Some Aspects of Hamilton-Jacobi Integrability and Separability

Workshop on Hamilton-Jacobi, Zaragoza, Enero 2008
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Some Aspects of
Hamilton-Jacobi Integrability and Separability
Mariano Santander Universidad de Valladolid

Basics on the Hamilton-Jacobi equation
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• History Hamilton attempt to formulate Mechanics as modelled after paraxial
geometric Optics
⋆ Optical axis as the time axis
⋆ Optical Image plane as Space at an instant
⋆ Simplectic structure behind (geometrical) optics appear as some
important structure in Mechanics: the shadow of the canonical symplectic
structure in Quantum Mechanics.
• Fermat principle
action
as Hamilton principle: optical lenght as the Hamilton’s
• Eikonal equation as the Hamilton-Jacobi equation.
W.R. Hamilton, Theory of systems of rays, Transactions Royal Irish
Academy, 15, 69-174 (1828)
W.R. Hamilton, On the application to Dynamics of a general Mathematical
Method previously applied to optics, British Association Report,
Edinburgh, 513-518 (1835)

Basics on the Hamilton-Jacobi equation
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• Visionary character of Hamilton contribution
⋆ The ‘fundamental analogy’
Geometric Optics : Ondulatory Optics
::
Classical Mechanics : Quantum Mechanics
• Hamilton-Jacobi equation as the shadow of Schrödinger equation.
• Classical mechanics as a double approximation from Quantum Mechanics
and from Relativity: short wavelenght λ → 0 and paraxial approximation.
⋆ Hamilton’s action is from a viewpoint the non-relativistic remnant of
the difference between proper and coordinate time S = mc 2 (τ − t)
⋆ And from another one, Hamilton’s action registers the phase of the
quantum amplitudes (Feynmann formulation, S = ¯hφ)

The History of Hamilton-Jacobi equation
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• Hamilton contribution Mechanics as a consequence of a extremal principle.
Action functional as the time integral of Lagrangian.
⋆ Consider the ‘Hamilton principal function’ defined as the value of
the action along the actual trajectories S(x, t). Now Hamilton saw that
this function is a solution of a partial differential equation, the mechanical
analogue of the optics eikonal equation.
1
⋆ The equation is obtained as a translation of
2m p2 + V (x) = E under
correspondence rules: p ↦→ ∇S, E ↦→ − ∂S
∂t
1
2m (∇S)2 + V (x) = − ∂S
∂t
• Jacobi contribution He realized that also the converse was true; any solution
of this equation can be used to solve the motion problem, by means of some
canonical transformation.
• Useful as an actual tool to solve specific problems? Indeed this is only
useful in practice when it is possible to separate variables.

The Mathematics of Hamilton-Jacobi equation
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• As a partial differential equation we note the essential appearance of
the Laplace operator ∇ 2
⋆ This the the only second-order differential operator invariant (scalar)
under the Euclidean isometry group
• Extension to more general configuration spaces Natural stages: Zero
curvature; Constant Curvature; General Riemannian space.
⋆ Eventually, also Minkowskian configuration spaces
• Is there some geometrical meaning of separability? Not an easy question;
because separability depends on the coordinate system.
• Now look to any classical book on Mechanics The odds are that they
use either the harmonic oscillator or the Kepler problem as an illustration of
HJ.
• Both systems are distinguished by the existence of an ‘abnormaly large’
number of constants of motion. And in both the HJ equation turns out
to be separable in several coordinate systems.

Constants of motion for the two ‘principal cases’
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• Harmonic Oscillator
(second order).
Angular momentum (first order) and Fradkin tensor
• Kepler-Coulomb problem Angular momentum (first order) and Laplace-
Runge-Lenz vector (second order)
• Fradkin conserved tensor and Runge-Lenz conserved vector appear
precisely as linked to irreducible sets of constants of motion coming from the
separability of Hamilton-Jacobi equation for the Harmonic oscillator and the
Kepler-Coulomb potentials in several coordinate systems:
⋆ Runge-Lenz vector is related to separability of the HJ equation for the
Kepler potential in a full 1-d family of parabolic coordinates, with a focus
at the origin
A 1 = Jẏ − k cos φ, A 2 = −Jẋ − k sin φ, (1)
⋆ Fradkin tensor related to separability of Harmonic oscillator potential in
a full 1-d family of cartesian coordinates with any axes orientation
F 11 = (ẋ) 2 + ω 2 0 x 2 , F 12 = F 21 = ẋẏ + ω 2 0 xy, F 22 = (ẏ) 2 + ω 2 0 y 2 . (2)

