## Hamilton-Jacobi equation in momentum space

Optics Express, Vol. 14, Issue 20, pp. 9083-9092 (2006)

http://dx.doi.org/10.1364/OE.14.009083

Acrobat PDF (198 KB)

### Abstract

The application of the Hamilton-Jacobi equation to isotropic optical materials leads to the well-known eikonal equation which provides the surfaces normal to the ray trajectories. The symmetry between the coordinates *
x
*=(

*x*

_{1},

*x*

_{2},

*x*

_{3}) and the momenta

*=(*

**p***p*

_{1},

*p*

_{2},

*p*

_{3}) in the Hamiltonian formulation of Geometrical Optics establishes a dual Hamilton-Jacobi equation for “wavefronts” in the momentum space. This equation is also an eikonal equation when the refractive index distribution has spherical symmetry. In this case, another spherical symmetric refractive index distribution may exist such that the ray trajectories in the coordinates and momentum space are exchanged (examples of this case are given: Maxwell fish-eye, Eaton lens and Luneburg lens). The relationship between the wavefronts in the coordinate and momentum space is also analyzed. Curved orthogonal coordinates are considered as well.

© 2006 Optical Society of America

*x*

_{i}and

*p*

_{i}are the unknown functions of the parameter

*t*, and they define a ray-trajectory. The

*x*

_{i}variables are called the configuration variables (or coordinates) and the

*p*

_{i}are called the conjugate variables (or momenta). A ray-trajectory in the phase space is a curve given by the functions

*x*

_{1}(

*t*),

*x*

_{2}(

*t*),

*x*

_{3}(

*t*),

*p*

_{1}(

*t*),

*p*

_{2}(

*t*),

*p*

_{3}(

*t*). We are usually only interested in the projection of this phase space trajectory in the coordinate space

*x*

_{1}-

*x*

_{2}-

*x*

_{3}. In the Hamiltonian formulation used here, not all of the solutions to Eq. (1) are possible ray trajectories, rather, only those consistent with

*H*(

*x*

_{i},

*p*

_{i})=0.

*H*is a function of the six variables

*x*

_{i},

*p*

_{i}called the Hamiltonian function. It can be proven [2, 3] that the solutions to Eq. (1) fulfil that

*H*=constant along their trajectories, i.e., the function

*H*is a first integral (also called a constant of motion) of the Hamiltonian system. The rays are merely those solutions to Eq. (1) contained in the subspace

*H*=0. The Hamiltonian function for an isotropic medium can be written as

### 1.1 Hamilton-Jacobi equation

*X*

_{i},

*P*

_{i}be the coordinates after a canonical transformation and let

*V*(

*x*

_{i},

*X*

_{i}) be its generating function. This type of generating function, which depends only on the old and new coordinates, is called a Type 1 generating function [5

5.
Wikipedia contributors, “Canonical transformation,” *Wikipedia, The Free Encyclopedia*, http://en.wikipedia.org/w/index.php?title=Canonical_transformation&oldid=65931629 (accessed June17, 2006).

*x*

_{i},

*X*

_{i}form a set of independent variables, which is needed for the existence the Type 1 generating function. There are also Type 2, Type 3, and Type 4 generating functions, for which we need to assume respectively that

*x*

_{i},

*P*

_{i};

*p*

_{i},

*X*

_{i}and

*p*

_{i},

*P*

_{i}, are sets of independent variables. According to the Canonical Transformation theory, Type 1 transformations from

*x*

_{i},

*p*

_{i}to

*X*

_{i},

*P*

_{i}are derived from the equations

*V*dependent on

*x*

_{i}and on 3 constants that are the

*X*

_{i}of the generating function theory [3, 6]. There are many possibilities to choose the integration constants some of which lead to the many possible characteristic functions of an optical system [7

7. M. A. Alonso and G. W. Forbes, “Generalization of Hamilton’s formalism for geometrical optics,” J. Opt. Soc. Am. A **12**, 2744- (1995), http://www.opticsinfobase.org/abstract.cfm?URI=josaa-12-12-2744. [CrossRef]

## 2. Hamilton-Jacobi equation in the momentum space

*W*(

*p*

_{i},

*X*

_{i}) instead of Type 1. In this case the transformations from

*x*

_{i},

*p*

_{i}to

*X*

_{i},

*P*

_{i}are given by [5

5.
Wikipedia contributors, “Canonical transformation,” *Wikipedia, The Free Encyclopedia*, http://en.wikipedia.org/w/index.php?title=Canonical_transformation&oldid=65931629 (accessed June17, 2006).

