Skip to main content
\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)
Mathematics LibreTexts

2.5: Hamilton-Jacobi Theory

  • Page ID
    2138
  • [ "article:topic" ]

    \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

    \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

    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

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

    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\\
    x_k'(\tau)&=&F_{p_k}=H_{p_k},\qquad k=1,\ldots,n\\
    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\\
    &=&-F_{x_k},\qquad k=1,\ldots,n.
    \end{eqnarray*}

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

    \begin{equation}
    \label{hamjac}
    u_t+H(x,t,\nabla_xu)=0
    \end{equation}
    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 \((r,\theta)\)
    \begin{equation}
    \label{keplerhj}
    \phi_t+\frac{1}{2}(\phi_r^2+\frac{1}{r^2}\phi_\theta^2)=\frac{k^2}{r}.
    \end{equation}
    Now we will seek a complete integral of (\ref{keplerhj}) by making the ansatz
    \begin{equation}
    \label{ansatzhj}
    \phi_t=-\alpha=const.\ \ \phi_\theta=-\beta=const.
    \end{equation}
    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