Integrability and separability of Hamilton-Jacobi equation in E 2
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• Is there any relation between existence of constants of motion and
separability of the HJ equation?
• Classical Integrability Existence of constants of motion. For n = 2 a single
constant ensures integrability, no restriction due to involution condition.
• Noether constant related with invariance under some one-parameter group of
symmetries (a subgroup of the Euclidean group). First order in the velocities.
⋆ These symmetries are clearly related to the separability of the HJ
equation for the Harmonic oscillator and for the Kepler potential in polar
coordinates.
⋆ Are there other Noether constants responsible of the separability of
HJ equation for harmonic oscillator in cartesian coordinates or of the HJ
for Kepler in parabolic coordinates? No

Integrability and separability of Hamilton-Jacobi equation in E 2 II
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• Are there other constants (not of Noether type) responsible of / linked
to the separability of HJ equation for harmonic oscillator in cartesian
coordinates or of the HJ for Kepler in parabolic coordinates? Yes
• Next type of constants of motion are required to be quadratic in the
velocities (similar to the energy). Precisely these constants are related to
separability of Hamilton-Jacobi equation.
• Separability of Hamilton Jacobi equation is possible only in some coordinate
systems, determined independently of the potential (i.e., for the
free Hamilton-Jacobi equation) and once a such system is fixed, it requires
additionally the potential to have some special form, called Stäckel or
separable form, in these coordinates.
• When these conditions are met there always exists a constant of motion
which is quadratic in the velocities. Exceptionally, this constant may come
as the square of a constant which is first order in the velocities.

Dynamics on a constant curvature 2d space
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• Motion of a particle in the configuration space under a natural mechanical
type Lagrangian
L = 1 2 g µν(q 1 , q 2 )v q µv q ν − V(q 1 , q 2 ).
⋆ Hamiltonian
H = 1 2 gµν (q 1 , q 2 )p µ p ν + V(q 1 , q 2 ).
⋆ Configuration space: either the Euclidean plane, or a space with
constant curvature, or a Riemannian space with a general metric
(or even a Lorentzian configuration space, with a non-definite metric)
• Restrict to the constant curvature case. The leading idea is to see how
properties change when the configuration space acquires non-zero curvature.
• Hamilton-Jacobi equation now involves the Laplace-Beltrami operator
of the metric

Systems allowing linear constants of motion, n = 2
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• Which is the most general system allowing constants of motion linear
in the velocities?
• The system must be invariant under a one-parameter subgroup of
(Euclidean, spherical, hyperbolic) isometries
⋆ Noether momenta Denote P 1 , P 2 , J 12 , the Noether momenta associated
to translations along two orthogonal lines l 1 , l 2 and rotations around O =
l 1 ∩ l 2 .
⋆ The most general constant of motion linear in the velocities is
I = a 12 J 12 + +a 1 P 1 + a 2 P 2
⋆ Free part of Hamilton-Jacobi equation (Laplace-Beltrami operator)
is automatically invariant under the subgroup generated by I No
restriction here.
⋆ Potential invariance condition singles out potentials rotationally invariant
(around a center, central potentials) and translationally invariant
(‘one-dimensional’ potentials). This works exactly alike in the non-flat case.

Linear constants of motion. Direct approach
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• Are there constants of motion which are linear in the velocities?
• Possible constants are of the form I K = K µ (q 1 , q 2 ) v q µ. This is actually
a constant of motion provided K µ (q 1 , q 2 ) satisfies some equations involving
the data g µν (q 1 , q 2 ), V(q 1 , q 2 ).
⋆ These equations split in two sets. The first is independent of the potential,
and restricts how the vector field K µ (q 1 , q 2 ) can depend on coordinates.
⋆ There is some geometric meaning in the equations in the first set?
K µ (q 1 , q 2 ) is a Killing vector field for the metric of the space.
I K = K µ (q 1 , q 2 ) v q µv q ν = a 0 J + a 1 P 1 + a 2 P 2
• Once a Killing vector field has been chosen the remaining equation to be
satisfied by the potential is the usual invariance requirement. Geometrically,
the potential must be invariant under the isometries generated by the Killing
vector field.

