Hamiltonian optics
[H. A. Buchdahl, ''An Introduction to Hamiltonian Optics'', Dover Publications, 1993, .] and Lagrangian optics
[Vasudevan Lakshminarayanan et al., ''Lagrangian Optics'', Springer Netherlands, 2011, .] are two formulations of
geometrical optics
Geometrical optics, or ray optics, is a model of optics that describes light propagation in terms of ''rays''. The ray in geometrical optics is an abstraction useful for approximating the paths along which light propagates under certain circumstan ...
which share much of the mathematical formalism with
Hamiltonian mechanics
Hamiltonian mechanics emerged in 1833 as a reformulation of Lagrangian mechanics. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities \dot q^i used in Lagrangian mechanics with (generalized) ''momenta ...
and
Lagrangian mechanics
In physics, Lagrangian mechanics is a formulation of classical mechanics founded on the stationary-action principle (also known as the principle of least action). It was introduced by the Italian-French mathematician and astronomer Joseph-Lou ...
.
Hamilton's principle
In
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
, Hamilton's principle states that the evolution of a system
described by
generalized coordinates
In analytical mechanics, generalized coordinates are a set of parameters used to represent the state of a system in a configuration space. These parameters must uniquely define the configuration of the system relative to a reference state.,p. 3 ...
between two specified states at two specified parameters ''σ''
''A'' and ''σ''
''B'' is a
stationary point
In mathematics, particularly in calculus, a stationary point of a differentiable function of one variable is a point on the graph of the function where the function's derivative is zero. Informally, it is a point where the function "stops" inc ...
(a point where the
variation is zero) of the
action
Action may refer to:
* Action (narrative), a literary mode
* Action fiction, a type of genre fiction
* Action game, a genre of video game
Film
* Action film, a genre of film
* ''Action'' (1921 film), a film by John Ford
* ''Action'' (1980 fil ...
functional
Functional may refer to:
* Movements in architecture:
** Functionalism (architecture)
** Form follows function
* Functional group, combination of atoms within molecules
* Medical conditions without currently visible organic basis:
** Functional sy ...
, or
where
and
is the
Lagrangian
Lagrangian may refer to:
Mathematics
* Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier
** Lagrangian relaxation, the method of approximating a difficult constrained problem with ...
. Condition
is valid if and only if the Euler-Lagrange equations are satisfied, i.e.,
with
.
The momentum is defined as
and the Euler–Lagrange equations can then be rewritten as
where
.
A different approach to solving this problem consists in defining a Hamiltonian (taking a
Legendre transform
In mathematics, the Legendre transformation (or Legendre transform), named after Adrien-Marie Legendre, is an involutive transformation on real-valued convex functions of one real variable. In physical problems, it is used to convert functions ...
of the
Lagrangian
Lagrangian may refer to:
Mathematics
* Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier
** Lagrangian relaxation, the method of approximating a difficult constrained problem with ...
) as
for which a new set of differential equations
can be derived by looking at how the
total differential
In calculus, the differential represents the principal part of the change in a function ''y'' = ''f''(''x'') with respect to changes in the independent variable. The differential ''dy'' is defined by
:dy = f'(x)\,dx,
where f'(x) is the ...
of the
Lagrangian
Lagrangian may refer to:
Mathematics
* Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier
** Lagrangian relaxation, the method of approximating a difficult constrained problem with ...
depends on parameter ''σ'', positions
and their derivatives
relative to ''σ''. This derivation is the same as in Hamiltonian mechanics, only with time ''t'' now replaced by a general parameter ''σ''. Those differential equations are the Hamilton's equations
with
. Hamilton's equations are first-order
differential equation
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
s, while Euler-Lagrange's equations are second-order.
Lagrangian optics
The general results presented above for
Hamilton's principle
In physics, Hamilton's principle is William Rowan Hamilton's formulation of the principle of stationary action. It states that the dynamics of a physical system are determined by a variational problem for a functional based on a single function ...
can be applied to optics.
[Roland Winston et al., ''Nonimaging Optics'', Academic Press, 2004, .] In
3D euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
the
generalized coordinates
In analytical mechanics, generalized coordinates are a set of parameters used to represent the state of a system in a configuration space. These parameters must uniquely define the configuration of the system relative to a reference state.,p. 3 ...
are now the coordinates of
euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
.
