
# 2.5: Hamilton-Jacobi Theory

The nonlinear equation (2.4.1) of previous section in one more dimension is

$$F(x_1,\ldots,x_n,x_{n+1},z,p_1,\ldots,p_n,p_{n+1})=0.$$

The content of the Hamilton1-Jacobi2 theory is the theory of the special case

\label{nonlinearham}
F\equiv p_{n+1}+H(x_1,\ldots,x_n,x_{n+1},p_1,\ldots,p_n)=0,

i. e., the equation is linear in $$p_{n+1}$$ and does not depend  on $$z$$ explicitly.

Remark. Formally, one can write equation (2.4.1)

$$F(x_1,\ldots,x_n,u,u_{x_1},\ldots,u_{x_n})=0$$

as an equation of type (\ref{nonlinearham}). Set $$x_{n+1}=u$$ and seek $$u$$ implicitly from

$$\phi(x_1,\ldots,x_n,x_{n+1})=const.,$$

where $$\phi$$ is a function which is defined by a differential equation.

Assume $$\phi_{x_{n+1}}\not=0$$, then

\begin{eqnarray*}
0&=&F(x_1,\ldots,x_n,u,u_{x_1},\ldots,u_{x_n})\\
&=&F(x_1,\ldots,x_n,x_{n+1},-\frac{\phi_{x_1}}{\phi_{x_{n+1}}},\ldots,-\frac{\phi_{x_n}}{\phi_{x_{n+1}}})\\
&=&:G(x_1,\ldots,x_{n+1},\phi_1,\ldots,\phi_{x_{n+1}}).
\end{eqnarray*}

Suppose that $$G_{\phi_{x_{n+1}}}\not=0$$, then

$$\phi_{x_{n+1}}=H(x_1,\ldots,x_n,x_{n+1},\phi_{x_1},\ldots,\phi_{x_{n+1}}).$$

The associated characteristic equations to (\ref{nonlinearham}) are

\begin{eqnarray*}
x_{n+1}'(\tau)&=&F_{p_{n+1}}=1\\
z'(\tau)&=&\sum_{l=1}^{n+1} p_lF_{p_l}=\sum_{l=1}^np_lH_{p_l}+p_{n+1}\\
&=&\sum_{l=1}^np_lH_{p_l}-H\\
p'_{n+1}(\tau)&=&-F_{x_{n+1}}-F_zp_{n+1}\\
&=&-F_{x_{n+1}}\\
p_k'(\tau)&=&-F_{x_k}-F_zp_k\\
\end{eqnarray*}

Set $$t:=x_{n+1}$$, then we can write partial differential equation (\ref{nonlinearham}) as

\label{hamjac}
u_t+H(x,t,\nabla_xu)=0

and $$2n$$ of the characteristic equations are
\begin{eqnarray}
\label{charhj1}
x'(t)&=&\nabla_pH(x,t,p)\\
\label{charhj2}
p'(t)&=&-\nabla_xH(x,t,p).
\end{eqnarray}
Here is

$$x=(x_1,\ldots,x_n),\ p=(p_1,\ldots,p_n).$$

Let $$x(t),\ p(t)$$ be a solution of (\ref{charhj1}) and (\ref{charhj2}), then it follows $$p_{n+1}'(t)$$ and $$z'(t)$$ from the characteristic equations

\begin{eqnarray*}
p'_{n+1}(t)&=&-H_t\\
z'(t)&=&p\cdot\nabla_pH-H.
\end{eqnarray*}

Definition. The function $$H(x,t,p)$$ is called Hamilton function, equation (\ref{nonlinearham}) Hamilton-Jacobi equation and the system (\ref{charhj1}), (\ref{charhj2}) canonical system to H.

There is an interesting interplay between the Hamilton-Jacobi equation and  the canonical system. According to the previous theory we can construct a solution of the Hamilton-Jacobi equation by using solutions of the canonical system. On the other hand, one obtains from  solutions of the Hamilton-Jacobi equation also solutions of the canonical system of ordinary differential equations.

