The purpose of this report is to give a brief overview of some results about the geometry of closed critical curves of a conformally invariant functional for space curve

Musso, Emilio

Journal of Physics : Conference Series

PORTO Publications Open Repository TOrino

Politecnico di Torino
Porto Institutional Repository
[Article] Closed Trajectories of the conformal arclength functional
Original Citation:
E. Musso (2013). Closed Trajectories of the conformal arclength functional. In: JOURNAL OF
PHYSICS. CONFERENCE SERIES, vol. 410 n. 012031, pp. 1-4. - ISSN 1742-6588
Availability:
This version is available at : http://porto.polito.it/2506109/ since: February 2013
Publisher:
JOP Publishing
Published version:
DOI:10.1088/1742-6596/410/1/012031
Terms of use:
This article is made available under terms and conditions applicable to Open Access Policy Article
("Public - All rights reserved") , as described at http://porto.polito.it/terms_and_conditions.
html
Porto, the institutional repository of the Politecnico di Torino, is provided by the University Library
and the IT-Services. The aim is to enable open access to all the world. Please share with us how
this access benefits you. Your story matters.
(Article begins on next page)
Closed trajectories of the conformal arclength functional
This article has been downloaded from IOPscience. Please scroll down to see the full text article.
2013 J. Phys.: Conf. Ser. 410 012031
(http://iopscience.iop.org/1742-6596/410/1/012031)
Download details:
IP Address: 2.34.64.155
The article was downloaded on 10/02/2013 at 09:20
Please note that terms and conditions apply.
View the table of contents for this issue, or go to the journal homepage for more
Home Search Collections Journals About Contact us My IOPscience
Closed trajectories of the conformal arclength
functional
Emilio Musso
Dipartimento di Scienze Matematiche, Politecnico di Torino, Corso Duca degli Abruzzi 24,
I-10129 Torino, Italy
E-mail: emilio.musso@polito.it
Abstract. The purpose of this report is to give a brief overview of some unpublished results
about the geometry of closed critical curves of a conformally invariant functional for space
curves.
1. Introduction
Conformal geometry of curves is a well studied subject in classical differential geometry
[4, 15, 16, 17]. In more recent times, the topic has been considered in connection with the
regularization of the Kepler problem [6, 9, 14], within the theory of integrable systems and
in the topology of knots [3, 1, 2, 5, 8, 10, 11]. In a previous paper, published several years
ago [12], I studied the variational problem defined by the conformal density of a space curve,
improperly called conformal arc-element. In that paper I computed the critical curves in terms
of elliptic integrals. This was not much surprising because the trajectories of the variational
problem can be obtained as projections of the integral curves of a collective completely integrable
contact Hamiltonian system, defined on a 13-dimensional momentum space. Recently, my work
was taken up and its results have been generalized to curves immersed in Euclidean spaces of
arbitrary dimension [13]. The problem that I will consider in this report is the existence and
the explicit determination of the closed trajectories of the variational problem. A more complete
and exhaustive analysis will be the subject of a forthcoming paper.
2. Conformal geometry of space curves and the period map
Consider an oriented, complete bi-regular curve γ : R → R3, parameterized by the Euclidean
arc-length. A vertex is a point γ(s) such that κ˙(s)2 + κ2(s)τ(s)2 = 0, where κ and τ are the
curvature and the torsion respectively. This notion is invariant by conformal transformations of
R3. The conformal arc element (or more appropriately the conformal density) of γ is the 1-form
ζ = 4
√
κ˙2 + κ2τ2ds. Assuming that the curve is generic (i.e., bi-regular and without vertices),
then the conformal density is nowhere vanishing and we can define the two conformal curvatures
k1 = r5
(
κ2τ3 + κκ˙τ˙ + τ(2κ˙2 − κκ¨)) ,
k2 =
1
2
(
r˙2 − 2rr¨ − r2κ2) . (1)
IC-MSQUARE 2012: International Conference on Mathematical Modelling in Physical Sciences IOP Publishing
Journal of Physics: Conference Series 410 (2013) 012031 doi:10.1088/1742-6596/410/1/012031
Published under licence by IOP Publishing Ltd 1
where r = (κ˙2 + κ2τ2)−1/4. The conformal density and the two curvatures determine the shape
of the curve, up to conformal transformations. This means that if two curves γ and γ˜ have the
same invariants, then there is a conformal transformation of R3 that sends one trajectory onto
the other. The conformal symmetry group of a curve γ is the group Gγ made up of all conformal
transformations which leave unchanged its trajectory. From now on we will assume that the
curves are bi-regular and without vertices.
