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15.4: Hamilton-Jacobi Theory

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    9652
  • Hamilton used the Principle of Least Action to derive the Hamilton-Jacobi relation (chapter \(15.3\))

    \[H(\mathbf{q},\mathbf{p}, t) + \frac{\partial S}{\partial t} = 0 \label{15.11}\]

    where \(\mathbf{q}, \mathbf{p}\) refer to the \(1 \leq i \leq n\) variables \(q_i, p_i\) and \(S(q_j (t_1), t_1, q_j (t_2), t_2)\) is the action functional. Integration of this first-order partial differential equation is non trivial which is a major handicap for practical exploitation of the Hamilton-Jacobi equation. This stimulated Jacobi to develop the mathematical framework for canonical transformation that are required to solve the Hamilton-Jacobi equation. Jacobi’s approach is to exploit generating functions for making a canonical transformation to a new Hamiltonian \(\mathcal{H}(\mathbf{Q}, \mathbf{P}, t)\) that equals zero.

    \[\mathcal{H}(\mathbf{Q},\mathbf{P}, t) = H(\mathbf{q},\mathbf{p}, t) + \frac{\partial S}{\partial t} = 0 \label{15.90}\]

    The generating function for solving the Hamilton-Jacobi equation then equals the action functional \(S\).

    The Hamilton-Jacobi theory is based on selecting a canonical transformation to new coordinates \((Q, P, t)\) all of which are either constant, or the \(Q_i\) are cyclic, which implies that the corresponding momenta \(P_i\) are constants. In either case, a solution to the equations of motion is obtained. A remarkable feature of Hamilton-Jacobi theory is that the canonical transformation is completely characterized by a single generating function, \(S\). The canonical equations likewise are characterized by a single Hamiltonian function, \(H\). Moreover, the generating function \(S\), and Hamiltonian function \(H\), are linked together by Equation \ref{15.11}. The underlying goal of Hamilton-Jacobi theory is to transform the Hamiltonian to a known form such that the canonical equations become directly integrable. Since this transformation depends on a single scalar function, the problem is reduced to solving a single partial differential equation.

    Time-dependent Hamiltonian

    Jacobi’s complete integral \(S(q_i, P_i, t)\)

    The principle underlying Jacobi’s approach to Hamilton-Jacobi theory is to provide a recipe for finding the generating function \(F = S\) needed to transform the Hamiltonian \(H(\mathbf{q}, \mathbf{p}, t)\) to the new Hamiltonian \(\mathcal{H}(\mathbf{Q}, \mathbf{P}, t)\) using Equation \ref{15.90}. When the derivatives of the transformed Hamiltonian \(\mathcal{H}(\mathbf{Q}, \mathbf{P}, t)\) are zero, then the equations of motion become

    \[\dot{Q}_i = \frac{\partial \mathcal{H}}{ \partial P_i} = 0 \label{15.91}\]

    \[\dot{P}_i = − \frac{\partial \mathcal{H}}{ \partial Q_i } = 0 \label{15.92}\]

    and thus \(Q_i\) and \(P_i\) are constants of motion. The new Hamiltonian \(\mathcal{H}\) must be related to the original Hamiltonian \(H\) by a canonical transformation for which

    \[\mathcal{H}(\mathbf{Q}, \mathbf{P}, t) = H(\mathbf{q}, \mathbf{p}, t) + \frac{\partial S}{ \partial t} \label{15.93}\]

    Equations \ref{15.91} and \ref{15.92} are automatically satisfied if the new Hamiltonian \(\mathcal{H} = 0\) since then Equation \ref{15.93} gives that the generating function \(S\) satisfies Equation \ref{15.90}.

    Any of the four types of generating function can be used. Jacobi chose the type 2 generating function as being the most useful for many practical cases, that is, \(S(q_i, P_i, t)\) which is called Jacobi’s complete integral.

    For generating functions \(F_1\) and \(F_2\) the generalized momenta are derived from the action by the derivative

    \[p_i = \frac{\partial S}{ \partial q_i} \label{15.4}\]

    Use this generalized momentum to replace \(p_i\) in the Hamiltonian \(H\), given in Equation \ref{15.93}, leads to the Hamilton-Jacobi equation expressed in terms of the action \(S\).

    \[H(q_1, ...q_n; \frac{\partial S}{ \partial q_1 }, ..., \frac{\partial S}{ \partial q_n} ;t) + \frac{\partial S}{ \partial t} = 0 \label{15.94}\]

    The Hamilton-Jacobi equation, \ref{15.94}, can be written more compactly using tensors \(\mathbf{q}\) and \(\boldsymbol{\nabla}S\) to designate \((q_1, ..q_n)\) and \(\frac{\partial S}{ \partial q_1 }, ..., \frac{\partial S}{ \partial q_n}\) respectively. That is

    \[H(\mathbf{q}, \boldsymbol{\nabla}S, t) + \frac{\partial S}{\partial t} = 0 \label{15.95}\]

    Equation \ref{15.95} is a first-order partial differential equation in \(n + 1\) variables which are the old spatial coordinates \(q_i\) plus time \(t\). The new momenta \(P_i\) have not been specified except that they are constants since \(\mathcal{H} = 0\).

