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{{classical mechanics|formulations}}
In [[mathematics]], the '''Hamilton–Jacobi equation''' ('''HJE''') is a [[Necessity and sufficiency#Necessity|necessary condition]] describing extremal [[geometry]] in generalizations of problems from the [[calculus of variations]], and is a special case of the [[Hamilton–Jacobi–Bellman equation]]. It is named for [[William Rowan Hamilton]] and [[Carl Gustav Jacob Jacobi]].

In [[physics]], the Hamilton–Jacobi equation is an alternative formulation of [[classical mechanics]], equivalent to other formulations such as [[Newton's laws of motion]]{{citation needed|date=August 2015}}, [[Lagrangian mechanics]] and [[Hamiltonian mechanics]].  The Hamilton–Jacobi equation is particularly useful in identifying [[conserved quantities]] for mechanical systems, which may be possible even when the mechanical problem itself cannot be solved completely.

The HJE is also the only formulation of mechanics in which the motion of a particle can be represented as a wave.  In this sense, the HJE fulfilled a long-held goal of theoretical physics (dating at least to [[Johann Bernoulli]] in the 18th century) of finding an analogy between the propagation of light and the motion of a particle.  The wave equation followed by mechanical systems is similar to, but not identical with, [[Schrödinger's equation]], as described below; for this reason, the HJE is considered the "closest approach" of [[classical mechanics]] to [[quantum mechanics]].<ref name=Goldstein_484>{{cite book |last=Goldstein |first=Herbert |authorlink=Herbert Goldstein |year=1980 |title= Classical Mechanics |edition=2nd |publisher=Addison-Wesley |location=Reading, MA |isbn= 978-0-201-02918-5 |pages=484–492|title-link=Classical Mechanics (textbook) }} (particularly the discussion beginning in the last paragraph of page 491)</ref><ref name=Sakurai_103>Sakurai, pp. 103–107.</ref>

==Notation==
Boldface variables such as <math>\mathbf{q}</math> represent a list of <math>N</math> [[generalized coordinates]],

:<math>
\mathbf{q} \equiv (q_1, q_2, \ldots, q_{N-1}, q_N)
</math>

A dot over a variable or list signifies the time derivative (see [[Newton's notation]]), e.g.,

:<math>
\dot{\mathbf{q}} \equiv \frac{d\mathbf{q}}{dt}.
</math>

The [[dot product]] notation between two lists of the same number of coordinates is a shorthand for the sum of the products of corresponding components, e.g.,

:<math>
\mathbf{p} \cdot \mathbf{q} \equiv \sum_{k=1}^N p_k q_k.
</math>

==Mathematical formulation==
Given the [[Hamiltonian mechanics|Hamiltonian]] <math>H(q,p,t)</math> of a mechanical system (where <math>q</math>, <math>p</math> are coordinates and momenta of the system and <math>t</math> is time) the Hamilton–Jacobi equation is written as a first-order, [[non-linear differential equation|non-linear]] [[partial differential equation]] for the '''Hamilton's principal function''' <math>S(q,t)</math><ref>{{cite book |first=L. N. |last=Hand |first2=J. D. |last2=Finch |title=Analytical Mechanics |location= |publisher=Cambridge University Press |year=2008 |isbn=978-0-521-57572-0 }}</ref>,

{{Equation box 1
|indent =:
|equation = <math> - \frac{\partial S}{\partial t} = H\left(q,\frac{\partial S}{\partial q},t\right) \;.</math>
|cellpadding
|border
|border colour = #50C878
|background colour = #ECFCF4}}

The Hamilton's principal function is defined as the function of the upper limit of the [[action (physics)|action integral]] taken along the minimal action trajectory of the system,
{{Equation box 1
|equation=<math> S(q,t) \doteq \int^{(q,t)}\mathcal{L}\,\mathrm{d}t \;, </math>
}}

where <math>\mathcal{L}</math> is the [[Lagrangian mechanics|Lagrangian]] of the system and where the trajectory satisfies the [[Euler–Lagrange equation]] of the system,

:<math>\frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial \mathcal{L}}{\partial\dot{q}}=\frac{\partial\mathcal{L}}{\partial q} \;.</math>

Calculating the variation of <math>S</math> with respect to variation of the end-point coordinate,
:<math>
\delta S = \int \mathrm{d}t\left(
\frac{\partial\mathcal{L}}{\partial q}\delta q +
\frac{\partial\mathcal{L}}{\partial\dot{q}}\delta\dot{q}
\right)
=\int \mathrm{d}t\left(
\frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial\mathcal{L}}{\partial \dot{q}}\delta q +
\frac{\partial\mathcal{L}}{\partial\dot{q}}\frac{\mathrm{d}}{\mathrm{d}t}\delta{q}
\right)
=\int \mathrm{d}t \frac{\mathrm{d}}{\mathrm{d}t}\left(\frac{\partial\mathcal{L}}{\partial \dot{q}}\delta q\right)
=\frac{\partial\mathcal{L}}{\partial \dot{q}}\delta q = p\delta q \;,
</math>
leads to
{{Equation box 1
|equation = <math>\frac{\partial S}{\partial q} = p \;.</math>
}}

