Newton's Off-Center Circular Orbits and the Magnetic Monopole
Introducing a radially dependent magnetic field into Newton's off-center circular orbits potential to preserve the = 0 dynamical symmetry leads to a unique choice of field that can be identified as the inclusion of a magnetic monopole in the inverse stereographically projected problem. One als...
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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| Дата: | 2023 |
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Інститут математики НАН України
2023
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| Цитувати: | Newton's Off-Center Circular Orbits and the Magnetic Monopole. Dipesh Bhandari and Michael Crescimanno. SIGMA 19 (2023), 099, 10 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860238771151175680 |
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| author | Bhandari, Dipesh Crescimanno, Michael |
| author_facet | Bhandari, Dipesh Crescimanno, Michael |
| citation_txt | Newton's Off-Center Circular Orbits and the Magnetic Monopole. Dipesh Bhandari and Michael Crescimanno. SIGMA 19 (2023), 099, 10 pages |
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| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | Introducing a radially dependent magnetic field into Newton's off-center circular orbits potential to preserve the = 0 dynamical symmetry leads to a unique choice of field that can be identified as the inclusion of a magnetic monopole in the inverse stereographically projected problem. One also finds a phenomenological correspondence with that of the linearly damped Kepler model. The presence of the monopole field deforms the symmetry algebra by a central extension, and the quantum mechanical version of this algebra reveals several zero modes equal to those counted using the index theorem of elliptic operators.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 19 (2023), 099, 10 pages
Newton’s Off-Center Circular Orbits
and the Magnetic Monopole
Dipesh BHANDARI a and Michael CRESCIMANNO b
a) Department of Physics and Astronomy, Texas A&M University-Commerce, TX, 75429, USA
E-mail: dbhandari@leomail.tamuc.edu
b) Department of Physics and Astronomy, Youngstown State University,
Youngstown, OH, 44555, USA
E-mail: dcphtn@gmail.com
Received July 31, 2023; in final form December 08, 2023; Published online December 17, 2023
https://doi.org/10.3842/SIGMA.2023.099
Abstract. Introducing a radially dependent magnetic field into Newton’s off-center circular
orbits potential so as to preserve the E = 0 dynamical symmetry leads to a unique choice
of field that can be identified as the inclusion of a magnetic monopole in the inverse stereo-
graphically projected problem. One finds also a phenomenological correspondence with that
of the linearly damped Kepler model. The presence of the monopole field deforms the sym-
metry algebra by a central extension, and the quantum mechanical version of this algebra
reveals a number of zero modes equal to that counted using the index theorem of elliptic
operators.
Key words: integrals of motion; magnetic monopole; zero modes
2020 Mathematics Subject Classification: 37J37; 19K56
1 Motivation
In Book 1, Proposition 7, Problem 2 of his 1687 Philosophiae Naturalis Principia Mathematica,
Isaac Newton finds the unique smooth, radially symmetric potential whose force results in off-
center circular orbits (non-relativistic) [21]. There are several remarkable features of these
circular orbits in this potential. Their orbital axis (length and angle) implies, just as in the
Kepler case, the existence of two constants of motion in addition to the angular momentum, as
was detailed in [22]. The Poisson algebra of these constants of motion closes on the E = 0 strata
into an so(3) algebra, as in that of the Runge–Lenz symmetry in the Kepler problem.
Here we explore a deeper algebraic and geometric connection between the Kepler case and
the E = 0 strata of Newton’s off-center circular orbit problem by first, as motivation, recalling
that Kepler orbits subject to frictional force proportional to the velocity evolve in E while
preserving their orbital eccentricity and orientation [12], a consequence of the ‘rigidity’ of the
so(3) Runge–Lenz symmetry (note this is also the case for the so(3) symmetry in the isotropic 2-d
harmonic oscillator with frictional forces linear in the velocity, a system also directly relatable to
the Kepler problem [7]). In seeking to test an analogue of this ‘rigidity’ in the off-center circular
orbit problem of Proposition 7, Problem 2 of the Principia, a frictional force proportional to the
velocity will not be appropriate since it will evolve into E ̸= 0 orbits for which there is no so(3)-
like symmetry. Giving the particles charge and subjecting them to a magnetic forces (also linear
in the velocity) on the other hand, will in general also modify the spectrum, but below we find
a unique choice of magnetic field that again results in an so(3)-like symmetry on E = 0 orbits.