Definition.
A solution $$\phi(a;x,t)$$ of the Hamilton-Jacobi equation, where $$a=(a_1,\ldots,a_n)$$ is an $$n$$-tuple of real parameters, is called a complete integral of the Hamilton-Jacobi equation if
$$\det (\phi_{x_ia_l})_{i,l=1}^n\not=0.$$

Remark. If $$u$$ is a solution of the Hamilton-Jacobi equation, then also $$u+const.$$

Theorem 2.4 (Jacobi). Assume

$$u=\phi(a;x,t)+c,\ c=const.,\ \phi\in C^2\ \mbox{in its arguments},$$

is a complete integral. Then one obtains by solving of

$$b_i=\phi_{a_i}(a;x,t)$$

with respect to $$x_l=x_l(a,b,t)$$, where $$b_i\ i=1,\ldots,n$$ are given real constants, and then by setting

$$p_k=\phi_{x_k}(a;x(a,b;t),t)$$

a 2n-parameter family of solutions of the canonical system.

Proof. Let

$$x_l(a,b;t),\ l=1,\ldots,n,$$

be the solution of the above system. The solution exists since $$\phi$$ is a complete integral by assumption. Set

$$p_k(a,b;t)=\phi_{x_k}(a;x(a,b;t),t),\ k=1,\ldots,n.$$

We will show that $$x$$ and $$p$$ solves the canonical system. Differentiating $$\phi_{a_i}=b_i$$ with respect to $$t$$ and the Hamilton-Jacobi equation $$\phi_t+H(x,t,\nabla_x\phi)=0$$ with respect to $$a_i$$, we obtain for $$i=1,\ldots,n$$

\begin{eqnarray*}
\phi_{ta_i}+\sum_{k=1}^n\phi_{x_ka_i}\frac{\partial x_k}{\partial t}&=&0\\
\phi_{ta_i}+\sum_{k=1}^n\phi_{x_ka_i}H{p_k}&=&0.
\end{eqnarray*}
Since $$\phi$$ is a complete integral it follows for $$k=1,\ldots,n$$

$$\frac{\partial x_k}{\partial t}=H_{p_k}.$$

Along a trajectory, i. e., where $$a,\ b$$ are fixed, it is $$\frac{\partial x_k}{\partial t}=x_k'(t)$$. Thus

$$x_k'(t)=H_{p_k}.$$

Now we differentiate $$p_i(a,b;t)$$ with respect to $$t$$ and $$\phi_t+H(x,t,\nabla_x\phi)=0$$ with respect to $$x_i$$, and obtain

\begin{eqnarray*}
p_i'(t)&=&\phi_{x_it}+\sum_{k=1}^n\phi_{x_ix_k}x_k'(t)\\
0&=&\phi_{x_it}+\sum_{k=1}^n\phi_{x_ix_k}H_{p_k}+H_{x_i}\\
0&=&\phi_{x_it}+\sum_{k=1}^n\phi_{x_ix_k}x_k'(t)+H_{x_i}
\end{eqnarray*}

It follows finally that $$p_i'(t)=-H_{x_i}$$.

$$\Box$$

Example 2.5.1: Kepler problem

The motion of a mass point in a central field takes place in a plane, say the $$(x,y)$$-plane, see Figure 2.5.1, and satisfies the system of ordinary differential equations of second order

$$x''(t)=U_x,\ y''(t)=U_y,$$

where

$$U(x,y)=\frac{k^2}{\sqrt{x^2+y^2}}.$$

Here we assume that $$k^2$$ is a positive constant and that the mass point is attracted of the origin. In the case that it is pushed one has to replace $$U$$ by $$-U$$. See Landau and Lifschitz [12], Vol 1, for instance, concerning the related physics.