    Assume the existence of a solution of \ref{15.95} of the form \(S(q_i, P_i, t) = S(q_1, ..q_n; \alpha_1, ..\alpha_{n+1};t)\) where the generalized momenta \(P_i = \alpha_1, \alpha_2, ....\alpha\) plus \(t\) are the \(n + 1\) independent constants of integration in the transformed frame. One constant of integration is irrelevant to the solution since only partial derivatives of \(S(q_i, P_i, t)\) with respect to \(q_i\) and \(t\) are involved. Thus, if \(S\) is a solution of the first-order partial differential equation, then so is \(S + \alpha\) where \(\alpha\) is a constant. Thus it can be assumed that one of the \(n + 1\) constants of integration is just an additive constant which can be ignored leading effectively to a solution

    \[S(q_i, P_i, t) = S(q_1, .....q_n;\alpha_1, .....\alpha_n;t) \label{15.96}\]

    where none of the \(n\) independent constants are solely additive. Such generating function solutions are called complete solutions of the first-order partial differential equations since all constants of integration are known.

    It is possible to assume that the \(n\) generalized momenta, \(P_i\) are constants \(\alpha_i\), where the \(\alpha_i\) are the constants. This allows the generalized momentum to be written as

    \[p_i = \frac{\partial S(\mathbf{q}, \boldsymbol{\alpha}, t)}{ \partial q_i } \label{15.97}\]

    Similarly, Hamilton’s equations of motion give the conjugate coordinate \(\mathbf{Q} = \boldsymbol{\beta}\), where \(\beta_i\) are constants. That is

    \[Q_i = \beta_i = \frac{\partial S(\mathbf{q}, \boldsymbol{\alpha}, t)}{ \partial \alpha_i} \label{15.98}\]

    The above procedure has determined the complete set of \(2n\) constants \((\mathbf{Q} = \boldsymbol{\beta}, \mathbf{P} = \boldsymbol{\alpha})\). It is possible to invert the canonical transformation to express the above solution, which is expressed in terms of \(Q_i = \beta_i\) and \(P_i = \alpha_i\), back to the original coordinates, that is, \(q_j = q_j (\alpha , \beta , t)\) and momenta \(p_j = p_j (\alpha , \beta , t)\) which is the required solution.

    Hamilton’s principle function \(S_H(\mathbf{q}_i, t; \mathbf{q}_o t_o)\)

    Hamilton’s approach to solving the Hamilton-Jacobi Equation \ref{15.95} is to seek a canonical transformation from variables \((\mathbf{p}, \mathbf{q})\) at time \(t\), to a new set of constant quantities, which may be the initial values \((\mathbf{q}_0, \mathbf{p}_0)\) at time \(t = 0\). Hamilton’s principle function \(S_H(q_i, t; q_ot_o)\) is the generating function for this canonical transformation from the variables \((\mathbf{q}, \mathbf{p})\) at time t to the initial variables \((\mathbf{q}_0, \mathbf{p}_0)\) at time \(t_0\). Hamilton’s principle function \(S_H(q_i, t; q_ot_o)\) is directly related to Jacobi’s complete integral \(S(q_i, P_i, t)\).

    Note that \(S_H\) is the generating function of a canonical transformation from the present time \((\mathbf{q}, \mathbf{p}, t)\) variables to the initial \((\mathbf{q}_0, \mathbf{p}_0, t_0)\), whereas Jacobi’s \(S\) is the generating function of a canonical transformation from the present \((\mathbf{q},\mathbf{p}, t)\) variables to the constant variables \((\mathbf{Q} = \boldsymbol{\beta}, \mathbf{P} = \boldsymbol{\alpha})\). For the Hamilton approach, the canonical transformation can be accomplished in two steps using \(S\) by first transforming from \((\mathbf{q}, \mathbf{p}, t)\) at time \(t\), to \((\boldsymbol{\beta}, \boldsymbol{\alpha})\), then transforming from \((\boldsymbol{\beta}, \boldsymbol{\alpha})\) to \((\mathbf{q}_0,\mathbf{p}_0, t_0)\). That is, this two-step process corresponds to

    \[S_H(\mathbf{q}, t; \mathbf{q}_ot_o) = S(\mathbf{q}, \boldsymbol{\alpha}, t) − S(\mathbf{q}_0, \boldsymbol{\alpha}, t_0) \label{15.99}\]

    Hamilton’s principle function \(S_H(\mathbf{q}, t; \mathbf{q}_ot_o)\) is related to Jacobi’s complete integral \(S(\mathbf{q}, \boldsymbol{\alpha}, t)\), and it will not be discussed further in this book.

    Time-independent Hamiltonian

    Frequently the Hamiltonian does not explicitly depend on time. For the standard Lagrangian with time-independent constraints and transformation, then \(H (\mathbf{q}, \mathbf{p},t) = E\) which is the total energy. For this case, the Hamilton-Jacobi equation simplifies to give

    \[\frac{\partial S}{ \partial t} = −H( \mathbf{ q}, \mathbf{ p}, t) = −E (\boldsymbol{\alpha}) \label{15.100}\]

    The integration of the time dependence is trivial, and thus the action integral for a time-independent Hamiltonian equals

    \[S(\mathbf{q}, \boldsymbol{\alpha},t) = W (\mathbf{q}, \boldsymbol{\alpha}) − E (\boldsymbol{\alpha})t \label{15.101}\]

    That is, the action integral has separated into a time independent term \(W (\mathbf{q}, \boldsymbol{\alpha})\) which is called Hamilton’s characteristic function plus a time-dependent term \(−E (\boldsymbol{\alpha})t\). Thus using equations \ref{15.97}, \ref{15.101} gives that the generalized momentum is

    \[p_i = \frac{\partial W(\mathbf{q}, \boldsymbol{\alpha})}{ \partial q_i} \label{15.102}\]

    The physical significance of Hamilton’s characteristic function \(W (\mathbf{q}, \boldsymbol{\alpha})\) can be understood by taking the total time derivative

    \[\frac{dW}{ dt} = \sum_i \frac{\partial W(\mathbf{q}, \boldsymbol{\alpha})}{ \partial q_i} \dot{q}_i = \sum_i p_i\dot{q}_i \nonumber\]

    Taking the time integral then gives

    \[W (\mathbf{q}, \boldsymbol{\alpha}) = \int \sum p_i\dot{q}_i dt =\int \sum p_idq_i \label{15.103}\]

    Note that this equals the abbreviated action described in chapter \(9.2.3\), that is \(W(\mathbf{q}, \boldsymbol{\alpha}) = S_0(\mathbf{q}, \boldsymbol{\alpha})\).