Linear constants of motion. Killing vector interpretation
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• Free motion allow for three linearly independent constants of motion
linear in the velocities. Suitable change of the basic l 1 , l 2 , O reduces any
I K = a 0 J + a 1 P 1 + a 2 P 2 to either P 1 and J .
⋆ Associated coordinate systems Rectifying coordinates for P 1 and J . In
the euclidean plane these are respectively cartesian coordinates and polar
coordinates. On the sphere and the hyperbolic plane these are its analogous
(geodesic parallel and geodesic polar coordinates)
⋆ Some authors call these coordinates ‘group coordinates’
⋆ Non-free systems with a given potential will keep a such linear
constant I = a 0 J + a 1 P 1 + a 2 P 2 as long as the potential is invariant
under the one-parameter subgroup generated by I.
⋆ In terms of the associated group coordinates, the condition can be
formulated geometrically as ‘the potential does depend only on a single
coordinate’. But beware!
• Close connection between Killing vectors and constant of motion linear in
the velocities.

Constants of motion quadratic in the velocities. Direct approach
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• Are there constants of motion I which are quadratic in the velocities?
⋆ Possible constants I K = K µν (q 1 , q 2 ) v q µv q ν + W(q 1 , q 2 ).
• The requirement for I K to be a constant of motion translates into
some conditions on K µν and W These conditions split in two subsets.
• First subset, independent of the potential V Thus this might be expected
to depend only on some property of the free Hamilton-Jacobi equation.
Which property?
• Not obvious at all These equations appear in several classical papers and
books.
⋆ Geometric interpretation: the tensor K µν should be a Killing tensor for
the metric g µν .
⋆ After all, a rather natural extension to the first order case
⋆ Most general solution to these equations :
K µν (q 1 , q 2 ) v q µv q ν = a 0 J 2 + a 1 P 2 1 + a 2 P 2 2 + 2a 01 J P 1 + 2a 02 J P 2 + 2a 12 P 1 P 2 (3)

Constants of motion quadratic in the velocities. Direct approach II
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• Once a Killing tensor K µν has been fixed , the second subset of conditions
(which would determine W) depend actually on the potential V. These are
a system of partial differential equations for W; its compatibility equation is
a single differential equation for the potential V.
⋆ Essentially this equation dates (in the euclidean case, in general
coordinates) from Levi-Civita
• Geometric meaning Not direct, but somehow a kind of ‘second order invariance’

Killing tensors and confocal coordinate systems
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• Connection between Killing vectors and group coordinates Each Killing
vector determines a coordinate web in the space (associated to the group
coordinates; e.g. for euclidean plane these are either cartesian or polar)
⋆ Potentials allowing for a Killing vector as a constant of motion have
a special dependence on these coordinates
• Each Killing tensor also determines a coordinate web in the configuration
space. But coordinates are not ‘group coordinates’
• Potentials allowing a I K -type quadratic constant of motion are precisely
those which are separable in the coordinate system associated to this
web.
⋆ W can be determined also starting from the separable form of V.
• Coordinates are secondary, the web is the important thing
• Main problem: find the coordinate webs associated to the more general
Killing tensor

Systems allowing quadratic constants of motion: elliptic coordinates
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• Most general coordinates associated to Killing tensors? These are
precisely coordinate systems allowing the (free) Hamilton-Jacobi equation to
be separated.
• In the euclidean case
⋆ For coordinates: Liouville Elliptic coordinates (Euler) and their three limiting
cases (polar coordinates, parabolic coordinates, cartesian coordinates).
Non-group coordinates versus group coordinates.
⋆ For the potential: Stäckel, Eisenhart The potential must be separable
in a coordinate system in the previous list.

Elliptic coordinates in E 2
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• For instance, elliptic coordinates with foci F 1 , F 2 symmetrically placed
relative to the origin, with separation 2f
Elliptic coordinates ( r 1+r 2
, r 1−r 2
).
2 2
Separable potentials: V = 1 ) + B( r 1+r 2
) }
2 2
Constant of motion:
(J 2 − f 2 P 2 1 )+ 1
r 1 r 2
{
A(
r 1 +r 2
{(
(
r 1 +r 2
) 2 − f 2) B( r 1−r 2
) − ( f 2 − ( r 1−r 2
) 2) A( r 1+r 2
) }
2 2 2 2
r 1 r 2
• The non-generic limits can be dealt with accordingly
⋆ Polar Both foci coincide
⋆ Parabolic One focus go to infinity
⋆ Cartesian Both foci go to infinity