Using this result and calculating the variation of <math>S</math> with respect to variation of the time of the end-point leads directly to the Hamilton–Jacobi equation,

:<math>
\delta S = \mathcal{L}\delta t + \frac{\partial\mathcal{L}}{\partial \dot{q}}\delta q
= \mathcal{L}\delta t - \frac{\partial\mathcal{L}}{\partial \dot{q}}\dot{q}\delta t
= -H\delta t \;,
</math>
or
{{Equation box 1
|equation = <math> \frac{\partial S}{\partial t}=-H \;,</math>
}}
where <math>\delta q = -\dot{q}\delta t</math> is the change of the trajectory at the old end-point due to the time shift and where
:<math>
H = \frac{\partial\mathcal{L}}{\partial \dot{q}}\dot{q}-\mathcal{L}
</math>
is the Hamiltonian of the system.

Alternatively, as described below, the Hamilton–Jacobi equation may be derived from [[Hamiltonian mechanics]] by treating ''S'' as the [[generating function (physics)|generating function]] for a [[canonical transformation]] of the classical Hamiltonian

:<math>H = H(q_1,q_2,\ldots, q_N;p_1,p_2,\ldots, p_N;t).</math>

The conjugate momenta correspond to the first derivatives of ''S'' with respect to the generalized coordinates

:<math>p_k = \frac{\partial S}{\partial q_k}.</math>

As a solution to the Hamilton–Jacobi equation, the principal function contains ''N'' + 1 undetermined constants, the first ''N'' of them denoted as ''α''<sub>1</sub>, ''α''<sub>2</sub> ... ''α<sub>N</sub>'', and the last one coming from the integration of <math>\scriptstyle \partial S/\partial t </math>.

The relationship between '''p''' and '''q''' then describes the orbit in [[phase space]] in terms of these [[constants of motion]]. Furthermore, the quantities

:<math>\beta_k=\frac{\partial S}{\partial\alpha_k},\quad k=1,2, \ldots, N </math>

are also constants of motion, and these equations can be inverted to find '''q''' as a function of all the α and β constants and time.<ref name=Goldstein_440>{{cite book |last=Goldstein |first=Herbert |authorlink=Herbert Goldstein |year=1980 |title= Classical Mechanics |edition=2nd |publisher=Addison-Wesley |location=Reading, MA |isbn= 978-0-201-02918-5 |pages=440|title-link=Classical Mechanics (Goldstein book) }}</ref>

==Comparison with other formulations of mechanics==

The HJE is a ''single'', first-order partial differential equation for the function ''S'' of the ''N'' [[generalized coordinate]]s ''q''<sub>1</sub>...''q<sub>N</sub>'' and the time ''t''.  The generalized momenta do not appear, except as derivatives of ''S''.  Remarkably, the function ''S'' is equal to the [[Action (physics)|classical action]].

For comparison, in the equivalent [[Euler–Lagrange equation|Euler–Lagrange equations of motion]] of [[Lagrangian mechanics]], the conjugate momenta also do not appear; however, those equations are a ''system'' of ''N'', generally second-order equations for the time evolution of the generalized coordinates. Similarly, [[Hamilton's equations|Hamilton's equations of motion]] are another ''system'' of 2''N'' first-order equations for the time evolution of the generalized coordinates and their conjugate momenta ''p''<sub>1</sub>...''p<sub>N</sub>''.

Since the HJE is an equivalent expression of an integral minimization problem such as [[Hamilton's principle]], the HJE can be useful in other problems of the [[calculus of variations]] and, more generally,  in other branches of [[mathematics]] and [[physics]], such as [[dynamical systems]], [[symplectic geometry]] and [[quantum chaos]].  For example, the Hamilton–Jacobi equations can be used to determine the [[geodesic]]s on a [[Riemannian manifold]], an important [[calculus of variations|variational problem]] in [[Riemannian geometry]].

==Derivation using canonical transformation==
{{hatnote|See the [[canonical transformation]] article for more details.}}

Any [[canonical transformation]] involving a type-2 [[generating function (physics)|generating function]] ''G''<sub>2</sub>('''q''', '''P''', ''t'') leads to the relations

:<math>
\mathbf{p} = {\partial G_2 \over \partial \mathbf{q}}, \quad
\mathbf{Q} = {\partial G_2 \over \partial \mathbf{P}}, \quad
K(\mathbf{Q},\mathbf{P},t) = H(\mathbf{q},\mathbf{p},t) + {\partial G_2 \over \partial t}
</math>

and Hamilton's equations in terms of the new variables '''P''', '''Q''' and new Hamiltonian ''K'' have the same form:

