📂Partial Differential Equations

Hamilton-Jacobi Equation and Hamiltonian Equation

There are two ways to derive the Hamilton equations. One is from the Euler-Lagrangian equations, and the other, which will be introduced in this article, is from the characteristic equations of the Hamilton-Jacobi equation.

Definition¹

The following partial differential equation is called the general Hamilton-Jacobi equation.

$$ G(Du, u_{t}, u, x, t)=u_{t}+H(Du, x)=0 $$

• $t >0 \in \mathbb{R}$
• $x \in \mathbb{R}^{n}$
• $u : \mathbb{R}^{n} \to \mathbb{R}$

Here, the differential operator $D$ follows the multi-index notation, and let us always consider differentiation with respect to the spatial variable $x$, that is, $D=D_{x}$, and $Du=D_{x}u=(u_{x_{1}}, \cdots, u_{x_{n}})$. And $H : \mathbb{R}^n \times \mathbb{R}^n \rightarrow \mathbb{R}$ is called the Hamiltonian.

Characteristic Equation

For convenience, assume that $H \in C^{\infty} \big(\mathbb{R}^{n} \times (0,\infty) \big)$. Given such a Hamilton-Jacobi equation, let’s simplify the expression by combining space-time variables into one as $y$.

$$ y=(x,t)=(x_{1}, \cdots, x_{n}, t) $$

Also, the time derivative and spatial derivative of $u$ are together represented as $q$.

$$ \begin{align*} q &=q(Du, u_{t}) =q(u_{x_{1}}, u_{x_{2}},\dots, u_{x_{n}}, u_{t}) \\ &= (p, p_{n+1}) =(p_{1}, p_{2}, \dots, p_{n}, p_{n+1}) \end{align*} $$

Lastly, if we say $z=u$, the Hamilton-Jacobi equation can be represented as below.

$$ \begin{equation} G(q, z, y)=p_{n+1}+H(p, x)=0 \quad \forall (q, z, y)\in\mathbb{R}^{n+1}\times \mathbb{R} \times \big( \mathbb{R}^n\times (0, \infty) \big) \label{eq1} \end{equation} $$

Solving for the derivative of $G$, we get the following.

$$ \begin{align} D_{q} G(q, z, y) &= (G_{p_{1}}, \cdots, G_{p_{n+1}})=\big(H_{p_{1}}(p,x), \dots , H_{p_{n}}(p,x), 1\big)=\big( D_{p} H(p,x), 1\big) \label{eq2} \\ D_{z} G(q, z, y) &= G_{z}=0 \label{eq3} \\ D_{y} G(q, z, y) &= \big( G_{y_{1}}, \cdots, G_{y_{n+1}} \big)=\big( H_{x_{1}}(p,x), \cdots, H_{x_{n}}(p,x), H_{t}(p,x) \big) =\big( D_{x}H (p,x), 0\big) \label{eq4} \end{align} $$

Moreover, the characteristic equation of $G(q,z,y)$ is as follows.

$$ \left\{ \begin{align*} \dot{q}(s) &= -D_{y} G\big(q(s), z(s), y(s) \big)-D_{z} G\big(q(s), z(s), y(s) \big)q(s) \\ \dot{z}(s) &= D_{q} G\big(q(s), z(s), y(s) \big) \cdot q(s) \\ \dot{y}(s) &= D_{q} G\big(q(s), z(s), y(s) \big) \end{align*} \right. $$

Then $\dot{q}(s)$ is as follows.

$$ \begin{align*} \dot{ q}(s) &= -D_{y} G\big(q(s), z(s), y(s) \big)-D_{z} G\big(q(s), z(s), y(s) \big)q(s) \\ &= -D_{y} G\big(q(s), z(s), y(s) \big) \\ &=- (D_{x} H(p,x), 0) \end{align*} $$

The second equality is by $\eqref{eq2}$, and the third equality is by $\eqref{eq4}$. Since $q=(p, p_{n+1})$, each component of $\dot{q}$ is as follows.

$$ \begin{align*} \dot{p}^{i}(s) &= -H_{x_{i}} \big( p(s), x(s) \big) &( i=1,\dots,n) \\ \dot{p}^{n+1}(s) &= 0 \end{align*} $$

$\dot{z}(s)$ is as follows.

$$ \begin{align*} \dot{z}(s) &= D_{q} G\big(q(s), z(s), y(s) \big) \cdot q(s) \\ &= \Big( D_{p}H\big(p(s), x(s) \big), 1 \Big)\cdot\big( p(s), p_{n+1}(s) \big) \\ &= D_{p} H\big( p(s), x(s)\big)\cdot p(s) +p_{n+1}(s) \\ &= D_{p} H\big( p(s), x(s)\big)\cdot p(s) -H\big( p(s), x(s)\big) \end{align*} $$

The second equality is by $\eqref{eq2}$, and the fourth equality is by $\eqref{eq1}$. $\dot{y}(s)$ is as follows.

$$ \begin{align*} \dot{y}(s) &= D_{q}G\big(q(s), z(s), y(s) \big) \\ &= \big(D_{p}H(p,x), 1 \big) \end{align*} $$

Since $y=(x,t)$, each component of $\dot{y}=(\dot{x}, \dot{t})$ is as follows.

$$ \begin{cases} \dot{x}(s) = D_{p}H\big( p(s), x(s) \big) \\ \dot{t}(s)=1 \end{cases} $$

From the results above, we can consider $s$ to be equivalent to $t$. By synthesizing everything we calculated so far, we obtain the characteristic equations of the Hamilton-Jacobi equation as follows.

$$ \begin{align*} \dot{p}(s) &= -D_{x}H \big( p(s), x(s) \big) \\ \dot{z}(s) &= D_{p} H\big( p(s), x(s)\big)\cdot p(s) -H\big( p(s), x(s)\big) \\ \dot{x}(s) &= D_{p}H\big( p(s), x(s) \big) \end{align*} $$

Here, specifically groups the first and third equations as Hamilton’s equations.

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