    Inserting the action \(S (\mathbf{q}, \boldsymbol{\alpha})\) into the Hamilton-Jacobi equation \((15.2.1)\) gives

    \[H(\mathbf{q}; \frac{\partial W(\mathbf{q}, \boldsymbol{\alpha})}{ \partial \mathbf{q}} ) = E (\boldsymbol{\alpha}) \label{15.104}\]

    This is called the time-independent Hamilton-Jacobi equation. Usually it is convenient to have \(E\) equal the total energy. However, sometimes it is more convenient to exclude the \(k^{th}\) energy \(E(\alpha_k)\) in the set, in which case \(E = E(\alpha_1, \alpha_2, ...\alpha_k−1)\); the Routhian exploits this feature.

    The equations of the canonical transformation expressed in terms of \(W (\mathbf{q}, \boldsymbol{\alpha})\) are

    \[p_i = \frac{\partial W(\mathbf{q}, \boldsymbol{\alpha}) }{\partial q_i } \quad \beta_i + \frac{\partial E(\boldsymbol{\alpha}) }{\partial \alpha_i} t = \frac{\partial W(\mathbf{q}, \boldsymbol{\alpha})}{ \partial \alpha_i} \label{15.105}\]

    These equations show that Hamilton’s characteristic function \(W (\mathbf{q}, \boldsymbol{\alpha})\) is itself the generating function of a time-independent canonical transformation from the old variables \((q, p)\) to a set of new variables

    \[Q_i = \beta_i + \frac{\partial E(\boldsymbol{\alpha})}{ \partial \alpha_i } t \quad P_i = \alpha_i \label{15.106}\]

    Table \(\PageIndex{1}\) summarizes the time-dependent and time-independent forms of the Hamilton-Jacobi equation.

    Hamiltonian Time dependent \(H(q, p, t)\) Time independent \(H(q, p)\)
    Transformed Hamiltonian \(\mathcal{H}= 0\) \(\mathcal{H}\) is cyclic
    Canonical transformed variables All \(Q_iP_i\) are constants of motion All \(P_i\) are constants of motion
    Transformed equations of motion

    \(\dot{Q}_i = \frac{\partial \mathcal{H}}{ \partial P_i} = 0\), therefore \(Q_i = \beta_i\)

    \(\dot{P}_i = − \frac{\partial \mathcal{H}}{ \partial Q_i} = 0\), therefore \(P_i = \alpha_i\)

    \(\dot{Q}_i = \frac{\partial \mathcal{H}}{ \partial P_i} = v_i\), therefore \(Q_i = v_i t + \beta_i\)

    \(\dot{P}_i = − \frac{\partial \mathcal{H}} {\partial Q_i} = 0\), therefore \(P_i = \alpha_i\)

    Generating function Jacobi’s complete integral \(S(\mathbf{q}, \mathbf{P}, t)\) Characteristic Function \(W(\mathbf{q}, \mathbf{P})\)
    Hamilton-Jacobi equation \(H(q_1, ...q_n; \frac{\partial S}{ \partial q_1 }, ..., \frac{\partial S} {\partial q_n} ;t)+\frac{\partial S}{ \partial t} = 0\) \(H(q_1, ...q_n; \frac{\partial W }{\partial q_1} , ..., \frac{\partial W}{ \partial q_n} ) = E\)
    Transformation equations

    \(p_i= \frac{\partial S}{ \partial q_i}\)

    \(Q_i= \frac{\partial S}{ \partial \alpha_i} = \beta_i\)

    \(p_i=\frac{\partial W}{ \partial q_i}\)

    \(Q_i=\frac{\partial W}{ \partial \alpha_i} = v_i t + \beta_i\)

    Table \(\PageIndex{1}\): Hamilton-Jacobi formulations

    Separation of variables

    Exploitation of the Hamilton-Jacobi theory requires finding a suitable action function \(S\). When the Hamiltonian is time independent, then Equation \ref{15.101} shows that the time dependence of the action integral separates out from the dependence on the spatial variables. For many systems, the Hamilton’s characteristic function \(W(\mathbf{q}, \mathbf{P})\) separates into a simple sum of terms each of which is a function of a single variable. That is,

    \[W(\mathbf{q}, \boldsymbol{\alpha}) = W_1(q_1) + W_2(q_2) + \cdots \cdot \cdot W_n(q_n) \label{15.107}\]

    where each function in the summation on the right depends only on a single variable. Then Equation \ref{15.100} reduces to

    \[H(q_1, ...q_n; \frac{\partial W }{\partial q_1} , ...,\frac{ \partial W}{ \partial q_n} ) = E \label{15.108}\]

    where \(E\) is the constant denoting the total energy.

    Hamilton’s characteristic function \(W( \mathbf{ q}, \mathbf{ P})\) can be used with equations \ref{15.101}, \ref{15.102}, \ref{15.91}, \ref{15.92}, and \ref{15.93} to derive

    \[p_i = \frac{\partial W( \mathbf{ q}, \boldsymbol{\alpha}) }{\partial q_i} \quad Q_i = \frac{\partial W( \mathbf{ q}, \boldsymbol{\alpha}) }{\partial P_i} \label{15.109}\]

    \[\dot{Q}_i = \frac{\partial \mathcal{H}}{ \partial P_i} = 0 \quad \dot{P}_i = \frac{\partial \mathcal{H}}{ \partial Q_i} = 0 \label{15.110}\]

    \[\mathcal{H} = H + \frac{\partial S}{\partial t} = H − E = 0 \label{15.111}\]

    which has reduced the problem to a simple sum of one-dimensional first-order differential equations.