In the Kepler problem, adding a magnetic field generally breaks the so(3) algebra, as all
orbits except concentric circular orbits now precess and the circular ones have E < 0. Adding
mailto:dbhandari@leomail.tamuc.edu
mailto:dcphtn@gmail.com
https://doi.org/10.3842/SIGMA.2023.099
2 D. Bhandari and M. Crescimanno
a magnetic field to the off-center circular orbit Newton problem has a markedly different effect.
Below we show that there is a unique magnetic field with that potential for which the Poisson
algebra of first integrals of motion for E = 0 satisfy an so(3)-like algebra. That algebra turns
out to be a central extension of that found in [22], again leading to off-center circular orbits.
The stereographic projection of the original Newton problem onto a sphere of radius R maps
the E = 0 off-center orbits into geodesics on that sphere [22]. In the magnetized problem as we
frame it below, the projection of the E = 0 orbits map onto the intersection of that sphere with
a plane not going through the origin. These orbits are those of a charge confined to the sphere
in which a magnetic monopole occupies its center. The charge of the monopole is exactly twice
the central extension of the Poisson algebra.
2 Algebra and dynamics
We consider all orbits as solutions of the equations of motion on the two-dimensional plane and
all vectors (denoted in bold) as two-dimensional unless explicitly indicated.
To fix notation, let the Hamiltonian of a non-relativistic particle of mass m = 1 in two-
dimensions with momentum p be written as
H =
p2
2
+ V (r). (2.1)
Throughout r, ϕ will refer to the usual polar co-ordinates on the plane, and x = r cosϕ,
y = r sinϕ the cartesian co-ordinates. Following Newton [21], in [22] for B = 0 (no magnetic
field), for the unique choice of V (r) = − α
(r2+R2)2
with a constant R, one has in addition to
L̃z = xpy − ypx, the two-dimensional vector I = L̃zr+ (r · p)ez × r+R2ez × p generating an
enveloping algebra that commutes with H on the H = 0 strata, where ez = ex × ey. To
recapitulate, as shown in [22] for B = 0, we have
{I, H} = −4Hez × r,
{
L̃z, H
}
= 0,
{
I, L̃z
}
= ez × I, {Ix, Iy} = 4R2L̃z. (2.2)
This observation greatly simplifies the determination of the orbits at H = 0. For H = 0 the I
are constants of motion and dividing them by L̃z = r2 dϕdt gives two first order, linear equations
in r(ϕ) and dr
dϕ . Eliminating the dr
dϕ gives an r(ϕ), an off-center circle whose radius R squared is
half the quadrature sum of Ix/L̃z and Iy/L̃z and whose angular orientation is given in terms of
arctan(Iy/Ix)
(
the distance between the orbital center and the origin being
√
R2 −R2
)
.
For B ̸= 0, we use minimal substitution, P = p +A, where p is the conjugate momentum
vector and A is the two-component vector potential, and write the Hamiltonian as
H =
P2
2
+ V (r) (2.3)
with the foregoing V . We define B, the magnetic field, here a pseudoscalar density on the
two-plane, through,
{Px, Py} = B. (2.4)
We limit our analysis to the case where B depends only on r, and define G = 2
∫
B(r)rdr. The
magnetic generalization of L̃z we denote Lz = xPy − yPx +G/2, which although it clearly does
not commute with the Hamiltonian in (2.1) it does with the one in (2.3), that is, {Lz, H} = 0.
The convenient magnetic generalization of the vector I we call J and define
J =
(
Lz +
G
2
)
r+ (r ·P)ez × r+R2ez ×P. (2.5)
Newton’s Off-Center Circular Orbits and the Magnetic Monopole 3
Requiring that {J, H} = −4Hez × r forces the magnetic field to have the unique solution
B(r) = −Q/
(
r2 +R2
)2
for some constant Q. Another brief calculation indicates
{J, H} = −4Hez × r, {Lz, H} = 0, {J, Lz} = ez × J,
{Jx, Jy} = 4R2Lz −Q, (2.6)
which is a central extension (by Q) of the B = 0 case in (2.2). This algebra again indicates that
the Lz, J are constants of the motion for H = E = 0 orbits. As in the B = 0 case, we use these
constants to find the shape of the E = 0 orbits. Resolving the expressions J · r/Lz and J× r/Lz
we find the periodic H = 0 orbits satisfy
J · r
Lz
= r2 −R2 +
Q
2Lz
,
J× r
Lz
= −
(
r2 +R2
)(r ·P)
Lz
. (2.7)
Note that the second equation does not involve Q. The first equation above is basically the
integral of the second one, in which term proportional to Q emerges as a constant of integration.