Killing tensors and elliptic coordinates on curved spaces
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• Main problem: In a space with any constant curvature, find the coordinate
webs associated to the more general Killing tensor
• Generic Confocal webs Curves with r 1 ±r 2 , ˜r 1 ± ˜r 2 , r 1 ± ˜r 2 equal to constant,
with r 1 , r 2 , . . . determined as:
⋆ A pair of focal points, r 1 , r 2 denoting the intrinsic distances to these foci
(Elliptic ).
⋆ A pair of focal lines, ˜r 1 , ˜r 2 denoting the intrinsic distances to these focal
lines (UltraElliptic ).
⋆ A pair made of a focal point and a focal line, with r 1 , ˜r 2 denoting the
intrinsic distances to these focal elements (Parabolic ).
• Particular or Limiting Confocal webs where focal elements either coincide
or go to infinity if possible at all.
⋆ The number of cases depend on whether there is an actual infinity (as
in H 2 ) or not (as in S 2 )

The elliptic coordinate system in the curved case
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• Space with constant curvature κ
• Consider the web associated to elliptic coordinates determined by two
foci F 1 , F 2 with separation 2f Basic generic system in all cases.
⋆ Elliptic coordinates ( r 1+r 2
, r 1−r 2
).
2 2 {
1
⋆ Separable potentials: V =
S κ1 (r 1 ) S κ1 (r 2 ) A(
r 1 +r 2
) + B( r 1+r 2
) }
2 2
⋆ Constant ( of motion: )
C
2
κ1
(f)J 2 − S 2 κ 1
(f)P1
2 +
1 {(
S
2
S κ1 (r 1 ) S κ1 (r 2 )
κ1
( r 1+r 2
) − S 2 2 κ 1
(f) ) B( r 1−r 2
) − ( S 2 2 κ 1
(f) − S 2 κ 1
( r 1−r 2
) ) A( r 1+r 2
) }
2 2
⋆ Labeled Trigonometric functions: ‘cosine’ C κ (x) and ‘sine’ S κ (x), ‘label’
κ:
⎧
⎪⎨ cos √ ⎧
κ x
√1
⎪⎨ κ
sin √ κ x κ > 0
C κ (x) := 1
⎪⎩
cosh √ S κ (x) := x κ = 0
−κ x
⎪⎩ √1
−κ
sinh √ . (4)
−κ x κ < 0

The classification of (real) confocal coordinate systems in H 2
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• Depicted in the Poincaré conformal Half plane model, doubled for
graphic convenience
⋆ Two copies of the hyperbolic plane glued by the circle at infinity

Non-obvious integrable systems in spaces of constant curvature
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• Basic classification in types of constants: two constants are of the
same type if their Killing tensor part is the same
• But constants of motion are linear If I 1 , I 2 are constants of motion, so it
is αI 1 + βI 2 .
• In particular this means that if a particular system admits two constants
of motion of quadratic type, with Killing tensors W 1 , W 2 , then it the
HJ equation is separable in two different coordinate systems.
• Now the system admits a one-parameter infinite family of Killing tensors
αW 1 + βW 2 . And in turn this means that the HJ eqution is separable
not only in two coordinate systems, but in any member of a infinite
family (depending essentially of a single parameter β/α).
• This elementary observation can be used to conclude, whitout any
computation, that some non-trivial systems are integrable and HJ separable.

An example of integrable system in spaces of constant curvature
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• Consider the problem with two Kepler-Coulomb centers Since Euler
this system on the Euclidean plane has been known to be integrable.
• Is this system integrable on a constant curvature space?
• The curved Kepler problem has constants of types J 2 , associated to polar
(double focus at the origin) and J P 2 , associated to parabolic coordinates
(a focus at the origin).
⋆ As constants are linear, it follows that Kepler has also a constant of
type J (J + λP 2 ). This is associated to a system of elliptic coordinates,
with a focus at origin and the other on the axis of the initial parabolic
system. All such interpolating elliptic systems appear by tuning λ.
• Thus Kepler has constants of the type associated to all elliptic coordinates
with a focus at the origin and is therefore separable in all these
coordinate systems.
• The problem with two Kepler-Coulomb centers is therefore separable,
for any constant curvature, in the elliptic coordinate system with the two
focus at the two centers.

Some recent references
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⋆ J.F. Cariñena, M.F. Rañada, M.S., Superintegrability in curved spaces,
orbits and momentum hodographs: revisiting a classical result by
Hamilton, (J. Phys. A)
⋆ J.F. Cariñena, M.F. Rañada, M.S., The harmonic oscillator on Riemannian
and Lorentzian configuration spaces of constant curvature, To
appear in J. Math. Phys, arXiv: 0709.2572 [math-ph]