    If the \(i^{th}\) variable is cyclic, then the Hamiltonian is not a function of \(q_i\) and the \(i^{th}\) term in Hamilton’s characteristic function equals \(W_i = \alpha_iq_i\) which separates out from the summation in Equation \ref{15.107}. That is, all cyclic variables can be factored out of \(W( \mathbf{ q}, \boldsymbol{\alpha})\) which greatly simplifies solution of the Hamilton-Jacobi equation. As a consequence, the ability of the Hamilton-Jacobi method to make a canonical transformation to separate the system into many cyclic or independent variables, which can be solved trivially, is a remarkably powerful way for solving the equations of motion in Hamiltonian mechanics.

    Example \(\PageIndex{1}\): Free particle

    Consider the motion of a free particle of mass \(m\) in a force-free region. Then Equation \ref{15.93} reduces to

    \[H(q_1, ...q_n; \frac{\partial S}{ \partial q_1} , ..., \frac{\partial S}{ \partial q_n} ;t) + \frac{\partial S}{\partial t} = 0 \nonumber\]

    Since no forces act, and the momentum \(\mathbf{p} = \boldsymbol{\nabla}S\), thus the Hamilton-Jacobi equation reduces to

    \[\frac{1}{ 2m } \nabla^2S + \frac{\partial S}{\partial t} = 0 \tag{A}\label{A}\]

    The Hamiltonian is time independent, thus Equation \ref{15.101} applies

    \[S(\mathbf{q}, t) = W(\mathbf{q}, \boldsymbol{\alpha}) − E(\boldsymbol{\alpha})t \nonumber\]

    Since the Hamiltonian does not explicitly depend on the coordinates \((x, y, z)\), then the coordinates are cyclic and separation of the variables, \ref{15.107}, gives that the action

    \[S = \boldsymbol{\alpha} \cdot \mathbf{ r} − Et \tag{B}\label{B}\]

    For Equation \ref{B} to be a solution of Equation \ref{A} requires that

    \[E = \frac{1}{ 2m} \boldsymbol{\alpha}^2 \tag{C}\label{C}\]

    Therefore

    \[S = \boldsymbol{\alpha} \cdot \mathbf{r} − \frac{1}{ 2m} \boldsymbol{\alpha}^2t \tag{D}\label{D}\]

    Since

    \[\mathbf{\dot{Q}} = \frac{\partial S }{\partial \boldsymbol{\alpha} } = \mathbf{r}− \frac{\boldsymbol{\alpha}}{ m }t \nonumber\]

    the equation of motion and the conjugate momentum are given by

    \[\mathbf{r} = \mathbf{\dot{Q}} + \frac{\boldsymbol{\alpha}}{ m} t \quad \mathbf{p} = \boldsymbol{\nabla}S = \boldsymbol{\alpha} \nonumber\]

    Thus the Hamilton-Jacobi relation has given both the equation of motion and the linear momentum \(\mathbf{p}\).

    Example \(\PageIndex{2}\): Point particle in a uniform gravitational field

    The Hamiltonian is

    \[H = \frac{1}{ 2m} (p^2_x + p^2_y + p^2_z) + mgz \nonumber\]

    Since the system is conservative, then the Hamilton-Jacobi equation can be written in terms of Hamilton’s characteristic function \(W\)

    \[E = \frac{1}{ 2m} \left[\left(\frac{\partial W}{ \partial x} \right)^2 + \left(\frac{\partial W}{ \partial y} \right)^2 + \left(\frac{\partial W}{ \partial z} \right)^2 \right] + mgz \nonumber\]

    Assuming that the variables can be separated \(W = X(x) + Y (y) + Z(z)\) leads to

    \[p_x = \frac{\partial X(x)}{ \partial x} = \alpha_x \nonumber\]

    \[p_y = \frac{\partial Y (y)}{ \partial y} = \alpha_y \nonumber\]

    \[p_z = \frac{\partial Z(z) }{\partial z} = \sqrt{ 2m(E − mgz) − \alpha^2_x − \alpha^2_y} \nonumber\]

    Thus by integration the total \(W\) equals

    \[W = \int^x_{x_0} \alpha_x dx + \int^y_{y_0} \alpha_ydy + \int^z_{z_0} \left(\sqrt{ 2m(E − mgz) − \alpha^2_x − \alpha^2_y }\right) dz \nonumber\]

    Therefore using \ref{15.106} gives

    \[\beta_z = t − t_0 = \int^z_{z_0} \frac{mdz}{ \sqrt{ 2m(E − mgz) − \alpha^2_x − \alpha^2_y } } \nonumber\]

    \[\beta_x = \text{ constant }= (x − x_0) − \int^z_{z_0} \frac{\alpha_xdz}{ \sqrt{ 2m(E − mgz) − \alpha^2_x − \alpha^2_y } } \nonumber\]

    \[\beta_y = \text{ constant } = (y − y_0) − \int^z_{z_0} \frac{\alpha_ydz }{\sqrt{ 2m(E − mgz) − \alpha^2_x − \alpha^2_y }} \nonumber\]

    If \(x_0, y_0, z_0\) is the position of the particle at time \(t = t_0\) then \(\beta_x = \beta_y = 0\), and from \ref{15.106}

    \[x − x_0 = \left(\frac{\alpha_x}{ m }\right) (t − t_0) \nonumber\]

    \[y − y_0 = \left(\frac{\alpha_y}{ m} \right) (t − t_0) \nonumber\]

    \[z − z_0 = \left( \frac{\sqrt{ 2m(E − mgz) − \alpha^2_x − \alpha^2_y }}{ m} \right) (t − t_0) − \frac{1}{ 2} g(t − t_0)^2 \nonumber\]

    This corresponds to a parabola as should be expected for this trivial example.