To see that in more detail, without loss of generality, take J = (J, 0) pointing in the x̂ direction
only and use the Hamilton equation, P = dr
dt . Writing Lz = r2 dϕdt on the right-hand side while
keeping Lz as a constant on the left-right side; using finally r · dr
dt =
dr2
2dt one can then integrate
the second equation to get the first. Note that for this choice of J the first equation is just the
equation for a circle centered about
(
J
2Lz
, 0
)
of radius squared equal to R2+ J2
4L2
z
− Q
2Lz
, requiring
the latter to be non-negative.
Varying the choices of J and Q it is clear that for a fixed R one sign (−) of Q
2Lz
corresponds to
circles whose chord through the origin extend beyond ±R, while for the other choice of sign (+)
is within ±R.
Finally, it is useful to compute the quadratic Casimir element of the Poisson algebra (2.6),
C2 =
J · J
4R2
+
(
Lz −
Q
4R2
)2
=
(
r2 +R2
)2
H
2R2
+
α
2R2
+
Q2
16R4
. (2.8)
For E = 0, the C2 must be a positive constant; this limits the magnitude of |J| and Lz for each Q.
Note also that combining C2 for E = 0 orbits with the (2.7), one arrives at a simple formula for
the radius R (in the plane) of the orbit, R =
√
α
2
1
Lz
, true for all Q. The value of |J | is directly
associated with the distance, l between the center of the orbit an the origin, l2 = R2 −R2 + Q
2Lz
.
For each fixed Q, the convexity of the Casimir element limits the magnitude of the Lz, and so
this indicates that there is a minimum possible R for that Q. It is only in the limit |Q| → ∞
that the Lz can grow without bound and the orbit can be of zero radius.
Historically, in dealing with a central potential combined with a magnetic monopole, some
authors have noted that including an additional potential term proportional to the square of the
magnetic charge and varying as 1/r2 can restore the dynamical symmetry [19] to what it was
before the addition of the magnetic monopole. That is, the gauge covariant integrals of motion
now commute with the new Hamiltonian and satisfy the same algebra as it was at Q = 0. At an
algebraic level, the analogous term to add to H of (2.3) would be −Q2/
[
8R2
(
r2 +R2
)2]
which
preserves the algebra and leads to the same Casimir element since it is just a change in α.
A simpler way to say this physically is that adding in a magnetic monopole field in the cases
studied in [19] causes the orbits to not close, and the additional potential term ∼ Q2/r2 added to
the H makes the orbits again close in those cases. The case of Newton’s off-center circular orbit
potential is quite different; here the inclusion of a magnetic monopole field preserves orbital
closure and the addition of the term analogous to “Q2/r2” simply keeps the problem in the
original family of Hamiltonians.
For completeness, we have also computed the orbital period for this potential. For E ̸= 0,
the period integral is elliptic functions, but as expected, one of the branchcuts degenerates in
4 D. Bhandari and M. Crescimanno
the E = 0 limit where the period integral reduces to a trigonometric function. One finds for
each E = 0 orbits the period
T =
πα
|L3
z|
+
πQ
2L2
z
.
For a fixed Q on the physical branch
(
on which the most negative Q/(2Lz) can be is −R2+R2
)
the maximum period occurs for Lz → 0, R → ∞. Note finally that the minimum possible period
is 0 in the limit of large |Q/Lz| only, as expected from the Larmor limit (R → 0).
3 The magnetic monopole via the stereographic projection
Note that the conformal transformation between the orbital plane and the stereographic repre-
sentation from [22] is given in that paper by
gplaneµν =
(
r2 +R2
2R2
)2
gS
2
µν . (3.1)
Since the field B is defined via the commutators of two momenta, in keeping with
Fµν = Bϵµν = {Pµ, Pν},
we see that the quantity B transforms as
Bplane =
( 2R2
r2 +R2
)2
BS2
,
and using the form of Bplane in the previous section we learn that the BS2
= − Q
4R4 , a constant,
that is, the unique B field that closes the algebra in the plane is a magnetic monopole as viewed
in the stereographic sphere, of dimensionless total flux = 2πM = −πQ
R2 (where we define the
monopole charge M , which is integer in the quantum case with ℏ = 1). Note also that as the
charge is a two-form integrated over a surface, it is a topological invariant and, so we of course
get the very same charge integrating Bplane in the area element dxdy of the plane over the entire
plane.
The free motion of charged mass point in three dimensions in the presence of a magnetic
monopole is a celebrated classical [9, 24] and quantum mechanical problem [1, 10, 15, 26] and
here, restricting the mass’s motion to an S2 with a central monopole breaks the symmetry
algebra to just so(3). The so(3) invariance of that problem indicates that geodesic motion in such
a case would of course have three independent constants of motion. In our problem the relaxed
constraint that the algebra only commute with the Hamiltonian on the single strata H = 0
allows us to evade the classification of superintegrable systems of this type in the 2-d plane [20].