*X*

_{i}=

*p*

_{i},

*P*

_{i}=-

*x*

_{i}and then using a Type 1 generating function to establish the classic Hamilton-Jacobi equation. Let’s call it a Hamilton-Jacobi equation in the momentum space.

*n*(

*) is the refractive-index distribution, then the dual Hamilton-Jacobi equation is*

**x***n*

^{2}(-∇

_{p}

*W*)-

**p**^{2}=0; where ∇

_{p}

*W*=(

*∂W*/

*∂p*

_{1},

*∂W*/

*∂p*

_{2},

*∂W*/

*∂p*

_{3}). The Hamilton-Jacobi equation is also an eikonal equation in Cartesian coordinates if the refractive index distribution has spherical symmetry (see Section 3) and is invertible,

*i*.

*e*., if

*n*is only a function of |

*|,*

**x***n*≡

*n*(|

*|), and it is a one-to-one correspondence between*

**x***n*and |

*|, which is fulfilled if*

**x***n*(|

*|) is a monotonic function. The existence of the inverse function of*

**x***n*guarantees that it is possible to extract the value of |∇

_{p}

*W*| with uniqueness from the dual Hamilton-Jacobi equation

*n*

^{2}(-∇

_{p}

*W*)-

**p**^{2}=0, and thus write it as the eikonal (∇

_{p}

*W*)

^{2}={

*n*

^{-1}(|

*|)}*

**p**^{2}, where

*n*

^{-1}is the inverse function of

*n*,

*i*.

*e*.,

*n*

^{-1}(

*n*(

*r*))=

*r*.

*V*(

*) and -*

**x***W*(

*) (we are now omitting their dependence on the variables*

**p***X*

_{i}) are a Legendre transform pair, since they are different generating function types of the same canonical transformation [5

5.
Wikipedia contributors, “Canonical transformation,” *Wikipedia, The Free Encyclopedia*, http://en.wikipedia.org/w/index.php?title=Canonical_transformation&oldid=65931629 (accessed June17, 2006).

8.
Wikipedia contributors, “Legendre transformation,” *Wikipedia, The Free Encyclopedia*, http://en.wikipedia.org/w/index.php?title=Legendre_transformation&oldid=66180751 (accessed May7, 2006).

**as a function of**

*x***with**

*p**=-∇*

**x**_{p}

*W*(

*) [which is the left-hand side equation in Eq. (5)], or*

**p***p*as a function of

**x**with

*p*=∇

*V*(

*) [which is the left-hand side equation in Eq. (3)]. This is a well-known result when functions*

**x***V*and

*W*are, respectively, Hamilton’s point and mixed characteristic functions [9].

*space (i.e.,*

**x***V*(

*)=constant surfaces) to the “wavefronts” in the*

**x***space (*

**p***W*(

*)=constant surfaces). For instance, a wavefront given by*

**p***W*(

*)=*

**p***W*

_{0}in the

*space is represented by the equation*

**p***V*(

*)-*

**x***·∇*

**x***V*(

*)=*

**x***W*

_{0}in the

*x*space.

*T*(

*p*

_{i},

*P*

_{i}) can also be used to get the dual Hamilton-Jacobi equation, for which the coordinate transformations from

*x*

_{i},

*p*

_{i}to

*X*

_{i},

*P*

_{i}are

*V*(

*,*

**x***) and -*

**X***T*(

*,*

**p***), where*

**P***=(*

**X***X*

_{i}) and

*=(*

**P***P*

_{i}), are a Legendre transform pair since they are different generating function types of the same canonical transformation [5

*Wikipedia, The Free Encyclopedia*, http://en.wikipedia.org/w/index.php?title=Canonical_transformation&oldid=65931629 (accessed June17, 2006).

*X*

_{i}and

*P*

_{i}in the Legendre transformation.