Fermat's principle
Fermat's principle states that the optical length of the path followed by light between two fixed points, A and B, is a stationary point. It may be a maximum, a minimum, constant or an
inflection point
In differential calculus and differential geometry, an inflection point, point of inflection, flex, or inflection (British English: inflexion) is a point on a smooth plane curve at which the curvature changes sign. In particular, in the case of ...
. In general, as light travels, it moves in a medium of variable
refractive index
In optics, the refractive index (or refraction index) of an optical medium is a dimensionless number that gives the indication of the light bending ability of that medium.
The refractive index determines how much the path of light is bent, or ...
which is a
scalar field
In mathematics and physics, a scalar field is a function (mathematics), function associating a single number to every point (geometry), point in a space (mathematics), space – possibly physical space. The scalar may either be a pure Scalar ( ...
of position in space, that is,
in
3D euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
. Assuming now that light travels along the ''x''
3 axis, the path of a light ray may be parametrized as
starting at a point
and ending at a point
. In this case, when compared to
Hamilton's principle
In physics, Hamilton's principle is William Rowan Hamilton's formulation of the principle of stationary action. It states that the dynamics of a physical system are determined by a variational problem for a functional based on a single function ...
above, coordinates
and
take the role of the generalized coordinates
while
takes the role of parameter
, that is, parameter ''σ'' =''x''
3 and ''N''=2.
In the context of
calculus of variations
The calculus of variations (or Variational Calculus) is a field of mathematical analysis that uses variations, which are small changes in functions
and functionals, to find maxima and minima of functionals: mappings from a set of functions t ...
this can be written as
where is an infinitesimal displacement along the ray given by
and
is the optical Lagrangian and
.
The
optical path length In optics, optical path length (OPL, denoted ''Λ'' in equations), also known as optical length or optical distance, is the product of the geometric length of the optical path followed by light and the refractive index of homogeneous medium through ...
(OPL) is defined as
where ''n'' is the local refractive index as a function of position along the path between points A and B.
The Euler-Lagrange equations
The general results presented above for
Hamilton's principle
In physics, Hamilton's principle is William Rowan Hamilton's formulation of the principle of stationary action. It states that the dynamics of a physical system are determined by a variational problem for a functional based on a single function ...
can be applied to optics using the Lagrangian defined in
Fermat's principle
Fermat's principle, also known as the principle of least time, is the link between ray optics and wave optics. In its original "strong" form, Fermat's principle states that the path taken by a ray between two given points is the pat ...
. The Euler-Lagrange equations with parameter ''σ'' =''x''
3 and ''N''=2 applied to Fermat's principle result in
with and where ''L'' is the optical Lagrangian and
.
Optical momentum
The optical momentum is defined as
and from the definition of the optical Lagrangian
this expression can be rewritten as
or in vector form
where
is a
unit vector
In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in \hat (pronounced "v-hat").
The term ''direction vecto ...
and angles ''α''
1, ''α''
2 and ''α''
3 are the angles p makes to axis ''x''
1, ''x''
2 and ''x''
3 respectively, as shown in figure "optical momentum". Therefore, the optical momentum is a vector of
norm
Naturally occurring radioactive materials (NORM) and technologically enhanced naturally occurring radioactive materials (TENORM) consist of materials, usually industrial wastes or by-products enriched with radioactive elements found in the envir ...
where ''n'' is the refractive index at which p is calculated. Vector p points in the direction of propagation of light. If light is propagating in a
gradient index optic the path of the light ray is curved and vector p is tangent to the light ray.
The expression for the optical path length can also be written as a function of the optical momentum. Having in consideration that
the expression for the optical Lagrangian can be rewritten as
and the expression for the optical path length is
Hamilton's equations
Similarly to what happens in
Hamiltonian mechanics
Hamiltonian mechanics emerged in 1833 as a reformulation of Lagrangian mechanics. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities \dot q^i used in Lagrangian mechanics with (generalized) ''momenta ...
, also in optics the Hamiltonian is defined by the expression given
above for corresponding to functions
and
to be determined
Comparing this expression with
for the Lagrangian results in
And the corresponding Hamilton's equations with parameter ''σ'' =''x''
3 and ''k''=1,2 applied to optics are
[Rudolf Karl Luneburg,''Mathematical Theory of Optics'', University of California Press, Berkeley, CA, 1964, p. 90.]
with
and
.