    Example \(\PageIndex{3}\): One-dimensional harmonic oscillator

    As discussed in example \(15.3.5\) the Hamiltonian for the one-dimensional harmonic oscillator can be written as

    \[H = \frac{1}{ 2m} ( p^2 + m^2\omega^2q^2) = E \nonumber\]

    assuming it is conservative and where \(\omega = \sqrt{\frac{k}{m}}\).

    Hamilton’s characteristic function \(W\) can be used where

    \[S (q, E, t) = W (q, E) − Et \nonumber\]

    \[p_i = \frac{\partial W}{ \partial q_i} \nonumber\]

    Inserting the generalized momentum \(p_i\) into the Hamiltonian gives

    \[\frac{1}{ 2m} \left(\left[ \frac{\partial W }{\partial q} \right]^2 + m^2\omega^2q^2 \right) = E \nonumber\]

    Integration of this equation gives

    \[W = \sqrt{ 2mE} \int dq \sqrt{1 − \frac{m\omega^2q^2}{ 2E}} \nonumber\]

    That is

    \[S = \sqrt{ 2mE } \int dq \sqrt{ 1 − \frac{m\omega^2q^2}{ 2E}} − Et \nonumber\]

    Note that

    \[\frac{\partial S(q, E, t)}{ \partial E} = \sqrt{\frac{2m }{E}} \int \frac{dq}{\sqrt{1 - \frac{ m\omega^2q^2}{ 2E}}} − t \nonumber\]

    This can be integrated to give

    \[t = \frac{1}{ \omega }\arcsin \left( q \sqrt{\frac{m\omega^2}{ 2E}}\right) + t_0 \nonumber\]

    That is

    \[q = \sqrt{\frac{2E}{m\omega^2}} \sin \omega (t − t_0) \nonumber\]

    This is the familiar solution of the undamped harmonic oscillator.

    Example \(\PageIndex{4}\): The central force problem

    The problem of a particle acted upon by a central force occurs frequently in physics. Consider the mass \(m\) acted upon by a time-independent central potential energy \(U(r)\). The Hamiltonian is time independent and can be written in spherical coordinates as

    \[H = \frac{1}{ 2m} \left( p^2_r + \frac{1}{r^2} p^2_{\theta} + \frac{1}{r^2 \sin^2 \theta} p^2_{\psi} \right) + U(r) = E \nonumber\]

    The time-independent Hamilton-Jacobi equation is conservative, thus

    \[\frac{1}{ 2m} \left[\left(\frac{\partial W}{ \partial r }\right)^2 + \frac{1}{ r^2} \left(\frac{\partial W }{\partial \theta} \right)^2 + \frac{1}{ r^2 \sin^2 \theta} \left(\frac{\partial W}{ \partial \phi} \right)^2 \right] + U(r) = E \nonumber\]

    Try a separable solution for Hamilton’s characteristic function \(W\) of the form

    \[W = R(r) + \Theta (\theta ) + \Phi (\phi ) \nonumber\]

    The Hamilton-Jacobi equation then becomes

    \[\frac{1}{ 2m} \left[\left(\frac{\partial R} {\partial r} \right)^2 + \frac{1}{ r^2} \left(\frac{\partial \Theta}{ \partial \theta} \right)^2 + \frac{1}{ r^2 \sin^2 \theta} \left(\frac{\partial \Phi}{ \partial \phi } \right)^2 \right] + U(r) = E \nonumber\]

    This can be rearranged into the form

    \[2mr^2 \sin^2 \theta \left\{ \frac{1}{ 2m} \left[\left(\frac{\partial R}{ \partial r} \right)^2 + \frac{1} {r^2} \left(\frac{\partial \Theta}{ \partial \theta} \right)^2 \right] + U(r) + E \right\} = − \left(\frac{\partial \Phi}{ \partial \phi} \right)^2 \nonumber\]

    The left-hand side is independent of \(\phi\) whereas the right-hand side is independent of \(r\) and \(\theta\). Both sides must equal a constant which is set to equal \(−L^2_z\), that is

    \[\frac{1}{2m} \left[\left(\frac{\partial R}{ \partial r} \right)^2 + \frac{1}{ r^2} \left(\frac{\partial \Theta}{ \partial \theta} \right)^2 \right] + U(r) + \frac{L^2_z }{2mr^2 \sin^2 \theta} = E \nonumber\]

    \[\left(\frac{\partial \Phi}{ \partial \phi} \right)^2 = L^2_z \nonumber\]

    The equation in \(r\) and \(\theta\) can be rearranged in the form

    \[2mr^2 \left[ \frac{1}{2m} \left(\frac{\partial R}{ \partial r} \right)^2 + U(r) − E \right] = − \left[\left(\frac{\partial \Theta}{ \partial \theta} \right)^2 + \frac{L^2_z}{ \sin^2 \theta} \right] \nonumber\]

    The left-hand side is independent of \(\theta\) and the right-hand side is independent of \(r\) so both must equal a constant which is set to be \(−L^2\)

    \[\frac{1}{2m} \left(\frac{\partial R}{ \partial r} \right)^2 + U(r) + \frac{L^2}{ 2mr^2} = E \nonumber\]

    \[\left(\frac{\partial \Theta}{ \partial \theta} \right)^2 + \frac{L^2_z}{ \sin^2 \theta} = L^2 \nonumber\]