Definiton. The critical curves of the functional defined by the integral of the conformal density
are called conformal geodesics. For simplicity, a closed conformal geodesic with non-constant
conformal curvatures is referred to as a conformal string.
The critical curves with periodic curvature functions are characterized by the Euler-Lagrange
equations [12]
dk1 + (k21 − a)(k21 − b)ζ = 0, k2 = −
3
2
k1 +
a+ b
a
, (2)
where a and b are two real constants, the natural parameters, such that a > 0, b 6= 0 and a > b.
An analysis of the momentum map shows that the natural parameters of a string belong to the
admissible region
Σ = {(a, b) : a > 1, a−1 < b < a}.
Using the conservation laws it can be checked (see [12]) that, up to conformal transformations,
the parametrization of a critical curve with natural parameters (a, b) ∈ Σ is given by
x(t) =
√
2
r(t)
µ
√
k(t)2 − υ2 cos(Θ2(t)),
y(t) =
√
2
r(t)
µ
√
k(t)2 − υ2 sin(Θ2(t)),
z(t) =
√
2
r(t)
υ
√
µ2 − k(t)2 sin(Θ1(t)),
(3)
where k(t) is the Jacobi elliptic function
√
a ·dn(√at, a−1(a− b)), µ and υ are the real constants
µ =
√
a+ b+
√
4 + (a− b)2
2
, υ =
√
a+ b−√4 + (a− b)2
2
, (4)
Θ1(t) and Θ2(t) are the incomplete elliptic integrals of the third kind
Θ1(t) =
∫ t
0
µ
µ2 − k(u)2du, Θ2(t) =
∫ t
0
υ
υ2 − k(u)2du.
and r(t) is the radial function
r(t) =
√
µ2 − υ2k(t) + υ
√
µ2 − k(t)2 cos(Θ1(t)).
The curve defined by the formula (3) is said to be the symmetrical configuration of the critical
curve with natural parameters (a, b). We denote by Π(m,n), −1 < n,m < 1, the complete
elliptic integral of the third kind
Π(m,n) =
∫ pi/2
0
dt
(1− n sin2(t))
√
1−m sin2(t)
,
IC-MSQUARE 2012: International Conference on Mathematical Modelling in Physical Sciences IOP Publishing
Journal of Physics: Conference Series 410 (2013) 012031 doi:10.1088/1742-6596/410/1/012031
2
and, for every (a, b) ∈ Σ we set
Φ1(a, b) = − µ(a, b)
pi
√
a(a− µ(a, b)2)Π
(
a− b
a− µ(a, b)2 ,
a− b
a
)
,
Φ2(a, b) =
υ(a, b)
pi
√
a(a− υ(a, b)2)Π
(
a− b
a− υ(a, b)2 ,
a− b
a
)
.
(5)
where µ(a, b) and υ(a, b) are defined as in (4). The real-analytic function
Φ = (a, b) ∈ Σ→ (Φ1(a, b),Φ2(a, b)) ∈ R2
is called the period map of the conformal arc-length functional. With a thorough analysis of the
behavior of the period map it can be proved that Φ is a real-analytic diffeomorphism of Σ onto
the circular domain
Ω = {(x, y) ∈ R2 : 1/2 < x < 1/
√
2, y > 0, x2 + y2 < 1/2}.