Applications
It is assumed that light travels along the ''x''
3 axis, in
Hamilton's principle
In physics, Hamilton's principle is William Rowan Hamilton's formulation of the principle of stationary action. It states that the dynamics of a physical system are determined by a variational problem for a functional based on a single function ...
above, coordinates
and
take the role of the generalized coordinates
while
takes the role of parameter
, that is, parameter ''σ'' =''x''
3 and ''N''=2.
Refraction and reflection
If plane ''x''
1''x''
2 separates two media of refractive index ''n''
''A'' below and ''n''
''B'' above it, the refractive index is given by a
step function
In mathematics, a function on the real numbers is called a step function if it can be written as a finite linear combination of indicator functions of intervals. Informally speaking, a step function is a piecewise constant function having only ...
and from
Hamilton's equations
Hamiltonian mechanics emerged in 1833 as a reformulation of Lagrangian mechanics. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities \dot q^i used in Lagrangian mechanics with (generalized) ''momenta ...
and therefore
or
for .
An incoming light ray has momentum p
''A'' before refraction (below plane ''x''
1''x''
2) and momentum p
''B'' after refraction (above plane ''x''
1''x''
2). The light ray makes an angle ''θ''
''A'' with axis ''x''
3 (the normal to the refractive surface) before refraction and an angle ''θ''
''B'' with axis ''x''
3 after refraction. Since the ''p''
1 and ''p''
2 components of the momentum are constant, only ''p''
3 changes from ''p''
3''A'' to ''p''
3''B''.
Figure "refraction" shows the geometry of this refraction from which
. Since
and
, this last expression can be written as
which is
Snell's law
Snell's law (also known as Snell–Descartes law and ibn-Sahl law and the law of refraction) is a formula used to describe the relationship between the angles of incidence and refraction, when referring to light or other waves passing through ...
of
refraction
In physics, refraction is the redirection of a wave as it passes from one medium to another. The redirection can be caused by the wave's change in speed or by a change in the medium. Refraction of light is the most commonly observed phenomeno ...
.
In figure "refraction", the normal to the refractive surface points in the direction of axis ''x''
3, and also of vector
. A unit normal
to the refractive surface can then be obtained from the momenta of the incoming and outgoing rays by
where i and r are unit vectors in the directions of the incident and refracted rays. Also, the outgoing ray (in the direction of
) is contained in the plane defined by the incoming ray (in the direction of
) and the normal
to the surface.
A similar argument can be used for
reflection Reflection or reflexion may refer to:
Science and technology
* Reflection (physics), a common wave phenomenon
** Specular reflection, reflection from a smooth surface
*** Mirror image, a reflection in a mirror or in water
** Signal reflection, in ...
in deriving the law of
specular reflection
Specular reflection, or regular reflection, is the mirror-like reflection of waves, such as light, from a surface.
The law of reflection states that a reflected ray of light emerges from the reflecting surface at the same angle to the surf ...
, only now with ''n''
''A''=''n''
''B'', resulting in ''θ''
''A''=''θ''
''B''. Also, if i and r are unit vectors in the directions of the incident and refracted ray respectively, the corresponding normal to the surface is given by the same expression as for refraction, only with ''n''
''A''=''n''
''B''
In vector form, if i is a unit vector pointing in the direction of the incident ray and n is the unit normal to the surface, the direction r of the refracted ray is given by:
with
If i⋅n<0 then −n should be used in the calculations. When
, light suffers
total internal reflection
Total internal reflection (TIR) is the optical phenomenon in which waves arriving at the interface (boundary) from one medium to another (e.g., from water to air) are not refracted into the second ("external") medium, but completely reflected b ...
and the expression for the reflected ray is that of reflection:
Rays and wavefronts
From the definition of optical path length
with ''k''=1,2 where the
Euler-Lagrange equations with ''k''=1,2 were used. Also, from the last of
Hamilton's equations
Hamiltonian mechanics emerged in 1833 as a reformulation of Lagrangian mechanics. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities \dot q^i used in Lagrangian mechanics with (generalized) ''momenta ...
and from
above
combining the equations for the components of momentum p results in
Since p is a vector tangent to the light rays, surfaces ''S''=Constant must be perpendicular to those light rays. These surfaces are called
wavefront
In physics, the wavefront of a time-varying ''wave field'' is the set (locus) of all points having the same ''phase''. The term is generally meaningful only for fields that, at each point, vary sinusoidally in time with a single temporal freque ...
s. Figure "rays and wavefronts" illustrates this relationship. Also shown is optical momentum p, tangent to a light ray and perpendicular to the wavefront.
