1 Hamilton-Jacobi Equation
1.1 Hamilton-Jacobi Equation
In physics the Hamilton-Jacobi equation is an alternative formulation of classical mechanics equivalent to [[newtonian-mechanics]], Lagrangian Mechanics and Hamiltonian Mechanics. Specifically, the Hamilton-Jacobi equation is a formulation of mechanics in which the motion of a particle can be represented as a wave. The wave equation followed by mechanical systems is similar but not identical to the [[schrodinger-equation]].
In mathematics, the Hamilton-Jacobi equation is a necessary condition describing extremal geometry in generalizations of problems from the Calculus of Variations. It can be understood as a special case of the [[hamilton-jacobi-bellman-equation]] from Dynamic Programming Principle.
The Hamilton-Jacobi Equation (HJE) is a non-linear first-order PDE with the form \[ \begin{cases}u_{t}+H(\nabla u)=0 & \text{in }\mathbb{R}^n\times(0,\infty) \\ u=g & \text{on } \mathbb{R}^n \times \{ t=0 \},\end{cases} \] where \(u=u(x,t):\mathbb{R}^n\times [0,\infty)\to \mathbb{R}\) is the function we are looking to solve out and \(\nabla u=(u_{x_{1}}, \dots, u_{x_{n}})\) is the gradient of \(u\) with respect to the \(n\)-dimensional space variable \(x:=(x_{1}, \dots, x_{n})\).
The Hamiltonian \(H:\mathbb{R}^n \times \mathbb{R}\) is given and the initial condition \(g:\mathbb{R}^n \to \mathbb{R}\) is given.
1.2 Link to Hamiltonian via Calculus of Variations
We highlight the link between the HJE and the [[hamilton-equation]] by considering a problem in Calculus of Variations. The problem is finding a best curve in the admissible class \[ \mathscr{A} := \{ w \in C^2,~w:[0,t]\to \mathbb{R}^n:w(0)=y,~w(t)=x \}, \] i.e. paths in \(\mathbb{R}^n\) with continuous second derivatives starting at \(y\in R^n\) and ending at \(x \in \mathbb{R}^n\) at time \(t>0\) or more intuitively, the moving trajectory of a particle. Under this intuition \(w'(s)\) is the speed of the particle.
The action [[functional|functional]] is defined as \[ I[w] = \int_{0}^{t}{\mathcal{L}(w',w)}~d{s} \] where \(\mathcal{L}:\mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}\) is a given smooth mapping function known as the Lagrangian. Our aim is to find a curve \(x(s)\in\mathscr{A}\) such that \[ I[x] = \inf_{w \in\mathscr{A}}{I[w]}. \] Assume \(L=L(v,x)\) for \(v,x \in \mathbb{R}^n\) (for notation convenience) and that we can achieve the infimum of \(I[w]\) by some optimal trajectory \(x(s)\in\mathscr{A}\). To build our PDE for \(x(s)\) we choose smooth \(y:[0,t] \to \mathbb{R}^n\) with \(y(s)=(y^1(s), \dots, y^n(s))\) such that \(y(0)=y(t)=0\) and consider perturbing \(x(s)\) by small multiples \(\tau \in \mathbb{R}\) of \(y(s)\) to get \[ w(s)=x(s)+\tau y(s). \] Since \(w(s) \in \mathscr{A}\) we see immediately that \[ I[w]\geq I[x]. \] Considering the action functional of the perturbed path we have that \[ i:\mathbb{R} \to \mathbb{R};\quad i(\tau)=I[x+\tau y]. \] Assuming differentiability we can compute \[ \begin{align} i'(\tau) & = \frac{d}{d\tau}\int_{0}^{t}{\mathcal{L}(x'(s)+\tau y'(s), x(s)+\tau y(s))}~d{s} \\ & =\int_{0}^{t}{y'(s)\cdot \mathcal{L}_{v}(x'(s)+\tau y'(s), x(s)+\tau y(s))+y(s) \cdot \mathcal{L}_{x}(x'(s)+\tau y'(s),x(s)+\tau y(s))}~d{s}, \end{align} \] where we have applied Leibniz Integral Rule taking the derivative inside the integral. Evaluating at \(\tau=0\) we find that \[ \begin{align} i'(0) & =\int_{0}^{t}{y'(s)\cdot \mathcal{L}_{v}(x'(s), x(s))+y(s) \cdot \mathcal{L}_{x}(x'(s), x(s))}~d{s} \\ & = \int_{0}^{t}\sum_{i=1}^{n}{([y^i(s)]'\cdot \mathcal{L}_{v_{i}}(x'(s), x(s)) + y^i(s) \cdot\mathcal{L}_{x_{i}}(x'(s), x(s)))}~d{s} \\ & =0. \end{align} \]