Figure 2.5.1: Motion in a central field

Set

$$p=x',\ q=y'$$

and

$$H=\frac{1}{2}(p^2+q^2)-U(x,y),$$

then

\begin{eqnarray*}
x'(t)&=&H_p,\ y'(t)=H_q\\
p'(t)&=&-H_x,\ q'(t)=-H_y.
\end{eqnarray*}
The associated Hamilton-Jacobi equation is
\begin{equation*}
\phi_t+\frac{1}{2}(\phi_x^2+\phi_y^2)=\frac{k^2}{\sqrt{x^2+y^2}}.
\end{equation*}
which is in polar coordinates $$(r,\theta)$$

\label{keplerhj}
\phi_t+\frac{1}{2}(\phi_r^2+\frac{1}{r^2}\phi_\theta^2)=\frac{k^2}{r}.

Now we will seek a complete integral of (\ref{keplerhj}) by making the ansatz

\label{ansatzhj}
\phi_t=-\alpha=const.\ \ \phi_\theta=-\beta=const.

and obtain from (\ref{keplerhj}) that
$$\phi=\pm\int_{r_0}^r\ \sqrt{2\alpha+\frac{2k^2}{\rho}-\frac{\beta^2}{\rho^2}}\ d\rho+c(t,\theta).$$
From ansatz (\ref{ansatzhj}) it follows
$$c(t,\theta)=-\alpha t-\beta\theta.$$
Therefore we have a two parameter family of solutions
$$\phi=\phi(\alpha,\beta;\theta,r,t)$$
of the Hamilton-Jacobi equation. This solution is a complete integral, see an exercise.
According to the theorem of Jacobi set
$$\phi_\alpha=-t_0,\ \ \phi_\beta=-\theta_0.$$
Then
$$t-t_0=-\int_{r_0}^r\ \frac{d\rho}{\sqrt{2\alpha+\frac{2k^2}{\rho}-\frac{\beta^2}{\rho^2}}}.$$
The inverse function $$r=r(t)$$, $$r(0)=r_0$$, is the $$r$$-coordinate depending on time $$t$$, and
$$\theta-\theta_0=\beta\int_{r_0}^r\ \frac{d\rho}{\rho^2\sqrt{2\alpha+\frac{2k^2}{\rho}-\frac{\beta^2}{\rho^2}}}.$$
Substitution $$\tau=\rho^{-1}$$ yields
\begin{eqnarray*}
\theta-\theta_0&=&-\beta\int_{1/r_0}^{1/r}\ \frac{d\tau}{\sqrt{2\alpha+2k^2\tau-\beta^2\tau^2}}\\
&=&-\arcsin\Bigg(\frac{\frac{\beta^2}{k^2}\frac{1}{r}-1}{\sqrt{1+\frac{2\alpha\beta^2}{k^4}}}\Bigg)
+
\arcsin\Bigg(\frac{\frac{\beta^2}{k^2}\frac{1}{r_0}-1}{\sqrt{1+\frac{2\alpha\beta^2}{k^4}}}\Bigg).
\end{eqnarray*}
Set
$$\theta_1=\theta_0+\arcsin\Bigg(\frac{\frac{\beta^2}{k^2}\frac{1}{r_0}-1}{\sqrt{1+\frac{2\alpha\beta^2}{k^4}}}\Bigg)$$
and
$$p=\frac{\beta^2}{k^2},\ \ \epsilon^2=\sqrt{1+\frac{2\alpha\beta^2}{k^4}},$$
then
$$\theta-\theta_1=-\arcsin\left(\frac{\frac{p}{r}-1}{\epsilon^2}\right).$$
It follows
$$r=r(\theta)=\frac{p}{1-\epsilon^2\sin(\theta-\theta_1)},$$
which is the polar equation of conic sections. It defines an ellipse if $$0\le\epsilon<1$$, a parabola if $$\epsilon=1$$ and a hyperbola if $$\epsilon>1$$, see Figure 2.5.2 for the case of an ellipse, where the origin of the coordinate system is one of the focal points of the ellipse.

Figure 2.5.2: The case of an ellipse

For another  application of the Jacobi theorem see Courant and Hilbert [4], Vol. 2, pp. 94,  where geodesics on an ellipsoid are studied.

1Hamilton, William Rowan, 1805--1865

2 Jacobi, Carl Gustav, 1805--1851

### Contributors

• Integrated by Justin Marshall.