*V*, while only

*∂W*/

*∂p*

_{1}appears in the dual Hamilton-Jacobi equation in the momentum space. This equation can be easily solved, giving

*f*(

*p*

_{2},

*p*

_{3}) is an arbitrary function. Now

*V*can be obtained with a Legendre transformation using Eq. (7) (

*V*(

*)=*

**x***·*

**x***+*

**p***W*(

*)) and*

**p***=-∇*

**x**_{p}

*W*(

*). Note that these 2 equations give the definition of the function*

**p***V*(

*) in parametric form (the parameters being the components of*

**x***p*).

## 3. Isotropic media with spherical symmetry

*-*

**x***for the Maxwell fish-eye lens (refractive index distribution*

**p***n*(|

*|)=2/(1+*

**x**

**x**^{2})). The projection of these trajectories in the coordinate space

**x**are what we usually call the ray trajectories. These projections in the

*space are the curves orthogonal to surfaces*

**x***V*(

*)=constant, where*

**x***V*is a solution of the eikonal equation (∇

*V*)

^{2}=

*n*

^{2}(|

*|)=4/(1+*

**x**

**x**^{2})

^{2}. The projection of the trajectories in the momentum space

*can also be calculated with an eikonal equation. In particular, the projection of the ray trajectories in the*

**p***space are orthogonal to surfaces*

**p***W*(

*)=constant, satisfying the eikonal equation (∇*

**p**_{p}

*W*)

^{2}={

*n*

^{-1}(|

*|)}*

**p**^{2}, where

*n*

^{-1}is the inverse function of

*n*,

*i*.

*e*.,

*n*

^{-1}(

*n*(

*r*))=

*r*. For the Maxwell fish-eye case this inverse function is

*n*

^{-1}(|

*|)=(2/|*

**p****|-1)**

*p*^{1/2}. This is the same function as the refractive index distribution of the Eaton lens, which is

*n*

_{E}(|

*|)=(2/|*

**x***|-1)*

**x**^{1/2}[10]. This means that the phase space trajectories of rays in the Maxwell fish eye, when projected into the

*space, follow the same tracks as the*

**p***space projection of rays of the Eaton lens and vice versa,*

**x***i*.

*e*., the

*space projection of phase space trajectories of the Eaton lens coincides with the*

**p***space projection of the rays in the Maxwell fish-eye lens. We will say that the Maxwell fish-eye and the Eaton lenses are a pair of dual lenses. Incidentally, the Hamiltonian function associated with the Eaton lens is identical to the Hamiltonian system of the Kepler problem in Mechanics, which establishes the connection between the Kepler problem in Mechanics and the Maxwell fish eye found by Buchdahl [11*

**x**11. H. A. Buchdahl, “Kepler problem and Maxwell fish-eye,” Am. J. Phys. , **46**, 840–843 (1978). [CrossRef]

*x*

_{3}=0 and

*p*

_{3}=0 without loss of generality. Fig. 1(left) shows the trajectories of rays on the

*x*

_{1},

*x*

_{2}plane and the trajectories of the same rays on the

*p*

_{1},

*p*

_{2}plane (right) for the Maxwell fish-eye lens. These trajectories coincide with the Eaton lens ray trajectories in the coordinate space (right) and in the momentum space (left).

*n*(|

*|)=(2/|*

**x***|-1)*

**x**^{1/2}inside the sphere

*x*

^{2}≤1 and equal to 1 outside it, while the dual lens of the Maxwell fish eye is the distribution

*n*(|

*|)=(2/|*

**x***|-1)*

**x**^{1/2}in the sphere

*x*

^{2}≤4 and it is undefined elsewhere. The Eaton lens has the property of retro-reflecting any ray in the incoming direction (Fig. 2).

*space then a 2 axis skew conservation must hold, and since they are also spherical symmetric in the*

**x***space, another equivalent 2-axis skew conservation must also hold for the trajectories in the momentum space. When calculated it is found that both conservations are the same. It can be written as*

**p***×*

**x***=constant along the ray trajectories (× denotes vector product), which is a symmetric expression with respect to*

**p***and*

**x***.*

**p***, as can be easily seen in with the Eq. (1) and Eq. (2), which establishes*

**p***d*/

**x***dt*=

*. Spherical symmetric systems also fulfil that ray trajectories in the momentum space are tangent to the vector*

**p***as can be seen with the same equations and noting that ∇*

**x***n*(|

*|) is parallel to*

**x***. Moreover, as well as the vector*

**x***is always*

**p***n*(

*) times the unit tangent to the ray trajectory in coordinate space, in spherical symmetric systems, the vector*

**x***is also*

**x***n*

^{-1}(

*) times the unit tangent to the ray trajectory in the momentum space.*

**p**### 3.1 Luneburg lens

*n*(|

*|)=(2-*

**x**

**x**^{2})

^{1/2}(inside the sphere

**x**^{2}=1) is such that its inverse function is itself,

*i*.