Under stereographic projection circles map to circles, and the three constants of motion
can be thought of as two moduli for the center of the circle and one for the radius (on either
the plane or the sphere). The radius of the orbit on the sphere can be conveniently written
in terms of the half angle γ of the cone whose apex is the center of the sphere. For a unit
charge moving uniformly in that orbit, by elementary consideration of Newton’s equation we
find its angular velocity would be ω = |Q|
4mR4 cos γ
when γ ̸= π/2. Note that the period of the
orbit in the plane and the period of a free charge bound to a sphere (see above) are quite
different, indicating that the stereographic projection relates a geometric similarity between the
two problems but is not a dynamical equivalence. We do however note that the sign of Q/Lz
does indicate whether γ < π/2 or γ > π/2, as the sign at large Q/Lz can localize orbits outside
of r = R or inside that circle in the plane. Finally, unlike the case with a charge moving on
a plane with constant B field, due to the curvature (sphere)/potential(plane) there is no simple
relation between the angular momentum of the orbit and the magnitude of the magnetic flux it
encompasses.
Newton’s Off-Center Circular Orbits and the Magnetic Monopole 5
Figure 1. Basic montage of the problem; red orbit and orbital, along with the stereographic sphere (light
blue) and the projection (green line) onto the sphere (orbit image is blue). For reference, the equator of
the sphere is shown in black with its center at ‘∗’ and the center of the orbit at ‘◦’. Poles are marked
with ‘N’ and ‘S’.
4 Discussion
In a sense, the findings described here are not surprising: the addition of the magnetic field has
caused some of the orbits that were precessing to stop precessing (and the ones that were not
precessing to start precessing). The spatial form of the field is such that these closed orbits all
happen at zero energy. Note that the general solution for the orbital shape at E ̸= 0 can be
given in terms of elliptic functions; they degenerate to trigonometric functions at E = 0.
From the algebraic point of view, much of the forgoing has also been known since the early
study of monopoles by Dirac and Schwinger [6]. For example, it is well known that in the presence
of a monopole the mechanical angular momentum and geometrical operator that causes rotations
about an axis are not the same, but are related by an overall term proportional to Qr̂, as was
also used here [10, 24]. Likewise the idea that the magnetic field results in a central extension
of the dynamical algebra is strongly reminiscent of the case of a constant field in the plane, the
celebrated Landau problem.
It is noteworthy to also compare the linear stability of the orbits. Instead of using the
effective potential approach we ask what bound (not closed) orbits nearby are accessible with an
arbitrary small change in the energy E and the angular momentum Lz. For concreteness, the
general bound orbit has an innermost and outermost excursion from the origin; let a represent
the outermost distance and va the particle’s velocity there. We presume that there are other
orbits nearby in a, va, so for this family locally we have E = 1
2v
2
a + V (a) and Lz = ava where V
is the potential energy function (mass m = 1). The general excursion (δE, δLz) can be rendered
as an excursion in (δa, δva) as
(δE, δLz)
t = M(δa, δva)
t, M =
[
V ′ va
va a
]
, (4.1)
where V ′ = dV
dr |a, indicating that the obstruction to linear response in orbital parameters a, va
under a general δE, δLz is 0 = det(M) =
(
aV ′−v2a
)
= (aV ′+2V −2E). This relation holds for
arbitrary magnetic field. For the V (r) = − α
(r2+R2)2
used here, the determinant det(M) is zero
only at a = R. Since for E = 0 that is a boundary case (the centered circle: for which δa must
change sign) this shows that all the off-center circular orbits of the Newton problem for any Q
are linearly stable. Note that for the family of orbits that are circles with centers at the origin
6 D. Bhandari and M. Crescimanno
(so E ̸= 0), the same computation gives det(M) = ωQa2
(a2+R2)2
where ω is the orbital frequency,
thus in the original Newton problem (Q = 0) orbits centered at the origin are linearly unstable.
In a direct sense, the existence of the closed Poisson algebra, because of the constants of motion
and their convexity due to the Casimir element, forces the orbits (in this case, restricted again
to the H = 0 strata) to be stable, perhaps even beyond the linear perturbative regime.
The foregoing brings us back to our original motivation, to see in this off-center circular po-
tential a generalization of the fact that the Kepler problem/isotropic harmonic oscillator system
subject to linear frictional damping preserves two of three orbital constants of motion. Recall
that in the Kepler case, linear friction—a non-Hamiltonian perturbation—causes the orbits to
shrink in such a way that the orbital axis and eccentricity remain constant [12]. Apparently,
a similar observation was made long ago in a somewhat different context by S. Lie [14, 17] for
the Kepler problem.