:<math> \dot{\mathbf{P}}  = -{\partial K \over \partial \mathbf{Q}},
\quad \dot{\mathbf{Q}}  = +{\partial K \over \partial \mathbf{P}}. </math>

To derive the HJE, we ''choose'' a generating function ''G''<sub>2</sub>('''q''', '''P''', ''t'') in such a way that, it will make the new Hamiltonian ''K'' = 0.
Hence, all its derivatives are also zero, and the transformed [[Hamilton's equations]] become trivial

:<math>\dot{\mathbf{P}} = \dot{\mathbf{Q}} = 0</math>

so the new generalized coordinates and momenta are [[constant of motion|''constants'' of motion]].  As they are constants, in this context the new generalized momenta '''P''' are usually denoted ''α''<sub>1</sub>, ''α''<sub>2</sub> ... ''α<sub>N</sub>'', i.e. ''P<sub>m</sub>'' = ''α<sub>m</sub>'', and the new [[generalized coordinates]] '''Q''' are typically denoted as ''β''<sub>1</sub>, ''β''<sub>2</sub> ... ''β<sub>N</sub>'', so ''Q<sub>m</sub>'' = ''β<sub>m</sub>''.

Setting the generating function equal to Hamilton's principal function, plus an arbitrary constant ''A'':

:<math>G_2(\mathbf{q},\boldsymbol{\alpha},t)=S(\mathbf{q},t)+A, </math>

the HJE automatically arises:

:<math>\mathbf{p}=\frac{\partial G_2}{\partial \mathbf{q}}=\frac{\partial S}{\partial \mathbf{q}} \, \rightarrow \,
H(\mathbf{q},\mathbf{p},t) + {\partial G_2 \over \partial t}=0 \, \rightarrow \,
H\left(\mathbf{q},\frac{\partial S}{\partial \mathbf{q}},t\right) + {\partial S \over \partial t}=0. </math>

Once we have solved for ''S''('''q''', '''α''', ''t''), these also give us the useful equations

:<math>
\mathbf{Q} = \boldsymbol\beta = 
{\partial S \over \partial \boldsymbol\alpha}
</math>

or written in components for clarity

:<math> Q_{m} = \beta_{m} =  \frac{\partial S(\mathbf{q},\boldsymbol\alpha, t)}{\partial \alpha_{m}}. </math>

Ideally, these ''N'' equations can be inverted to find the original [[generalized coordinates]] '''q''' as a function of the constants '''α''', '''β''' and ''t'', thus solving the original problem.

==Action and Hamilton's functions==
Hamilton's principal function ''S'' and classical function ''H'' are both closely related to [[action (physics)|action]]. The [[total differential]] of ''S'' is:

:<math> \mathrm{d}S =\sum_i \frac{\partial S}{\partial q_i} \mathrm{d}q_i + \frac{\partial S}{\partial t}\mathrm{d}t </math>

so the [[time derivative]] of ''S'' is

:<math>\frac{\mathrm{d}S}{\mathrm{d}t} =\sum_i\frac{\partial S}{\partial q_i}\dot{q}_i+\frac{\partial S}{\partial t} =\sum_ip_i\dot{q}_i-H = L. </math>

Therefore,

:<math>S=\int L\,\mathrm{d}t ,</math>

so ''S'' is actually the classical action plus an undetermined constant.

When ''H'' does not explicitly depend on time,

:<math>W=S+Et=S+Ht=\int(L+H)\,\mathrm{d}t=\int\mathbf{p}\cdot\mathrm{d}\mathbf{q}, </math>

in this case ''W'' is the same as '''[[Symplectic action|abbreviated action]]'''.

==Separation of variables==
The HJE is most useful when it can be solved via [[separation of variables|additive separation of variables]], which directly identifies [[constant of motion|constants of motion]].  For example, the time ''t'' can be separated if the Hamiltonian does not depend on time explicitly.  In that case, the time derivative <math>\scriptstyle \partial S/\partial t </math> in the HJE must be a constant, usually denoted (–''E''), giving the separated solution

:<math> S = W(q_1,q_2, \ldots, q_N) - Et </math>

where the time-independent function ''W''('''q''') is sometimes called '''Hamilton's characteristic function'''. The reduced Hamilton–Jacobi equation can then be written

:<math> H\left(\mathbf{q},\frac{\partial S}{\partial \mathbf{q}} \right) = E. </math>

To illustrate separability for other variables, we assume that a certain [[generalized coordinate]] ''q<sub>k</sub>'' and its derivative <math>\scriptstyle \partial S/\partial q_k </math> appear together as a single function

:<math>\psi \left(q_k, \frac{\partial S}{\partial q_k} \right)</math>

in the Hamiltonian

:<math> H = H(q_1,q_2,\ldots, q_{k-1}, q_{k+1},\ldots, q_N; p_1,p_2,\ldots, p_{k-1}, p_{k+1},\ldots, p_N; \psi; t). </math>

In that case, the function ''S'' can be partitioned into two functions, one that depends only on ''q<sub>k</sub>'' and another that depends only on the remaining [[generalized coordinate]]s

:<math>S = S_k(q_k) + S_\text{rem}(q_1,\ldots, q_{k-1}, q_{k+1}, \ldots, q_N, t). </math>

Substitution of these formulae into the Hamilton–Jacobi equation shows that the function ''ψ'' must be a constant (denoted here as Γ<sub>''k''</sub>), yielding a first-order [[ordinary differential equation]] for ''S''<sub>''k''</sub>(''q''<sub>''k''</sub>).