*e*.,

*n*

^{-1}(|

*|)=*

**x***n*(|

*|). As a consequence, the phase space trajectories of rays in the Luneburg lens follow the same tracks when projected on the*

**x***space as when projected on the*

**p***space because both eikonal equations (∇*

**x**_{p}

*W*)

^{2}={

*n*

^{-1}(|

*|)}*

**p**^{2}and (∇

*V*)

^{2}=

*n*

^{2}(|

*|) are the same equation. (see Fig. 3). Even more remarkable is that the projection of a ray on a plane*

**x***x*

_{i}-

*x*

_{j}or a plane

*p*

_{i}-

*p*

_{j}follows the same ellipse, although not in the same part of the ellipse, as can be easily understood by noting that

**p**^{2}>1 when

*2<1 because*

**x***n*(|

*|)>1 for*

**x**

**x**^{2}<1. The vector

*and the vector*

**x***at a given point of the trajectory in one of these planes are conjugate diameters of the same ellipse. One property of the ellipse (due to Apollonius of Perga) establishes that the sum of the squares of conjugate diameters is constant. For the case of the Luneburg lens this is*

**p**

**x**^{2}+

**p**^{2}=2, which is simply the eikonal equation for the Luneburg lens.

*x*

_{1},

*x*

_{2}plane and the trajectories of the same rays on the

*p*

_{1},

*p*

_{2}plane (right). The rays leave from point

*x*

_{1}=-1,

*x*

_{2}=0 on the

*x*

_{1},

*x*

_{2}plane and follow an elliptical trajectory. They cross the circle (

*x*

_{1})

^{2}+(

*x*

_{2})

^{2}=1 parallel to the

*x*

_{1}axis,

*i*.

*e*., with

*p*

_{2}=0. In the

*p*

_{1},

*p*

_{2}plane, the rays are the same ellipses, and thus they are parallel to the abscissa axis at the circle of radius 1. All these trajectories converge at the point

*p*

_{1}=1,

*p*

_{2}=0.

**x**^{2}=1 is

*n*=1 for the Luneburg lens, so that the ray trajectories are straight lines outside that sphere (Fig. 4). If the medium outside this sphere also follows the law

*n*(|

*|)=(*

**x****2**-

**x**^{2})

^{1/2}, then the trajectories of the rays are closed ellipses in both spaces. The ray trajectories can be easily calculated with Eq. (1), when

*H*≡

**2**-

**x**^{2}-

**p**^{2}:

**x**^{2}≤1 in the Luneburg lens and valid everywhere else when the law

*n*(|

*|)=(*

**x****2**-

**x**^{2})

^{1/2}extends to all points. The solutions,

*i*.

*e*, the phase space ray trajectories in parametric form can be written as

*t*is a parameter along the ray trajectories and

*x*

_{i0},

*p*

_{i0}are the initial conditions. Equation (15) can be written shorter using complex vector notation as

*+*

**x****i**

*=(*

**p**

**x**_{0}+

**i**

**p**_{0})exp(-i

*t*), where i

^{2}=-1 (thus, the ray trajectories in coordinate space are simply Re{(

**x**_{0}+

**i**

*p*_{0})exp(-i

*t*)}). The 6 initial conditions are not completely free, since they must fulfil that

*H*=0, i.e., they must fulfil 2-(

*x*

_{10})

^{2}-(

*x*

_{20})

^{2}-(

*x*

_{30})

^{2}-(

*p*

_{10})

^{2}-(

*p*

_{20})

^{2}-(

*p*

_{30})

^{2}=0. Rays in Fig. 4 depart (

*t*=0) from the point

*x*

_{0}=(-1,0,0) with

*p*

_{0}ranging from (0,-1,0) to (0,1,0) and fulfilling (

*p*

_{0})

^{2}=1. Equation (15) for

*t*=

*π*/2 establishes

*x*(

*π*/2)=

*p*

_{0}and

*p*(

*π*/2)=-

*x*

_{0}. Then,

*x*(

*π*/2) is in the circle

*x*

^{2}=1 ranging from (0,-1,0) to (0,1,0) and

*p*(

*π*/2)=(1,0,0). These are the coordinates of the rays in Fig. 4 when they leave the circle

*x*

^{2}=1.