Vector field
is
conservative vector field
In vector calculus, a conservative vector field is a vector field that is the gradient of some function. A conservative vector field has the property that its line integral is path independent; the choice of any path between two points does not c ...
. The
gradient theorem
The gradient theorem, also known as the fundamental theorem of calculus for line integrals, says that a line integral through a gradient field can be evaluated by evaluating the original scalar field at the endpoints of the curve. The theorem is ...
can then be applied to the optical path length (as given
above) resulting in
and the optical path length ''S'' calculated along a curve ''C'' between points A and B is a function of only its end points A and B and not the shape of the curve between them. In particular, if the curve is closed, it starts and ends at the same point, or A=B so that
This result may be applied to a closed path ABCDA as in figure "optical path length"
for curve segment AB the optical momentum p is perpendicular to a displacement ''d''s along curve AB, or
. The same is true for segment CD. For segment BC the optical momentum p has the same direction as displacement ''d''s and
. For segment DA the optical momentum p has the opposite direction to displacement ''d''s and
. However inverting the direction of the integration so that the integral is taken from A to D, ''d''s inverts direction and
. From these considerations
or
and the optical path length ''S''
BC between points B and C along the ray connecting them is the same as the optical path length ''S''
AD between points A and D along the ray connecting them. The optical path length is constant between wavefronts.
Phase space
Figure "2D phase space" shows at the top some light rays in a two-dimensional space. Here ''x''
2=0 and ''p''
2=0 so light travels on the plane ''x''
1''x''
3 in directions of increasing ''x''
3 values. In this case
and the direction of a light ray is completely specified by the ''p''
1 component of momentum
since ''p''
2=0. If ''p''
1 is given, ''p''
3 may be calculated (given the value of the refractive index ''n'') and therefore ''p''
1 suffices to determine the direction of the light ray. The refractive index of the medium the ray is traveling in is determined by
.
For example, ray ''r''
''C'' crosses axis ''x''
1 at coordinate ''x''
''B'' with an optical momentum p
''C'', which has its tip on a circle of radius ''n'' centered at position ''x''
''B''. Coordinate ''x''
''B'' and the horizontal coordinate ''p''
1''C'' of momentum p
''C'' completely define ray ''r''
''C'' as it crosses axis ''x''
1. This ray may then be defined by a point ''r''
''C''=(''x''
''B'',''p''
1''C'') in space ''x''
1''p''
1 as shown at the bottom of the figure. Space ''x''
1''p''
1 is called
phase space
In dynamical system theory, a phase space is a space in which all possible states of a system are represented, with each possible state corresponding to one unique point in the phase space. For mechanical systems, the phase space usually ...
and different light rays may be represented by different points in this space.
As such, ray ''r''
''D'' shown at the top is represented by a point ''r''
''D'' in phase space at the bottom. All rays crossing axis ''x''
1 at coordinate ''x''
''B'' contained between rays ''r''
''C'' and ''r''
''D'' are represented by a vertical line connecting points ''r''
''C'' and ''r''
''D'' in phase space. Accordingly, all rays crossing axis ''x''
1 at coordinate ''x''
''A'' contained between rays ''r''
''A'' and ''r''
''B'' are represented by a vertical line connecting points ''r''
''A'' and ''r''
''B'' in phase space. In general, all rays crossing axis ''x''
1 between ''x''
''L'' and ''x''
''R'' are represented by a volume ''R'' in phase space. The rays at the boundary ∂''R'' of volume ''R'' are called edge rays. For example, at position ''x''
''A'' of axis ''x''
1, rays ''r''
''A'' and ''r''
''B'' are the edge rays since all other rays are contained between these two. (A ray parallel to x1 would not be between the two rays, since the momentum is not in-between the two rays)
In three-dimensional geometry the optical momentum is given by
with
. If ''p''
1 and ''p''
2 are given, ''p''
3 may be calculated (given the value of the refractive index ''n'') and therefore ''p''
1 and ''p''
2 suffice to determine the direction of the light ray. A ray traveling along axis ''x''
3 is then defined by a point (''x''
1,''x''
2) in plane ''x''
1''x''
2 and a direction (''p''
1,''p''
2). It may then be defined by a point in four-dimensional
phase space
In dynamical system theory, a phase space is a space in which all possible states of a system are represented, with each possible state corresponding to one unique point in the phase space. For mechanical systems, the phase space usually ...