:<math> \psi \left(q_k, \frac{\mathrm{d} S_k}{\mathrm{d} q_k} \right) = \Gamma_k. </math>

In fortunate cases, the function ''S'' can be separated completely into ''N'' functions ''S<sub>m</sub>''(''q<sub>m</sub>'')

:<math> S=S_1(q_1)+S_2(q_2)+\cdots+S_N(q_N)-Et. </math>

In such a case, the problem devolves to ''N'' [[ordinary differential equation]]s.

The separability of ''S'' depends both on the Hamiltonian and on the choice of [[generalized coordinate]]s.  For [[orthogonal coordinates]] and Hamiltonians that have no time dependence and are [[Quadratic function|quadratic]] in the generalized momenta, ''S'' will be completely separable if the potential energy is additively separable in each coordinate, where the potential energy term for each coordinate is multiplied by the coordinate-dependent factor in the corresponding momentum term of the Hamiltonian (the '''Staeckel conditions''').  For illustration, several examples in [[orthogonal coordinates]] are worked in the next sections.

==Examples in various coordinate systems==
{{further|Coordinate system|Orthogonal coordinates|Curvilinear coordinates}}

===Spherical coordinates===
In [[spherical coordinates]] the Hamiltonian of a free particle moving in a conservative potential ''U'' can be written

:<math> H = \frac{1}{2m} \left[ p_{r}^{2} + \frac{p_{\theta}^{2}}{r^{2}} + \frac{p_{\phi}^{2}}{r^{2} \sin^{2} \theta} \right] + U(r, \theta, \phi). </math>

The Hamilton–Jacobi equation is completely separable in these coordinates provided that there exist functions ''U<sub>r</sub>''(''r''), ''U<sub>θ</sub>''(''θ'') and ''U<sub>ϕ</sub>''(''ϕ'') such that ''U'' can be written in the analogous form

:<math> U(r, \theta, \phi) = U_{r}(r) + \frac{U_{\theta}(\theta)}{r^{2}} + \frac{U_{\phi}(\phi)}{r^{2}\sin^{2}\theta} . </math>

Substitution of the completely separated solution

:<math>S = S_{r}(r) + S_{\theta}(\theta) + S_{\phi}(\phi) - Et</math>

into the HJE yields

:<math>
\frac{1}{2m} \left( \frac{\mathrm{d}S_{r}}{\mathrm{d}r} \right)^{2} + U_{r}(r) +
\frac{1}{2m r^{2}} \left[ \left( \frac{\mathrm{d}S_{\theta}}{\mathrm{d}\theta} \right)^{2} + 2m U_{\theta}(\theta) \right] +
\frac{1}{2m r^{2}\sin^{2}\theta} \left[ \left( \frac{\mathrm{d}S_{\phi}}{\mathrm{d}\phi} \right)^{2} + 2m U_{\phi}(\phi) \right]  = E.
</math>

This equation may be solved by successive integrations of [[ordinary differential equation]]s, beginning with the equation for ''ϕ''

:<math> \left( \frac{\mathrm{d}S_{\phi}}{\mathrm{d}\phi} \right)^{2} + 2m U_{\phi}(\phi) = \Gamma_{\phi} </math>

where Γ<sub>''ϕ''</sub> is a [[constant of motion|constant of the motion]] that eliminates the ''ϕ'' dependence from the Hamilton–Jacobi equation

:<math> \frac{1}{2m} \left( \frac{\mathrm{d}S_{r}}{\mathrm{d}r} \right)^{2} + U_{r}(r) + \frac{1}{2m r^{2}} \left[ \left( \frac{\mathrm{d}S_{\theta}}{\mathrm{d}\theta} \right)^{2} + 2m U_{\theta}(\theta) + \frac{\Gamma_{\phi}}{\sin^{2}\theta} \right] = E. </math>

The next [[ordinary differential equation]] involves the ''θ'' [[generalized coordinate]]

:<math> \left( \frac{\mathrm{d}S_{\theta}}{\mathrm{d}\theta} \right)^{2} + 2m U_{\theta}(\theta) + \frac{\Gamma_{\phi}}{\sin^{2}\theta} = \Gamma_{\theta} </math>

where Γ<sub>''θ''</sub> is again a [[constant of motion|constant of the motion]] that eliminates the ''θ'' dependence and reduces the HJE to the final [[ordinary differential equation]]

:<math> \frac{1}{2m} \left( \frac{\mathrm{d}S_{r}}{\mathrm{d}r} \right)^{2} + U_{r}(r) + \frac{\Gamma_{\theta}}{2m r^{2}} = E </math>

whose integration completes the solution for ''S''.