In our problem, the perturbation (magnetic field) is Hamiltonian, but, as above, associated
with forces linear in the velocity. As in the Kepler case, changing Q will cause the constants
associated with the motion to evolve; we now show that the rigidity of the Poisson algebra
restricts the flow of the orbital parameters in a way strongly reminiscent of the “Kepler with
linear damping” case; the orbits preserve their shape and orientation but the orbit’s center moves.
We also observe from direct integrations of the equations of motion that under a gradual change
of Q, the centers of our circular orbits on the xy plane move in a straight line, corresponding to
a shrinking of the length of the orbital axis while not disturbing the other constants of motion,
just as in the Kepler case with linear damping (see Figure 2 (a)). Part of this phenomenology can
be understood from the global structure of the space of orbits. Note that in the Q = 0 case, since
the negative of the potential is less than 1/r2 at large distances, the usual stability arguments
indicate that there will be a range of angular momentum L ∈ [0, Lmax]
(
with Lmax =
√
α
2R2
)
that are associated with bound orbits. Note also that the effective potential in these cases has
a positive value at its outermost local maximum, indicating the spectrum of finite orbits is an
interval containing 0. Now including the effect of a magnetic field, the Q shifts the Lmax but the
global picture remains the same. For one sign of Q then an arbitrarily large value of Q can be
accommodated, whereas for the other sign of Q there is a maximum, beyond which the orbital
sense (i.e., sign of Lz) itself must flip to the other branch of the solutions (see Figure 2 (b)).
For l2 = α
2L2
z
−R2 + Q
2Lz
, where l is the location of the centers of the orbits on the xy plane,
imposing the l2 > 0 condition implies that |Q/2Lz| < α/2L2
z − R2, i.e., Q/Lz cannot be too
negative for a fixed value of Lz. This allows us to plot Figures 2 (a) and 2 (b) with physical
solutions of our system for both large and small values of Q and Lz which helps us further
elucidate the global features of the space of orbits. For large Lz, the Q → ∞ limit results in
a zero radius circle both on the xy plane and on the stereographic sphere. Moreover, the gradual
change in Q allows us to see that the space of orbits forms something akin to the double covering
of a 2-sphere because as we increase Q in the direction where arbitrarily large values cannot be
assigned, inverse projected trajectories on the sphere “meet” the oppositely oriented solution on
the other side of the North pole on the sphere, i.e., the Lz must change in sign (by having the
orbits cross the North pole) such that more solutions on the sphere are obtainable, as described
earlier.
Of historical note, the Newton off-center circular orbit potential with or without magnetic
charge as fastened above admits a hodograph [13, 18] that is in a sense the inverse of that of
the Kepler problem. The hodograph is a dynamically relevant version of the Gauss map of
a curve [8]. In the Gauss map, each point of a closed plane curve is mapped to the unit tangent
vector there, the collection of which sweep out a complete circle some number of times. In
the same sense, the hodograph of a closed orbit is the map from the points of the orbit to the
velocity vector there; for the case of Kepler although the orbit is an ellipse, the velocity vectors
sweep out a circle displaced from the origin (for a condensed yet easy-to-read rendition of the
Newton’s Off-Center Circular Orbits and the Magnetic Monopole 7
Figure 2. Illustrating the change in the orbits on the (a) xy plane (b) stereographic sphere caused by
a gradual change in the monopole charge Q (indicated here by colors). In (a), every cross section parallel
to the xy plane (a specific value of Q) yields a circular orbit on the xy plane. The orientation of the
sphere in (b) is the same as that of Figure 1.
Kepler hodograph, see [27]). As expected, the ‘axis’ of the off-center hodograph circle and the
major axis of the elliptical orbit are normal.
In the case Newton off-center circular orbit potential, the E = 0 orbits for any Q are circles
but their hodograph is an ellipse. One finds that the eccentricity of these ‘velocity vector ellipses’
is ϵ = 2Rl
R2+R2+l2
where l is the distance of the center of the orbit from the origin and R is the
radius of the orbit in the plane, and that their axis is again normal to the ‘axis’ of the orbit.
Finally, note that the correspondence in both cases of conic sections only occur for the position
and momenta; other higher order dynamical quantities of interest like acceleration, etc. do not
form a conic section. This is presumably due to the fact that the algebra of the constants of
motion we find in both Kepler and the off-center potential are linear in the momenta.