''x''
1''x''
2''p''
1''p''
2.
Conservation of etendue
Figure "volume variation" shows a volume ''V'' bound by an area ''A''. Over time, if the boundary ''A'' moves, the volume of ''V'' may vary. In particular, an infinitesimal area ''dA'' with outward pointing unit normal n moves with a velocity v.
This leads to a volume variation
. Making use of
Gauss's theorem
In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, reprinted in is a theorem which relates the ''flux'' of a vector field through a closed surface (mathematics), surface to the ''divergence'' o ...
, the variation in time of the total volume ''V'' volume moving in space is
The rightmost term is a
volume integral
In mathematics (particularly multivariable calculus), a volume integral (∭) refers to an integral over a 3-dimensional domain; that is, it is a special case of multiple integrals. Volume integrals are especially important in physics for many ap ...
over the volume ''V'' and the middle term is the
surface integral
In mathematics, particularly multivariable calculus, a surface integral is a generalization of multiple integrals to integration over surfaces. It can be thought of as the double integral analogue of the line integral. Given a surface, one may ...
over the boundary ''A'' of the volume ''V''. Also, v is the velocity with which the points in ''V'' are moving.
In optics coordinate
takes the role of time. In phase space a light ray is identified by a point
which moves with a "
velocity
Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity is a ...
"
where the dot represents a derivative relative to
. A set of light rays spreading over
in coordinate
,
in coordinate
,
in coordinate
and
in coordinate
occupies a volume
in phase space. In general, a large set of rays occupies a large volume
in phase space to which
Gauss's theorem
In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, reprinted in is a theorem which relates the ''flux'' of a vector field through a closed surface (mathematics), surface to the ''divergence'' o ...
may be applied
and using
Hamilton's equations
Hamiltonian mechanics emerged in 1833 as a reformulation of Lagrangian mechanics. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities \dot q^i used in Lagrangian mechanics with (generalized) ''momenta ...
or
and
which means that the phase space volume is conserved as light travels along an optical system.
The volume occupied by a set of rays in phase space is called
etendue
Etendue or étendue (; ) is a property of light in an optical system, which characterizes how "spread out" the light is in area and angle. It corresponds to the beam parameter product (BPP) in Gaussian beam optics. Other names for etendue include a ...
, which is conserved as light rays progress in the optical system along direction ''x''
3. This corresponds to
Liouville's theorem, which also applies to
Hamiltonian mechanics
Hamiltonian mechanics emerged in 1833 as a reformulation of Lagrangian mechanics. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities \dot q^i used in Lagrangian mechanics with (generalized) ''momenta ...
.
Imaging and nonimaging optics
Figure "conservation of etendue" shows on the left a diagrammatic two-dimensional optical system in which ''x''
2=0 and ''p''
2=0 so light travels on the plane ''x''
1''x''
3 in directions of increasing ''x''
3 values.
Light rays crossing the input aperture of the optic at point ''x''
1=''x''
''I'' are contained between edge rays ''r''
''A'' and ''r''
''B'' represented by a vertical line between points ''r''
''A'' and ''r''
''B'' at the phase space of the input aperture (right, bottom corner of the figure). All rays crossing the input aperture are represented in phase space by a region ''R''
''I''.
Also, light rays crossing the output aperture of the optic at point ''x''
1=''x''
''O'' are contained between edge rays ''r''
''A'' and ''r''
''B'' represented by a vertical line between points ''r''
''A'' and ''r''
''B'' at the phase space of the output aperture (right, top corner of the figure). All rays crossing the output aperture are represented in phase space by a region ''R''
''O''.
Conservation of etendue in the optical system means that the volume (or area in this two-dimensional case) in phase space occupied by ''R''
''I'' at the input aperture must be the same as the volume in phase space occupied by ''R''
''O'' at the output aperture.
In imaging optics, all light rays crossing the input aperture at ''x''
1=''x''
''I'' are redirected by it towards the output aperture at ''x''
1=''x''
''O'' where ''x''
''I''=''m x''
''O''. This ensures that an image of the input is formed at the output with a magnification ''m''. In phase space, this means that vertical lines in the phase space at the input are transformed into vertical lines at the output. That would be the case of vertical line ''r''
''A'' ''r''
''B'' in ''R''
''I'' transformed to vertical line ''r''
''A'' ''r''
''B'' in ''R''
''O''.