===Elliptic cylindrical coordinates===
The Hamiltonian in [[elliptic cylindrical coordinates]] can be written

:<math> H = \frac{p_{\mu}^{2} + p_{\nu}^{2}}{2ma^{2} \left( \sinh^{2} \mu + \sin^{2} \nu\right)} +  \frac{p_{z}^{2}}{2m}  + U(\mu, \nu, z) </math>

where the [[Focus (geometry)|foci]] of the [[ellipse]]s are located at <math>\pm a</math> on the <math>x</math>-axis.  The Hamilton–Jacobi equation is completely separable in these coordinates provided that ''U'' has an analogous form

:<math> U(\mu, \nu, z) = \frac{U_{\mu}(\mu) + U_{\nu}(\nu)}{\sinh^{2} \mu + \sin^{2} \nu} + U_{z}(z) </math>

where : <math> U_\mu(\mu)</math>, <math>U_\nu(\nu)</math> and <math>U_z(z)</math> are arbitrary functions.  Substitution of the completely separated solution

:<math>S = S_{\mu}(\mu) + S_{\nu}(\nu) + S_{z}(z) - Et</math> into the HJE yields

:<math>
\frac{1}{2m} \left( \frac{\mathrm{d}S_{z}}{\mathrm{d}z} \right)^{2} + U_{z}(z) +
\frac{1}{2ma^{2} \left( \sinh^{2} \mu + \sin^{2} \nu\right)} \left[ \left( \frac{\mathrm{d}S_{\mu}}{\mathrm{d}\mu} \right)^{2} + \left( \frac{\mathrm{d}S_{\nu}}{\mathrm{d}\nu} \right)^{2} + 2m a^{2} U_{\mu}(\mu) + 2m a^{2} U_{\nu}(\nu)\right] = E.
</math>

Separating the first [[ordinary differential equation]]

:<math> \frac{1}{2m} \left( \frac{\mathrm{d}S_{z}}{\mathrm{d}z} \right)^{2} + U_{z}(z) = \Gamma_{z} </math>

yields the reduced Hamilton–Jacobi equation (after re-arrangement and multiplication of both sides by the denominator)

:<math> \left( \frac{\mathrm{d}S_{\mu}}{\mathrm{d}\mu} \right)^{2} + \left( \frac{\mathrm{d}S_{\nu}}{\mathrm{d}\nu} \right)^{2} + 2m a^{2} U_{\mu}(\mu) + 2m a^{2} U_{\nu}(\nu) = 2ma^{2} \left( \sinh^{2} \mu + \sin^{2} \nu\right) \left( E - \Gamma_{z} \right) </math>

which itself may be separated into two independent [[ordinary differential equations]]

:<math> \left( \frac{\mathrm{d}S_{\mu}}{\mathrm{d}\mu} \right)^{2} + 2m a^{2} U_{\mu}(\mu) + 2ma^{2} \left(\Gamma_{z} - E \right) \sinh^{2} \mu = \Gamma_{\mu} </math>

:<math> \left( \frac{\mathrm{d}S_{\nu}}{\mathrm{d}\nu} \right)^{2} + 2m a^{2} U_{\nu}(\nu) + 2ma^{2} \left(\Gamma_{z} - E \right) \sin^{2} \nu  = \Gamma_{\nu} </math>

that, when solved, provide a complete solution for ''S''.

===Parabolic cylindrical coordinates===
The Hamiltonian in [[parabolic cylindrical coordinates]] can be written

:<math> H = \frac{p_{\sigma}^{2} + p_{\tau}^{2}}{2m \left( \sigma^{2} + \tau^{2}\right)} + \frac{p_{z}^{2}}{2m}  + U(\sigma, \tau, z). </math>

The Hamilton–Jacobi equation is completely separable in these coordinates provided that ''U'' has an analogous form

:<math> U(\sigma, \tau, z) = \frac{U_{\sigma}(\sigma) + U_{\tau}(\tau)}{\sigma^{2} + \tau^{2}} + U_{z}(z) </math>

where ''U<sub>σ</sub>''(''σ''), ''U<sub>τ</sub>''(''τ'') and ''U<sub>z</sub>''(''z'') are arbitrary functions.  Substitution of the completely separated solution

:<math>S = S_{\sigma}(\sigma) + S_{\tau}(\tau) + S_{z}(z) - Et + const.</math>

into the HJE yields

:<math>
\frac{1}{2m} \left( \frac{\mathrm{d}S_{z}}{\mathrm{d}z} \right)^{2} + U_{z}(z) +
\frac{1}{2m \left( \sigma^{2} + \tau^{2} \right)} \left[ \left( \frac{\mathrm{d}S_{\sigma}}{\mathrm{d}\sigma} \right)^{2} + \left( \frac{\mathrm{d}S_{\tau}}{\mathrm{d}\tau} \right)^{2} + 2m U_{\sigma}(\sigma) + 2m U_{\tau}(\tau)\right] = E.
</math>