The existence of an everywhere attractive potential on the plane, leading to special clas-
sical zero energy orbits associated with a dynamical symmetry that survives the inclusion of
magnetic flux of a magnetic monopole on the stereographic sphere is evocative of the Atiyah–
Patodi–Singer index theorem in the quantum version of this problem [2, 3, 25]. In the inter-
ests of brevity of this note, our intention here is not to give a complete, detailed account of
the quantum mechanical problem associated with the off-center circular orbit potential with
a monopole field, but to just focus on a few phenomenological consequences of the dynamical
symmetry.
Consider what happens to the spectrum of the model as we go from the classical to quantum
version of this problem. In the Q = 0 classical case, the spectrum consists of two overlapping
continua, one that starts below zero and ends above zero (“bounded orbits”) and one that starts
at zero and is unbounded positive (“unbound orbits”). Note that some of the bounded orbits are
classically unstable. In the associated quantum problem, the spectrum always contains a discrete
spectrum of bound states for any α > 0, that terminates for generic values of α below E = 0,
followed by a continuum of unbound states that starts at E = 0 (Note this is easy to establish via
variational considerations, and a physically and historically important example of the existence
of bound states in two spatial dimensions for an everywhere attractive potential is [5]). The
continuity of the spectral flow in α indicates that at special values of α > 0 we should expect to
get E = 0 bound states.
For simplicity consider first the Q = 0 solution of the associated quantum problem (here we
take m = 1 for notational convenience). Let the eigenvalue of the Lz = I. We have[
− 1
2r
∂rr∂r +
I2
2r2
− α(
r2 +R2
)2]Ψ = EΨ, (4.2)
8 D. Bhandari and M. Crescimanno
which for E = 0 and only when α = 2R2I(I + 1) admits the solution
Ψ±(r) = A
rIe±iIϕ(
r2 +R2
)I (4.3)
from which we see that, on physical grounds, that I must be integer (hermiticity of Lz) and I > 0
(normalizability). Note the familiar appearance of the analytic z = reiϕ and the leading zI
dependence of this solution, all strongly reminiscent of the analytic structure of the wave function
in the integer quantum hall state [11, 16] (for a recent treatment, see [23]).
We anticipate the quantum version of the operators (2.5) will satisfy an operator algebra
where the Poisson bracket in (2.6) is replaced by the commutator, becoming a state-classifying
algebra in the H = 0 subspace of which (for Q = 0) (4.3) is a member. Viewed from the
stereographic sphere, there are no E = 0 sections except constant functions, which lead to
non-normalizable solutions on the plane (the conformal dimension of Ψ is 0, and in two di-
mensions the kinetic energy, here the Laplace–Beltrami operator, is conformal dimension −2).
The α = 2R2I(I + 1) condition on the zero energy section indicates that the conformal trans-
formation of the problem from the plane to the sphere is not ES2
= 0 but ES2
= const, as
can also be seen from the quadratic Casimir element (2.8), which, as a consequence of re-
quiring the commutator algebra to have a linear representation on a finite number of states,
we have C2 = I(I + 1), for some I ∈ Z. Rewriting Ψ± in the S2 coordinates reveals it is as
the Y ±I
I (θ, ϕ) (in weight space notation, |I,±I >) state of spherical harmonics.
The operator algebra is universal to the sphere, and up to a shift in Lz by a central element,
is unchanged by the inclusion of monopole magnetic field. As noted long ago by Dirac [6], in
the quantum problem requiring the wave functions to be single valued implies the quantization
of the product, M , of the magnetic and electric charges; in our case this implies (ℏ = 1)
M = Q
2R2 ∈ Z. The quadratic Casimir element then indicates that there is a model (a particular
value of α) having E = 0 states for each choice of I, M , though clearly not all choices of the
later are possible. Let m be the eigenvalue of the Lz in the Q ̸= 0 case; by hermiticity of Lz we
have m ∈ Z. Thus, the quantum version of (2.8) indicates that |m−M/2| is bounded by I.
It is now revealing to count the possible E = 0 normalizable quantum solutions consistent
with the operator algebra in this 2-d monopole background [4]. For example, at large I the
possible |M | are limited to be less than 2I (since we require α > 0). We know that for Q = 0
there are two zero modes on the plane if α is large enough; to simplify the discussion here we
limit ourselves to that case in which in the absence of Q there would be no zero modes, that is, we
specialize to the largest possible value, |M | = 2I (the smallest possible non-zero α) and take M
even and positive. By the forgoing constraint the possible m values now range from 0 to 2I.