*t*in a pair of Eq. (15) we can easily find the integral invariants in the Luneburg ray trajectories [13

13. H. A. Buchdahl, “Luneburg lens: unitary invariance and point characteristic,” J. Opt. Soc. Am. **73**, 490- (1983), http://www.opticsinfobase.org/abstract.cfm?URI=josa-73-4-490. [CrossRef]

*x*

_{i}-

*p*

_{i}plane are circles:

*x*

_{i}-

*x*

_{j}or conjugate variables

*p*

_{i}-

*p*

_{j}or a combination of variables

*x*

_{i}-

*p*

_{j},

*i*≠

*j*are the ellipses given by

## 4. Non Cartesian orthogonal coordinates

*ξ*=(

*ξ*

_{1},

*ξ*

_{2},

*ξ*

_{3}) given by 3 functions

*ξ*

_{i}(

*). For any coordinate transformation from*

**x***to ξ, it is possible to find a set of conjugate momenta*

**x***ζ*=(

*ζ*

_{1},

*ζ*

_{2},

*ζ*

_{3}) such that the phase space transformation from (

*,*

**x***) to variables (ξ, ζ) is canonical [3]. Unlike the new coordinates ξ, which only depend on the old ones*

**p***, the new momenta ζ depend in general of both old coordinates and momenta. The Hamiltonian function in the new phase space coordinates is given by [1]*

**x***h*

_{1},

*h*

_{2},

*h*

_{3}are the three scale factors of the coordinate transformation, i.e.,

*h*

_{i}=1/|∇ξ

_{i}|. The function

*n*is the refractive index distribution in the new coordinates

*n*(

*ξ*)=

*n*(

*(*

**x***ξ*)). In general

*h*

_{i}are functions of configuration coordinates ξ. We are going to assume that the scale factors are equal for at least one point, let’s say the point ξ

_{10},ξ

_{20},ξ

_{30}, i.e.,

*h*

_{1}(ξ

_{10},ξ

_{20},ξ

_{30})=

*h*

_{2}(ξ

_{10},ξ

_{20},ξ

_{30})=

*h*

_{3}(ξ

_{10},ξ

_{20},ξ

_{30}). Note that if there is no point ξ

_{10},ξ

_{20},ξ

_{30}, with this property we can do an additional canonical transformation scaling up or down each one of the coordinates ξ

_{1},ξ

_{2},ξ

_{3}(since we want the transformation to be canonical, the corresponding conjugate momenta would result inversely scaled with respect to ζ

_{1}, ζ

_{2}, ζ

_{3}.)

*,*

**x***) to variables (ξ, ζ) is canonical, the eikonal equation is obtained by replacing (ζ*

**p**_{1}, ζ

_{2}, ζ

_{3}) by (

*∂V*/

*∂ξ*

_{1},

*∂V*/

*∂ξ*

_{2},

*∂V*/

*∂ξ*

_{3}) in the expression

*H*(

*ξ, ζ*)=0,

*i*.

*e*.,

*V*)

^{2}=

*n*

^{2}(ξ), because the gradient operator in ξ-coordinates is ∇

*V*=((1/

*h*

_{1})

*∂V*/

*∂ξ*

_{1}, (1/

*h*

_{2})

*∂V*/

*∂ξ*

_{2}, (1/

*h*

_{3})

*∂V*/

*∂ξ*

_{3}) [14

14. E. W. Weisstein “Gradient.” From *MathWorld*. http://mathworld.wolfram.com/Gradient.html.

_{1},ξ

_{2},ξ

_{3}) by (-

*∂W*/

*∂ζ*

_{1}, -

*∂W*/

*∂ζ*

_{2}, -

*∂W*/

*∂ζ*

_{3}) in the expression

*H*(ξ, ζ)=0:

*and -*

**x***variables found in spherical symmetric systems seems to be dependent on the particular coordinate system (Cartesian). It seems also that other refractive index distributions may have a coordinate system in which both Hamilton-Jacobi equations are eikonal equations (similarly as for given coordinate system there are refractive index distributions which make the eikonal equation separable [15*