Separating the first [[ordinary differential equation]]

:<math>
\frac{1}{2m} \left( \frac{\mathrm{d}S_{z}}{\mathrm{d}z} \right)^{2} + U_{z}(z) = \Gamma_{z}
</math>

yields the reduced Hamilton–Jacobi equation (after re-arrangement and multiplication of both sides by the denominator)

:<math>
\left( \frac{\mathrm{d}S_{\sigma}}{\mathrm{d}\sigma} \right)^{2} + \left( \frac{\mathrm{d}S_{\tau}}{\mathrm{d}\tau} \right)^{2} + 2m U_{\sigma}(\sigma) + 2m U_{\tau}(\tau) = 2m \left( \sigma^{2} + \tau^{2} \right) \left( E - \Gamma_{z} \right)
</math>

which itself may be separated into two independent [[ordinary differential equations]]

:<math>
\left( \frac{\mathrm{d}S_{\sigma}}{\mathrm{d}\sigma} \right)^{2} + 2m U_{\sigma}(\sigma) + 2m\sigma^{2} \left(\Gamma_{z} - E \right) = \Gamma_{\sigma}
</math>

:<math>
\left( \frac{\mathrm{d}S_{\tau}}{\mathrm{d}\tau} \right)^{2} + 2m U_{\tau}(\tau) + 2m \tau^{2} \left(\Gamma_{z} - E \right) = \Gamma_{\tau}
</math>

that, when solved, provide a complete solution for ''S''.

==Eikonal approximation and relationship to the Schrödinger equation==
{{further|Eikonal approximation|Non-linear Schrödinger equation}}
The [[isosurface]]s of the function ''S''('''q'''; ''t'') can be determined at any time ''t''.  The motion of an ''S''-isosurface as a function of time is defined by the motions of the particles beginning at the points '''q''' on the isosurface.  The motion of such an isosurface can be thought of as a ''[[wave]]'' moving through '''q''' space, although it does not obey the [[wave equation]] exactly.  To show this, let ''S'' represent the [[phase (waves)|phase]] of a wave

:<math> \psi = \psi_{0} e^{iS/\hbar} </math>

where ''ħ'' is a constant ([[Planck's constant]]) introduced to make the exponential argument dimensionless; changes in the [[amplitude]] of the [[wave]] can be represented by having ''S'' be a [[complex number]].  We may then rewrite the Hamilton–Jacobi equation as

:<math> \frac{\hbar^{2}}{2m} \nabla^2 \psi - U\psi = \frac{\hbar}{i} \frac{\partial \psi}{\partial t} </math>

which is a ''nonlinear'' variant of the [[Schrödinger equation]].

Conversely, starting with the Schrödinger equation and our [[ansatz]] for ''ψ'', we arrive at
<ref name=Goldstein_490>{{cite book |last=Goldstein |first=Herbert |authorlink=Herbert Goldstein |year=1980 |title= Classical Mechanics |edition=2nd |publisher=Addison-Wesley |location=Reading, MA |isbn= 978-0-201-02918-5 |pages=490–491|title-link=Classical Mechanics (textbook) }}</ref>

:<math> \frac{1}{2m} \left( \nabla S \right)^{2} + U + \frac{\partial S}{\partial t} = \frac{i\hbar}{2m} \nabla^{2} S. </math>

The classical limit (''ħ'' → 0) of the Schrödinger equation above becomes identical to the following variant of the Hamilton–Jacobi equation,

:<math> \frac{1}{2m} \left( \nabla S \right)^{2} + U + \frac{\partial S}{\partial t} = 0. </math>

==HJE in a gravitational field==

Using the [[energy–momentum relation]] in the form;<ref>{{cite book|title=Gravitation|author1=J.A. Wheeler |author2=C. Misner |author3=K.S. Thorne |publisher=W.H. Freeman & Co|url=https://books.google.com/books?id=w4Gigq3tY1kC&dq=isbn%3A0716703440&q=Hamilton-Jacobi#v=snippet&q=Hamilton-Jacobi&f=false|pages=649, 1188|year=1973|isbn=978-0-7167-0344-0}}</ref>

:<math>g^{\alpha\beta}P_\alpha P_\beta - (mc)^2 = 0 \,, </math>

for a particle of [[rest mass]] ''m'' travelling in curved space, where ''g<sup>αβ</sup>'' are the [[Covariance and contravariance of vectors|contravariant]] coordinates of the [[metric tensor]] (i.e., the [[Metric tensor#Inverse metric|inverse metric]]) solved from the [[Einstein field equations]], and ''c'' is the [[speed of light]], setting the [[four-momentum]] ''P<sub>α</sub>'' equal to the [[four-gradient]] of the action ''S'';

:<math>P_\alpha =-\frac{\partial S}{\partial x^\alpha}</math>

gives the Hamilton–Jacobi equation in the geometry determined by the metric ''g'':

:<math>g^{\alpha\beta}\frac{\partial S}{\partial x^\alpha}\frac{\partial S}{\partial x^\beta} -(mc)^2 = 0\,,</math>

in other words, in a [[gravitational field]].