Since the m = 2I mode is not normalizable in the plane that leaves exactly M zero modes in this
case, as expected by the application of the index theorem. Technically, the index theorem counts
the difference between zero modes of different chirality, but in two dimensions the chirality is
manifest in the complex analytic factor in the wavefunction. In typical applications (as here)
the zero modes are all of the same chirality; going to M < 0 doesn’t change the above counting
argument but flips the chirality of the zero modes.
5 Conclusion
Newton’s off-center circular orbit potential is ideal for a study at E = 0 that includes dynamical
symmetry, their consequence for perturbation linear in the velocity, hodography and the index
theorem of elliptic operators. In this brief note we’ve sought to display some aspects of these
connections with the dynamical symmetry and kinematic consequence of the associated situation
in the Kepler problem. The E = 0 problem has a conformal image as motion on a two sphere that
Newton’s Off-Center Circular Orbits and the Magnetic Monopole 9
containing a magnetic monopole at its center. The confluence of the rigidity of the dynamical
algebra and the topology of the monopole field leads to a simple counting of possible quantum
solutions with E = 0, the ‘zero modes’, that can be understood as a straightforward consequence
of the simplest version of the index theorem.
Acknowledgements
The authors acknowledge partial support via NSF grant DMR-2226956. We thank Don Priour
for conversations and assistance with the figures. We are thankful to a referee for bringing
reference [19] to our attention.
References
[1] Andrade e Silva R., Jacobson T., Particle on the sphere: group-theoretic quantization in the presence of
a magnetic monopole, J. Phys. A 54 (2021), 235303, 33 pages, arXiv:2011.04888.
[2] Atiyah M.F., Bott R., Patodi V.K., On the heat equation and the index theorem, Invent. Math. 19 (1973),
279–330.
[3] Atiyah M.F., Singer I.M., The index of elliptic operators on compact manifolds, Bull. Amer. Math. Soc. 69
(1963), 422–433.
[4] Bardakci K., Crescimanno M., Monopole backgrounds on the world sheet, Nuclear Phys. B 313 (1989),
269–292.
[5] Cooper L.N., Bound electron pairs in a degenerate Fermi gas, Phys. Rev. 104 (1956), 1189–1190.
[6] Dirac P., Quantised singularities in the electromagnetic field, Proc. Roy. Soc. Lond. A 133 (1931), 60–72.
[7] Faure R., Transformations conformes en mécanique ondulatoire, C. R. Acad. Sci. Paris 237 (1953), 603–605.
[8] Gauss C.F., General investigations of curved surfaces of 1827 and 1825, Nature 66 (1902), 316–317.
[9] Golo V., Dynamic SO(3, 1) symmetry of the Dirac magnetic monopol, JETP Lett. 35 (1982), 663–665.
[10] Grossman B., A 3-cocyle in quantum mechanics, Phys. Lett. B 152 (1985), 93–97.
[11] Haldane F.D.M., Rezayi E.H., Spin-singlet wave function for the half-integral quantum Hall effect, Phys.
Rev. Lett. 60 (1988), 1886–1886.
[12] Hamilton B., Crescimanno M., Linear frictional forces cause orbits to neither circularize nor precess,
J. Phys. A 41 (2008), 235205, 13 pages, arXiv:0708.3827.
[13] Hamilton W.R., The Hodograph or a new method of expressing in symbolic language the Newtonian law of
attraction, Proc. R. Ir. Acad. 3 (1847), 344–353.
[14] Ince E.L., Ordinary differential equations, Dover Publications, New York, 1956.
[15] Kemp G.M., Veselov A.P., On geometric quantization of the Dirac magnetic monopole, J. Nonlinear Math.
Phys. 21 (2014), 34–42, arXiv:1103.6242.
[16] Laughlin R.B., Quantized Hall conductivity in two dimensions, Phys. Rev. B. 23 (1981), 5632–5633.
[17] Lie S., Vorlesung über Differentialgleichungen mit bekannten infinitesimalen Transformationen, B.G. Teub-
ner Verlag, 1891.
[18] Maxwell J.C., Matter and motion, Cambridge Library Collect. Phys. Sci., Cambridge University Pres, 2010.
[19] McIntosh H.V., Cisneros A., Degeneracy in the presence of a magnetic monopole, J. Math. Phys. 11 (1970),
896–916.
[20] McSween E., Winternitz P., Integrable and superintegrable Hamiltonian systems in magnetic fields, J. Math.
Phys. 41 (2000), 2957–2967.
[21] Newton I., Newton’s Principia. The mathematical principles of natural philosophy, Cambridge Library Col-
lect. Phys. Sci., New-York, D. Adee, 1848.
