poisson commuting first integrals. The Liouville-Arnold theorem, anyway, requires additional topological conditions to find a transformation which leads to action-angle…
ablagi
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In the differential geometric view of Hamiltonian mechanics, why do the phase space coordinates come from the cotangent bundle $T^* Q$?

I have following question about the Hamiltonian mechanics from differential geometrical viewpoint: We start with a physical system parametrized by generalized (position) coordinates $(q^i)$ providing under given restrictions the configuration space…
user267839
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When is separation of variables possible?

In classical PDE courses it is common to learn to perform a change of variables without really learning how to find the adequate equations of the change (polar, cylindrical or spherical coordinates are just plain easy to detect). Now, is it always…
nabla
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Why are viscosity solutions to Hamilton-Jacobi equations interesting?

This is a soft question. For my background, I am an undergraduate math major just finishing my first course in PDE. In my PDE class, we have been learning about viscosity solutions. I am following the definitions in these notes We say a continuous…
msm
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Compute the Hopf-Lax formula for a given PDE

in Evans' book (pag. 135) there's the following example (already asked in this site, but never answered): Consider the initial-value problem \begin{cases} u_t+\frac 12|Du|^2=0 & \text{in }\mathbb{R}^n \times (0,\infty) \\ \qquad \qquad \, \, \,…
VoB
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Solving the Hamilton-Jacobi equation (Evans)

I can comprehend some but not all of the proofs. I do not understand how the limit definition of a derivative is derived in this context, and those are highlighted in $\color{#009900}{\text{green}}$. This is from PDE Evans, 2nd edition, pages…
Cookie
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IVP for nonlinear PDE $u_t + \frac{1}{3}{u_x}^3 = -cu$

I'm trying to solve the following partial differential equations: $$ u_t + \frac{1}{3}{u_x}^3 = 0 \tag{a} $$ $$ u_t + \frac{1}{3}{u_x}^3 = -cu \tag{b} $$ with the initial value problem $$ u(x,0)=h(x)= \left\lbrace \begin{aligned} &e^{x}-1 &…
Infinite_28
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Understanding of the Hopf-Lax formula

This is an exercise in the book Partial Differential Equations (2nd edition) by Evans: Here $L^*(q)=\max_{y\in {\Bbb R}^n}\{q\cdot y-L(y)\}$ and $L$ is assumed to be such that it is convex and…
user9464
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Finding mimimum of the unique, weak solution to an IVP

PDE Evans, 2nd edition, page 134 Consider the initial-value problem \begin{cases} u_t+\frac 12|Du|^2=0 & \text{in }\mathbb{R}^n \times (0,\infty) \\ \qquad \qquad \, \, \, \, u = |x| & \text{on } \mathbb{R}^n \times \{t=0\} \tag{49} \end{cases} …
Cookie
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References for finding coordinates in which some natural Hamiltonian becomes separable

Given a natural Hamiltonian, $$H = \frac{1}{2} g^{\mu \nu}(x) p_{\mu} p_{\nu} + V(x)$$ and a general potential $V(x)$ I want to be able to find the coordinates, given an expression for $V(x)$, such that the Hamiltonian is separable. I had previously…
user1887919
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Is the Hamiltonian just a mnemonic for the Lagrangian?

I've been studying the calculus of variations and optimal control theory, and I often read about the "Hamiltonian formalism". I've been hearing about "Hamiltonian mechanics" for a while, and I don't really understand what the advantage is of…
user56834
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Hopf-Lax formula motivation

I'm studying the Hamilton-Jacobi equation by the Evans's "PDE" book and I got a little confused with the way the author got to the definition of the Hopf-Lax formula. We are considering the classical case, with convex Hamiltonian…
Proton
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Understanding HJB equation for the infinite horizon consumption control problem

Context Given the following maximization problem as well as wealth dynamics $$\max\mathbb{E}\left[\int_0^{\infty} \frac{1}{\gamma} e^{-\beta t} c_t^\gamma \mathrm{d} t\right]$$ $$\mathrm{d} X_t=X_t\left(r+\pi_t(\alpha-r)\right) \mathrm{d} t-c_t…
student
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