In
nonimaging optics Nonimaging optics (also called anidolic optics)Roland Winston et al., ''Nonimaging Optics'', Academic Press, 2004 R. John Koshel (Editor), ''Illumination Engineering: Design with Nonimaging Optics'', Wiley, 2013 is the branch of optics concerned wi ...
, the goal is not to form an image but simply to transfer all light from the input aperture to the output aperture. This is accomplished by transforming the edge rays ∂''R''
''I'' of ''R''
''I'' to edge rays ∂''R''
''O'' of ''R''
''O''. This is known as the
edge ray principle.
Generalizations
Above it was assumed that light travels along the ''x''
3 axis, in
Hamilton's principle
In physics, Hamilton's principle is William Rowan Hamilton's formulation of the principle of stationary action. It states that the dynamics of a physical system are determined by a variational problem for a functional based on a single function ...
above, coordinates
and
take the role of the generalized coordinates
while
takes the role of parameter
, that is, parameter ''σ'' =''x''
3 and ''N''=2. However, different parametrizations of the light rays are possible, as well as the use of
generalized coordinates
In analytical mechanics, generalized coordinates are a set of parameters used to represent the state of a system in a configuration space. These parameters must uniquely define the configuration of the system relative to a reference state.,p. 3 ...
.
General ray parametrization
A more general situation can be considered in which the path of a light ray is parametrized as
in which ''σ'' is a general parameter. In this case, when compared to
Hamilton's principle
In physics, Hamilton's principle is William Rowan Hamilton's formulation of the principle of stationary action. It states that the dynamics of a physical system are determined by a variational problem for a functional based on a single function ...
above, coordinates
,
and
take the role of the generalized coordinates
with ''N''=3. Applying
Hamilton's principle
In physics, Hamilton's principle is William Rowan Hamilton's formulation of the principle of stationary action. It states that the dynamics of a physical system are determined by a variational problem for a functional based on a single function ...
to optics in this case leads to
where now
and
and for which the Euler-Lagrange equations applied to this form of Fermat's principle result in
with ''k''=1,2,3 and where ''L'' is the optical Lagrangian. Also in this case the optical momentum is defined as
and the Hamiltonian ''P'' is defined by the expression given
above for ''N''=3 corresponding to functions
,
and
to be determined
And the corresponding Hamilton's equations with ''k''=1,2,3 applied optics are
with
and
.
The optical Lagrangian is given by
and does not explicitly depend on parameter ''σ''. For that reason not all solutions of the Euler-Lagrange equations will be possible light rays, since their derivation assumed an explicit dependence of ''L'' on ''σ'' which does not happen in optics.
The optical momentum components can be obtained from
where
. The expression for the Lagrangian can be rewritten as
Comparing this expression for ''L'' with that for the Hamiltonian ''P'' it can be concluded that
From the expressions for the components
of the optical momentum results
The optical Hamiltonian is chosen as
although other choices could be made.
The Hamilton's equations with ''k'' = 1, 2, 3 defined above together with
define the possible light rays.
Generalized coordinates
As in
Hamiltonian mechanics
Hamiltonian mechanics emerged in 1833 as a reformulation of Lagrangian mechanics. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities \dot q^i used in Lagrangian mechanics with (generalized) ''momenta ...
, it is also possible to write the equations of Hamiltonian optics in terms of
generalized coordinates
In analytical mechanics, generalized coordinates are a set of parameters used to represent the state of a system in a configuration space. These parameters must uniquely define the configuration of the system relative to a reference state.,p. 3 ...
, generalized momenta
and Hamiltonian ''P'' as
where the optical momentum is given by
and
,
and
are
unit vectors
In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in \hat (pronounced "v-hat").
The term ''direction vecto ...
. A particular case is obtained when these vectors form an
orthonormal basis
In mathematics, particularly linear algebra, an orthonormal basis for an inner product space ''V'' with finite dimension is a basis for V whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For example, ...
, that is, they are all perpendicular to each other. In that case,
is the cosine of the angle the optical momentum
makes to unit vector
.
See also
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Hamiltonian mechanics
Hamiltonian mechanics emerged in 1833 as a reformulation of Lagrangian mechanics. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities \dot q^i used in Lagrangian mechanics with (generalized) ''momenta ...
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Calculus of variations
The calculus of variations (or Variational Calculus) is a field of mathematical analysis that uses variations, which are small changes in functions
and functionals, to find maxima and minima of functionals: mappings from a set of functions t ...
References
{{Reflist, 2
Geometrical optics