**p**15. H. A. Buchdahl, “Rays in gradient-index media: separable systems,” J. Opt. Soc. Am. **63**, 46- (1973), http://www.opticsinfobase.org/abstract.cfm?URI=josa-63-1-46. [CrossRef]

*f*an arbitrary function. Is this possible?. Many solutions can be found if the refractive index is anisotropic, but this is not our case of interest. For the isotropic case, at least one solution occurs when the scale factors are such that

*h*

_{1}=

*h*

_{2}/

*k*

_{2}=

*h*

_{3}/

*k*

_{3}where

*k*

_{2}and

*k*

_{3}are 2 constants. These constants

*k*

_{2}and

*k*

_{3}must be equal to 1, because

*h*

_{1}(ξ

_{10},ξ

_{20},ξ

_{30})=

*h*

_{2}(ξ

_{10},ξ

_{20},ξ

_{30})=

*h*

_{3}(ξ

_{10},ξ

_{20},ξ

_{30}),

*i*.

*e*., the 3 scale factors are the same function. Then the Hamilton-Jacobi Eq. (21) can be written as

*η*is the function

*η*(ξ

_{1},ξ

_{2},ξ

_{3})=

*n*(ξ

_{1},ξ

_{2},ξ

_{3})·

*h*

_{1}(ξ

_{1},ξ

_{2},ξ

_{3}). The Hamilton-Jacobi equation in the momentum space Eq. (22) is

*η*(ξ

_{1},ξ

_{2},ξ

_{3}) is a function solely of the variable ξ

^{2}=

*h*

_{1}=

*h*

_{2}=

*h*

_{3}the coordinate transformation from

*to ξ, is conformal [16]. Inversions and similarities are the only possible conformal transformations in 3D Euclidean space [17]. We conclude that the Hamilton-Jacobi equation in momentum space can also be an eikonal equation in a curved orthogonal coordinate system, when this coordinate system can be obtained from Cartesian coordinates by means of an inversion, a similarity or a combination of both with an additional constant scaling factor for each configuration variable if necessary. Moreover, the refractive index distribution times the scale factor function*

**x***h*

_{1}(ξ) must be a function solely of the variable ξ

^{2}, and must be invertible.

## 5. Conclusions

*i*.

*e*., the ray trajectories in the coordinate and the momentum spaces are the same.

*i*.

*e*., any of these equations can be solved through the other plus a Legendre transformation. The Hamilton-Jacobi equation in the momentum space is also an eikonal equation in the new variables, whenever the curved coordinate system can be obtained by inversions and similarities from the Cartesian coordinate system plus an arbitrary constant factor for each of the new variables.

## Acknowledgments

## References and links

1. | R. Winston, J. C. Miñano, and P. Benítez, |

2. | V. I. Arnold, |

3. | J. V. José and E. J. Saletan, |

4. | G. W. Forbes, “On variational problems in parametric form,” Am. J. Phys. |

5. |
Wikipedia contributors, “Canonical transformation,” |

6. | O. N. Stravoudis, |

7. | M. A. Alonso and G. W. Forbes, “Generalization of Hamilton’s formalism for geometrical optics,” J. Opt. Soc. Am. A |

8. |
Wikipedia contributors, “Legendre transformation,” |

9. | M. Born and E. Wolf, |

10. | S. Cornbleet, |

11. | H. A. Buchdahl, “Kepler problem and Maxwell fish-eye,” Am. J. Phys. , |

12. | R. K. Luneburg, |

13. | H. A. Buchdahl, “Luneburg lens: unitary invariance and point characteristic,” J. Opt. Soc. Am. |

14. | E. W. Weisstein “Gradient.” From |

15. | H. A. Buchdahl, “Rays in gradient-index media: separable systems,” J. Opt. Soc. Am. |

16. | D. J. Struik, |

17. | D. E. Blair, |

**OCIS Codes**

(080.2710) Geometric optics : Inhomogeneous optical media

(080.2720) Geometric optics : Mathematical methods (general)

(080.2740) Geometric optics : Geometric optical design

**History**

Original Manuscript: June 30, 2006

Revised Manuscript: August 9, 2006

Manuscript Accepted: September 4, 2006

Published: October 2, 2006

**Citation**

Juan C. Miñano, Pablo Benítez, and Asunción Santamaría, "Hamilton-Jacobi equation in momentum space," Opt. Express **14**, 9083-9092 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-20-9083

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