[22] Olshanii M., A novel potential featuring off-center circular orbits, SIGMA 19 (2023), 001, 8 pages,
arXiv:2207.09606.
[23] Senthil T., Levin M., Integer quantum Hall effect for bosons, Phys. Rev. Lett. 110 (2013), 046801, 5 pages.
[24] Shnir Y.M., Magnetic monopoles, Texts Monogr. Phys., Springer, Berlin, 2005.
https://doi.org/10.1088/1751-8121/abf961
https://arxiv.org/abs/2011.04888
https://doi.org/10.1007/BF01425417
https://doi.org/10.1090/S0002-9904-1963-10957-X
https://doi.org/10.1016/0550-3213(89)90319-2
https://doi.org/10.1103/PhysRev.104.1189
https://doi.org/10.1098/rspa.1931.0130
https://doi.org/10.1038/066316b0
https://doi.org/10.1016/0370-2693(85)91146-3
https://doi.org/10.1103/PhysRevLett.60.1886
https://doi.org/10.1103/PhysRevLett.60.1886
https://doi.org/10.1088/1751-8113/41/23/235205
https://arxiv.org/abs/0708.3827
https://doi.org/10.1080/14029251.2014.894719
https://doi.org/10.1080/14029251.2014.894719
https://arxiv.org/abs/1103.6242
https://doi.org/10.1103/PhysRevB.23.5632
https://doi.org/10.14463/GBV:1013269691
https://doi.org/10.14463/GBV:1013269691
https://doi.org/10.1017/CBO9780511709326
https://doi.org/10.1063/1.1665227
https://doi.org/10.1063/1.533283
https://doi.org/10.1063/1.533283
https://doi.org/10.3842/SIGMA.2023.001
https://arxiv.org/abs/2207.09606
https://doi.org/10.1103/PhysRevLett.110.046801
https://doi.org/10.1007/3-540-29082-6
10 D. Bhandari and M. Crescimanno
[25] Singer I.M., Future extensions of index theory and elliptic operators, in Prospects in Mathematics, Ann. of
Math. Stud., Vol. 70, Princeton University Press, Princeton, NJ, 1971, 171–185.
[26] Song H., Jo S.G., Quantum mechanics on S1, S2 and Lorentz group, J. Korean Phys. Soc. 59 (2011),
3314–3320.
[27] Suzuki M.S., Suzuki I.S., Laplace–Runge–Lenz triangles in Feynman hodograph diagram: the Kepler’s
model and Sommerfeld’s model, Binghamton, New York, 2022.
https://doi.org/10.1515/9781400881697-006
1 Motivation
2 Algebra and dynamics
3 The magnetic monopole via the stereographic projection
4 Discussion
5 Conclusion
References
|
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| spelling | Bhandari, Dipesh Crescimanno, Michael 2026-01-23T10:09:02Z 2023 Newton's Off-Center Circular Orbits and the Magnetic Monopole. Dipesh Bhandari and Michael Crescimanno. SIGMA 19 (2023), 099, 10 pages 1815-0659 2020 Mathematics Subject Classification: 37J37; 19K56 arXiv:2307.15222 https://nasplib.isofts.kiev.ua/handle/123456789/212032 https://doi.org/10.3842/SIGMA.2023.099 Introducing a radially dependent magnetic field into Newton's off-center circular orbits potential to preserve the = 0 dynamical symmetry leads to a unique choice of field that can be identified as the inclusion of a magnetic monopole in the inverse stereographically projected problem. One also finds a phenomenological correspondence with that of the linearly damped Kepler model. The presence of the monopole field deforms the symmetry algebra by a central extension, and the quantum mechanical version of this algebra reveals several zero modes equal to those counted using the index theorem of elliptic operators. The authors acknowledge partial support via NSF grant DMR-2226956. We thank Don Priour for conversations and assistance with the figures. We are thankful to a referee for bringing reference [19] to our attention. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Newton's Off-Center Circular Orbits and the Magnetic Monopole Article published earlier |
| spellingShingle | Newton's Off-Center Circular Orbits and the Magnetic Monopole Bhandari, Dipesh Crescimanno, Michael |
| title | Newton's Off-Center Circular Orbits and the Magnetic Monopole |
| title_full | Newton's Off-Center Circular Orbits and the Magnetic Monopole |
| title_fullStr | Newton's Off-Center Circular Orbits and the Magnetic Monopole |
| title_full_unstemmed | Newton's Off-Center Circular Orbits and the Magnetic Monopole |
| title_short | Newton's Off-Center Circular Orbits and the Magnetic Monopole |
| title_sort | newton's off-center circular orbits and the magnetic monopole |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/212032 |
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