==HJE in electromagnetic fields==

For a particle of [[rest mass]] <math>m</math> and electric charge <math>e</math>  moving in electromagnetic field with [[four-potential]]  <math>A_i  =  (\phi,\Alpha)</math> in vacuum, the Hamilton–Jacobi equation in geometry determined by the metric tensor <math>g^{ik} = g_{ik}</math> has a form

: <math>g^{ik}\left ( \frac{\partial S}{\partial x^i} + \frac {e}{c}A_i \right ) \left ( \frac{\partial S}{\partial x^k} + \frac {e}{c}A_k \right ) = m^2 c^2</math>

and can be solved for the Hamilton Principal Action function <math>S</math> to obtain further solution for the particle trajectory and momentum:<ref>{{cite book|title=THE CLASSICAL THEORY OF FIELDS|author=[[L. Landau]] and E. Lifshitz|publisher=ADDISON-WESLEY PUBLISHING COMPANY, INC., Reading, Massachusetts, USA 1959|volume= |issue= |pages= |isbn=}}</ref>

: <math>x = - \frac {e}{c \gamma}\int A_z \,d\xi,</math>
: <math>y = - \frac {e}{c \gamma} \int A_y \,d\xi,</math>
: <math>z = - \frac {e^2}{2c^2 \gamma^2}\int (\Alpha^2 - \overline {\Alpha^2 }) \, d \xi,</math>
: <math>\xi = ct - \frac{e^2}{2 \gamma^2 c^2}\int (\Alpha^2 - \overline {\Alpha^2})  \, d \xi, </math>
: <math>p_x = - \frac{e}{c}A_x </math>, <math>p_y = - \frac{e}{c}A_y,</math>
: <math>p_z = \frac{e^2}{2\gamma c}(\Alpha^2 - \overline {\Alpha^2}),</math>
: <math>\mathcal{E}= c\gamma + \frac{e^2}{2 \gamma c}(\Alpha^2 - \overline {\Alpha^2}),</math>

where <math>\xi = ct - z</math> and <math>\gamma^2 = m^2 c^2 + \frac{e^2}{c^2} \overline{A}^2 </math> with <math>\overline{\mathbf{A}}</math> the cycle average of the vector potential. Therefore:

'''a)''' For a wave with the [[circular polarization]]:

: <math>E_x = E_0 \sin \omega \xi_1 </math>, <math>E_y = E_0 \cos \omega \xi_1, </math>
: <math>A_x = \frac{ cE_0 }{\omega} \cos \omega \xi_1 </math>, <math>A_y = - \frac{ cE_0 }{\omega} \sin \omega \xi_1. </math>

Hence

: <math>x = - \frac{ecE_0} \omega \sin \omega \xi_1, </math>
: <math>y = - \frac{ecE_0} \omega \cos \omega \xi_1, </math>
: <math>p_x = - \frac{eE_0} \omega \cos \omega \xi_1, </math>
: <math>p_y =  \frac{eE_0}{\omega} \sin \omega \xi_1, </math>

where <math>\xi_1 = \xi /c </math>,
implying the particle moving along a circular trajectory with a permanent radius <math>ecE_0 / \gamma \omega^2 </math> and an invariable value of momentum <math>eE_0 / \omega^2 </math> directed along a magnetic field vector.

'''b)''' For the flat, monochromatic, linearly polarized wave with a field <math>E</math> directed along the axis <math>y</math>

: <math>E_y = E_0 \cos \omega \xi_1,</math>
: <math>A_y = - \frac {cE_0}{\omega} \sin \omega \xi_1,</math>

hence

: <math>x = \text{const},</math>
: <math>y_0 = -\frac{ecE_0}{\gamma \omega^2},</math>
: <math>y = y_0 \cos \omega \xi_1</math>, <math>z = C_z y_0 \sin 2\omega \xi_1,</math>
: <math>C_z = \frac{eE_0}{8\gamma \omega}</math>, <math>\gamma^2 = m^2 c^2  + \frac{e^2 E_0^2}{2 \omega^2}, </math>
: <math>p_x = 0,</math>
: <math>p_{y,0} = \frac{eE_0}{\omega},</math>
: <math>p_y = p_{y,0} \sin \omega \xi_1, </math>
: <math>p_z = - 2C_z p_{y,0} \cos 2\omega \xi_1 </math>

implying the particle figure-8 trajectory with a long its axis oriented along the electric field <math>E</math> vector.