    The variables now are completely separated and, by rearrangement plus integration, one obtains

    \[R(r) = \sqrt{2m} \int \sqrt{ E − U(r) − \frac{L^2 }{2mr^2}} dr \nonumber\]

    \[\Theta (\theta ) = \int \sqrt{ L^2 − \frac{L^2_z}{ \sin^2 \theta}} d\theta \nonumber\]

    \[\Phi (\phi ) = L_z \phi \nonumber\]

    Substituting these into \(W = R(r) + \Theta (\theta ) + \Phi (\phi )\) gives

    \[W = \sqrt{2m} \int \sqrt{ E − U(r) − \frac{L^2 }{2mr^2}} dr + \int \sqrt{ L^2 − \frac{L^2_z}{ \sin^2 \theta}} d\theta + L_z \phi \nonumber\]

    Hamilton’s characteristic function \(W\) is the generating function from coordinates \((r, \theta , \phi , p_r, p_{\theta} , p_{\phi} )\) to new coordinates, which are cyclic, and new momenta that are constant and taken to be the separation constants \(E, L, L_z \).

    \[p_r = \frac{\partial W}{ \partial r} = \sqrt{2m} \sqrt{ E − U(r) − \frac{L^2 }{2mr^2}} \nonumber\]

    \[p_{\theta} = \frac{\partial W}{ \partial \theta} = \sqrt{ L^2 − \frac{L^2_z}{ \sin^2 \theta}} \nonumber\]

    \[p_{\phi} = \frac{\partial W}{ \partial \phi} = L_z \nonumber\]

    Similarly, using \ref{15.109} gives the new coordinates \(E, L, L_z\)

    \[\beta_E + t = \frac{\partial W}{ \partial E} = \sqrt{\frac{m}{ 2}} \int \frac{dr}{\sqrt{ E − U(r) − \frac{L^2}{ 2mr^2}}} \nonumber\]

    \[\beta_L = \frac{\partial W}{ \partial L} = \sqrt{2m} \int \frac{dr}{\sqrt{E − U(r) − \frac{L^2}{ 2mr^2}}} \left( \frac{−L }{2mr^2} \right) + \int \frac{Ld\theta}{\sqrt{ L^2 − \frac{L^2_z }{\sin^2 \theta}}} \nonumber\]

    \[\beta_{L_z} = \frac{\partial W}{ \partial L_z} = \int \frac{d\theta}{\sqrt{ L^2 − \frac{L^2_z}{ \sin^2 \theta}}} \left( \frac{−L}{ 2mr^2} \right) + \phi \nonumber\]

    These equations lead to the elliptical, parabolic, or hyperbolic orbits discussed in chapter \(11\).

    Example \(\PageIndex{5}\): Linearly-damped, one-dimensional, harmonic oscillator

    A canonical treatment of the linearly-damped harmonic oscillator provides an example that combines use of non-standard Lagrangian and Hamiltonians, a canonical transformation to an autonomous system, and use of Hamilton-Jacobi theory to solve this transformed system. It shows that Hamilton-Jacobi theory can be used to determine directly the solutions for the linearly-damped harmonic oscillator.

    Non-standard Hamiltonian:

    In chapter \(3.5\), the equation of motion for the linearly-damped, one-dimensional, harmonic oscillator was given to be

    \[\frac{m}{ 2} [ \ddot{q}+ \Gamma \dot{q} + \omega^2_0 q ] = 0 \tag{a}\label{a} \]

    Example \(10.5.1\) showed that three non-standard Lagrangians give equation of motion \(\alpha\) when used with the standard Euler-Lagrange variational equations. One of these was the Bateman[Bat31] time-dependent Lagrangian

    \[L_2 (q, \dot{q}, t ) = \frac{m}{ 2} e^{\Gamma t} [ \dot{q}^2 − \omega^2_0 q^2] \tag{b}\label{b} \]

    This Lagrangian gave the generalized momentum to be

    \[p = \frac{\partial L^2}{ \partial \dot{q}} = m\dot{q} e^{\Gamma t} \tag{c}\label{c} \]

    which was used with equation \((15.1.3)\) to derive the Hamiltonian

    \[H_2(q, p, t) = p\dot{q} − L_2(q, \dot{q}, t ) = e^{−\Gamma t} \frac{p^2}{ 2m} + \frac{1}{ 2} m\omega^2_0 q^2e^{\Gamma t} \tag{d}\label{d1} \]

    Note that both the Lagrangian and Hamiltonian are explicitly time dependent and thus they are not conserved quantities. This is as expected for this dissipative system.

    Hamilton-Jacobi theory:

    The form of the non-autonomous Hamiltonian \ref{d1} suggests use of the generating function for a canonical transformation to an autonomous Hamiltonian, for which \(H\) is a constant of motion.

    \[S(q, P, t) = F_2(q, P, t) = qPe^{ \frac{\Gamma t}{ 2}} = QP \tag{d}\label{d2} \]

    Then the canonical transformation gives

    \[p = \frac{\partial S}{ \partial q} = P e^{\frac{ \Gamma t}{ 2}} \label{e}\tag{e} \]

    \[Q = \frac{\partial S}{ \partial P} = qe^{\frac{ \Gamma t}{ 2}} \nonumber\]

    Insert this canonical transformation into the above Hamiltonian leads to the transformed Hamiltonian that is autonomous.

    \[\mathcal{H}(Q, P, t) = H_2(q, p, t) + \frac{\partial F_2}{ \partial t} = \frac{P^2}{ 2m} + \frac{\Gamma}{ 2} QP + \frac{m\omega^2_0}{ 2} Q^2 \tag{f}\label{f} \]

    That is, the transformed Hamiltonian \(\mathcal{H}(Q, P, t)\) is not explicitly time dependent, and thus is conserved. Expressed in the original canonical variables \((q, p)\), the transformed Hamiltonian \(\mathcal{H}(Q, P, t)\)

    \[\mathcal{H}(Q, P, t)= \frac{p^2}{ 2m } e^{−\Gamma t }+ \frac{\Gamma }{2} qp + \frac{m\omega^2_0 }{2} q2e^{\Gamma t }\nonumber\]

    is a constant of motion which was not readily apparent when using the original Hamiltonian. This unexpected result illustrates the usefulness of canonical transformations for solving dissipative systems. The Hamilton-Jacobi theory now can be used to solve the equations of motion for the transformed variables \((Q, P)\) plus the transformed Hamiltonian \(\mathcal{H}(Q, P, t)\). The derivative of the generating function

    \[\frac{\partial S}{ \partial Q} = P \label{g}\tag{g} \]

    Use Equation \ref{g} to substitute for \(P\) in the Hamiltonian \(\mathcal{H}(Q, P, t)\) (Equation \ref{f}), then the Hamilton-Jacobi method gives

    \[\frac{1}{2m} \left( \frac{\partial S }{\partial Q} \right)^2 + \frac{\Gamma}{ 2} Q \frac{\partial S}{ \partial Q} + \frac{m\omega^2_0}{ 2} Q^2 + \frac{\partial S}{\partial t} = 0 \nonumber\]

    This equation is separable as described in \ref{15.107} and thus let

    \[S(Q, \alpha , t) = W(Q, \alpha ) − \alpha t \nonumber\]

    where \(\alpha\) is a separation constant. Then

    \[\left[ \frac{1}{2m} \left(\frac{\partial W }{\partial Q} \right)^2 + \Gamma Q\frac{\partial W}{ \partial Q} + \frac{m\omega^2_0}{ 2} Q^2 \right] = \alpha \tag{h}\label{h} \]

    To simplify the equations define the variable x as

    \[x \equiv \sqrt{m\omega_0} Q \label{i}\tag{i}\]

    then Equation \ref{h} can be written as

    \[\left(\frac{\partial W}{ \partial x} \right)^2 + Ax\frac{\partial W}{ \partial x} + (x^2 − B ) = 0 \tag{j}\label{j} \]

    where \(A = \frac{\Gamma}{ \omega_0}\) and \(B = \frac{2\alpha}{\omega_0}\). Assume initial conditions \(q(0) = q_0\) and \(\dot{q}(0) = 0 \)

    For this case the separation constant \(\alpha > 0\), therefore \(B > 0\). Note that Equation \ref{j} is a simple second-order algebraic relation, the solution of which is

    \[\frac{\partial W}{ \partial x} = −\frac{\alpha x}{ 2} \pm \sqrt{B − \left[ 1 − \left(\frac{A}{ 2} \right)^2 \right] x^2} \label{k}\tag{k} \]

    The choice of the sign is irrelevant for this case and thus the positive sign is chosen. There are three possible cases for the solution depending on whether the square-root term is real, zero, or imaginary.

    Case 1: \(\frac{A}{ 2} < 1\), that is, \(\frac{\lambda }{2m\omega_0 }< 1 \)

    Define \(C = \sqrt{\left[ 1 − ( \frac{A}{ 2} )^2 \right]}\) Then Equation \ref{k} can be integrated to give

    \[S = −\alpha t − \frac{Ax^2}{ 4} + \int \sqrt{(B − C^2x^2)}dx \tag{l}\label{l} \]

    and

    \[\beta = \frac{\partial S}{ \partial \alpha} = −t + \frac{1}{ \omega_0 } \int \frac{ dx }{\sqrt{(B − C^2x^2)}} \nonumber\]

    This integral gives

    \[sin^{−1} \left( \frac{Cx}{ \sqrt{B}} \right) = C\omega_0 (t + \beta ) \equiv \omega t + \delta \nonumber\]

    where

    \[\omega = \omega_0 C = \omega_0 \sqrt{1 − \left( \frac{\Gamma }{2\omega_0} \right)^2} = \sqrt{ \omega^2_0 − \left(\frac{\Gamma }{2} \right)^2} \label{m}\tag{m}\]

    Transforming back to the original variable \(q\) gives

    \[q(t) = Ge^{−\frac{ \Gamma t }{2}} \sin (\omega t + \delta ) \tag{n}\label{n}\]

    where \(G\) and \(\delta\) are given by the initial conditions. Equation \ref{m} is identical to the solution for the underdamped linearly-damped linear oscillator given previously in equation \((3.5.12)\).

    Case 2: \(\frac{A}{ 2} = 1\), that is, \(\frac{\Gamma }{2\omega_0} = 1\)

    In this case \(C = \sqrt{\left[ 1 − ( \frac{A}{ 2} )^2 \right]} = 0\) and thus Equation \ref{k} simplifies to

    \[S = −\alpha t − \frac{Ax^2}{ 4} + x \sqrt{B} \nonumber\]

    and

    \[\beta = \frac{\partial S}{ \partial \alpha} = −t + \frac{x}{ \omega_0 \sqrt{B}} \nonumber\]

    Therefore the solution is

    \[q(t) = e^{− \frac{\Gamma t}{ 2}} (F + Gt) \label{o}\tag{o} \]

    where \(F\) and \(G\) are constants given by the initial conditions. This is the solution for the critically-damped linearly-damped, linear oscillator given previously in equation \((3.5.15)\).