'''c)''' For the electromagnetic wave with axial (solenoidal) magnetic field:<ref>{{cite journal|title=Inductively Coupling Plasma Reactor With Plasma Electron Energy Controllable in the Range from ~6 to ~100 eV|journal=IEEE Transactions on Plasma Science |author1=E. V. Shun'ko |author2=D. E. Stevenson |author3=V. S. Belkin |volume= 42, part II|issue= 3|pages= 774–785|doi=10.1109/TPS.2014.2299954|bibcode = 2014ITPS...42..774S |year=2014 }}</ref>

: <math>E = E_\phi = \frac{\omega \rho_0}{c} B_0 \cos \omega \xi_1, </math>
: <math>A_\phi = - \rho_0 B_0 \sin \omega \xi_1 =  - \frac{L_s}{\pi \rho_0 N_s} I_0 \sin \omega \xi_1,</math>

hence

: <math>x = \text{constant},</math>
: <math>y_0 = -\frac{e \rho_0 B_0}{\gamma \omega},</math>
: <math>y = y_0 \cos \omega \xi_1,</math>
: <math>z = C_z y_0 \sin 2\omega \xi_1,</math>
: <math>C_z = \frac{e \rho_0 B_0}{8c \gamma},</math>
: <math>\gamma^2 = m^2 c^2 + \frac{e^2 \rho_0^2 B_0^2}{2c^2},</math>
: <math>p_x = 0,</math>
: <math>p_{y,0} = \frac{e \rho_0 B_0}{c},</math>
: <math>p_y = p_{y,0} \sin \omega \xi_1,</math>
: <math>p_z = - 2C_z p_{y,0} \cos 2 \omega \xi_1,</math>

where <math>B_0</math> is the magnetic field magnitude in a solenoid with the effective radius <math>\rho_0</math>, inductivity <math>L_s</math>, number of windings <math>N_s</math>, and an electric current magnitude <math>I_0</math> through the solenoid windings. The particle motion occurs along the figure-8 trajectory in <math>yz</math> plane set perpendicular to the solenoid axis with arbitrary azimuth angle <math>\varphi</math> due to axial symmetry of the solenoidal magnetic field.

==See also==
{{div col}}
* [[Canonical transformation]]
* [[Constant of motion]]
* [[Hamiltonian vector field]]
* [[Hamilton–Jacobi–Bellman equation]] in control theory
* [[Hamilton–Jacobi–Einstein equation]]
* [[WKB approximation]]
* [[William Rowan Hamilton]]
* [[Carl Gustav Jacob Jacobi]]
* [[Action-angle coordinates]]
{{div col end}}

==References==

{{reflist}}

==Further reading==
*{{Cite journal |last=Hamilton |first=W. |year=1833 |title=On a General Method of Expressing the Paths of Light, and of the Planets, by the Coefficients of a Characteristic Function |journal=Dublin University Review |pages=795–826|url=http://www.emis.de/classics/Hamilton/CharFun.pdf }}
*{{Cite journal |last=Hamilton |first=W. |year=1834 |title=On the Application to Dynamics of a General Mathematical Method previously Applied to Optics |journal=British Association Report |pages=513–518|url=http://www.emis.de/classics/Hamilton/BARep34A.pdf }}
*{{cite book |first=Herbert |last=Goldstein |authorlink=Herbert Goldstein |title=Classical Mechanics |edition=3rd |publisher=Addison Wesley |location= |year=2002 |isbn=978-0-201-65702-9 }}
*{{cite book |first=A. |last=Fetter |lastauthoramp=yes |first2=J. |last2=Walecka |title=Theoretical Mechanics of Particles and Continua |publisher=Dover Books |year=2003 |isbn=978-0-486-43261-8 }}
*{{Cite book |last=[[L. Landau|Landau]] |first=L. D. |last2=Lifshitz |first2=E. M. |title=Mechanics |publisher=Elsevier |location=Amsterdam |year=1975 }}
*{{cite book |first=J. J. |last=Sakurai |title=Modern Quantum Mechanics |url=https://archive.org/details/modernquantummec0000saku |url-access=registration |publisher=Benjamin/Cummings Publishing |year=1985 |isbn=978-0-8053-7501-5 }}
*{{Citation |first=C. G. J.|last=Jacobi |title=Vorlesungen über Dynamik |publisher=G. Reimer |series=C. G. J. Jacobi's Gesammelte Werke|year=1884|location=Berlin|language=German|
url=http://openlibrary.org/books/OL14009561M/C._G._J._Jacobi%27s_Vorlesungen_u%CC%88ber_Dynamik }}
*{{cite journal |first=Michiyo |last=Nakane|first2=Craig G. |last2=Fraser |title=The Early History of Hamilton-Jacobi Dynamics |journal=Centaurus|volume=44|issue=3–4|pages=161–227|year=2002|
doi=10.1111/j.1600-0498.2002.tb00613.x|pmid=17357243}}

{{DEFAULTSORT:Hamilton-Jacobi equation}}
[[Category:Hamiltonian mechanics]]
[[Category:Symplectic geometry]]
[[Category:Partial differential equations]]
[[Category:William Rowan Hamilton]]