    Case 3: \(\frac{A}{ 2} > 1\), that is, \(\frac{\Gamma }{2\omega_0} > 1\)

    Define a real constant \(D\) where \(D = \sqrt{\left[ ( \frac{A}{ 2} )^2 - 1\right]} = iC\), then

    \[S = −\alpha t − \frac{Ax^2}{ 4} + \int \sqrt{(B + D^2x^2)}dx \nonumber\]

    Then

    \[\beta = \frac{\partial S}{ \partial \alpha} = −t + \frac{1}{ \omega_0 } \int \frac{dx}{\sqrt{(B + D^2x^2)}} \nonumber\]

    This last integral gives

    \[\sinh^{−1} \left( \frac{Dx}{ \sqrt{B}} \right) = D\omega_0 (t + \beta ) \equiv \omega t + \delta \nonumber\]

    where

    \[\omega = \omega_0C = \omega_0 \sqrt{\left( \frac{\lambda }{2m\omega_0} \right)^2 − 1} \nonumber\]

    Then the original variable gives

    \[q(t) = Ge^{− \frac{\Gamma t}{ 2 }} \sinh (\omega t + \delta ) \tag{l}\label{l2} \nonumber\]

    This is the classic solution of the overdamped linearly-damped, linear harmonic oscillator given previously in equation \((3.5.14)\)​​​​​​. The canonical transformation from a non-autonomous to an autonomous system allowed use of Hamiltonian mechanics to solve the damped oscillator problem.

    Note that this example used Bateman’s non-standard Lagrangian, and corresponding Hamiltonian, for handling a dissipative linear oscillator system where the dissipation depends linearly on velocity. This nonstandard Lagrangian led to the correct equations of motion and solutions when applied using either the time-dependent Lagrangian, or time-dependent Hamiltonian, and these solutions agree with those given in chapter \(3.5\) which were derived using Newtonian mechanics.

    Visual representation of the action function \(S\).

    15.4.11.PNG
    Figure \(\PageIndex{1}\): Surfaces of constant action integral S (dashed lines) and the corresponding particle momenta (solid lines) with arrows showing the direction.

    The important role of the action integral \(S\) can be illuminated by considering the case of a single point mass \(m\) moving in a time independent potential \(U(r)\). Then the action reduces to

    \[S(q, \alpha , t) = W(q, \alpha ) − Et \label{15.112}\]

    Let \(q_1 = x, q_2 = y, q_3 = z, p_1 = p_x, p_2 = p_y, p_3 = p_z\). The momentum components are given by

    \[p_i = \frac{\partial W(q, \alpha ) }{\partial q_i} \label{15.113}\]

    which corresponds to

    \[\mathbf{p} = \boldsymbol{\nabla}W = \boldsymbol{\nabla}S \label{15.114}\]

    That is, the time-independent Hamilton-Jacobi equation is

    \[\frac{1}{2m} |\boldsymbol{\nabla}W|^2 + U(r) = E \label{15.115}\]

    This implies that the particle momentum is given by the gradient of Hamilton’s characteristic function and is perpendicular to surfaces of constant \(W\) as illustrated in Figure \(\PageIndex{1}\). The constant \(W\) surfaces are time dependent as given by Equation \ref{15.101}. Thus, if at time \(t = 0\) the equi-action surface \(S_0(q, t) = W_0(q, P_i)=0\), then at \(t = 1\) the same surface \(S_0(q, t)=0\) now coincides with the \(S_0(q, t) = E\) surface etc. That is, the equi-action surfaces move through space separately from the motion of the single point mass.

    The above pictorial representation is analogous to the situation for motion of a wavefront for electromagnetic waves in optics, or matter waves in quantum physics where the wave equation separates into the form \(\phi = \phi_0 e^{\frac{ iS}{ \hbar }} = \phi_0 e^{i(\mathbf{k} \cdot \mathbf{r}−\omega t)}\). Hamilton’s goal was to create a unified theory for optics that was equally applicable to particle motion in classical mechanics. Thus the optical-mechanical analogy of the Hamilton-Jacobi theory has culminated in a universal theory that describes wave-particle duality; this was a Holy Grail of classical mechanics since Newton’s time. It played an important role in development of the Schrödinger representation of quantum mechanics.

    Advantages of Hamilton-Jacobi theory

    Initially, only a few scientists, like Jacobi, recognized the advantages of Hamiltonian mechanics. In 1843 Jacobi made some brilliant mathematical developments in Hamilton-Jacobi theory that greatly enhanced exploitation of Hamiltonian mechanics. Hamilton-Jacobi theory now serves as a foundation for contemporary physics, such as quantum and statistical mechanics. A major advantage of Hamilton-Jacobi theory, compared to other formulations of analytic mechanics, is that it provides a single, first-order partial differential equation for the action \(S\), which is a function of the \(n\) generalized coordinates \(\mathbf{q}\) and time \(t\). The generalized momenta no longer appear explicitly in the Hamiltonian in equations \ref{15.94}, \ref{15.95}. Note that the generalized momentum do not explicitly appear in the equivalent Euler-Lagrange equations of Lagrangian mechanics, but these comprise a system of \(n\) second-order, partial differential equations for the time evolution of the generalized coordinate \(\mathbf{q}\). Hamilton’s equations of motion are a system of \(2n\) first-order equations for the time evolution of the generalized coordinates and their conjugate momenta.

    An important advantage of the Hamilton-Jacobi theory is that it provides a formulation of classical mechanics in which motion of a particle can be represented by a wave. In this sense, the Hamilton-Jacobi equation fulfilled a long-held goal of theoretical physics, that dates back to Johann Bernoulli, of finding an analogy between the propagation of light and the motion of a particle. This goal motivated Hamilton to develop Hamiltonian mechanics. A consequence of this wave-particle analogy is that the Hamilton-Jacobi formalism featured prominently in the derivation of the Schrödinger equation during the development of quantum-wave mechanics.