3. Closed trajectories and their phenomenological invariants
From the previous discussion it follows that a critical curve with natural parameters (a, b) ∈ Σ
is closed if and only if Φ(a, b) is a rational point of the domain Ω. This shows that, up to
conformal transformations, the conformal strings are in one to one correspondence with the
pairs (q1, q2) of rational numbers belonging to Ω. More precisely, for every rational point q of
the domain Ω there is a unique conformal string in its symmetrical configuration, with natural
parameters (a, b) ∈ Σ such that q = Φ(a, b). The rational points of Ω are called the modules
of the conformal strings. To clarify the geometric meaning of the modules we must say a few
words about the maximal tori of the conformal group. The conformal group of the Euclidean
space is isomorphic to the pseudo-orthogonal group SO(4, 1). Its maximal tori are conjugates
to the subgroup SO(2) × SO(2) ⊂ SO(4, 1). The infinitesimal generators for the action of
SO(2)× SO(2) on R3 are
ξ1 = −y∂x + x∂y, ξ2 = − xz√
2
∂x − yz√
2
∂y +
x2 + y2 − z2 − 2
2
√
2
∂z.
The first vector field generates the rotations around the Oz-axis, while the second one generates
the toroidal rotations around the Clifford circle, that is the circle of the Oxy-plane, centered at
the origin, with radius
√
2. The geometric meaning of the modules is clarified by the following
Theorem.
Theorem. Let γ : R → R3 be a conformal string in its symmetrical configuration. Denote by
q1 = m1/n1 and q2 = m2/n2 the modules of γ and assume that gcd(m1, n1) = gcd(m2, n2) = 1.
Let n be the least common multiple of n1 and n2. Then, the conformal symmetry group of γ is
the cyclic subgroup Gγ ⊂ SO(2)× SO(2) of order n generated by the composition of a rotation
R1(2piq1) of an angle 2piq1 around the Oz-axis and a toroidal rotation R2(2piq2) of an angle
2piq2 around the Clifford circle. Moreover, if we denote by h1 and h2 the coprime integers n/n1
and n/n2, then m1h1 is the linking number of γ with the Clifford circle and m2h2 is the linking
number of γ with the Oz-axis.
This result implies that, up to the action of the conformal group, the conformal strings are are
uniquely determined by three numerical invariants: the order of the conformal symmetry group
and the linking numbers with the Oz-axis and the Clifford circle.
The inverse of the period map can be calculated by numerical methods. Therefore, once that
we know the three phenomenological invariants (order of symmetry and linking numbers), the
IC-MSQUARE 2012: International Conference on Mathematical Modelling in Physical Sciences IOP Publishing
Journal of Physics: Conference Series 410 (2013) 012031 doi:10.1088/1742-6596/410/1/012031
3
symmetrical configuration of the string can be determined explicitly. The numerical experiments
suggest that the strings are simple curves. However, we do not have a mathematically rigorous
proof of this fact. Another interesting problem is to find an asymptotic estimate of the cardinality
ρ(n) of the set of the equivalence classes of conformal strings with order of symmetry n. Our
expectation is that the growth of ρ(n) is quadratic.
4. Prospectives
It is likely that the techniques and the results obtained so far can be extended to time-like curves
in the conformal completion of the Minkowski space (and more generally, in any conformally
flat Lorenzian 4-manifold). In this framework, it is important to analyze the physical meaning
of the scattering data of the critical curves with non-periodic conformal curvatures.
References
[1] S. Bryson, M. H. Freedman, Z-X. He, Z. Wang, Mœbius invariance of knot energy Bull. of Am.Math.Soc.,
28 (1993), pp 99–103.
[2] K. S. Chou, C. Qu, Integrable equations arising from motions of plane curves. II, J. Nonlinear Sci. 13 (2003),
487–517.
[3] M. Eastwood, G. Mari-Beffa, Geometric Poisson brackets on Grassmannians and Conformal Spheres, to
appear in Proceedings Royal Society of Edimburg, (2012), arXiv:1006.5753, 30 June 2010.
[4] A. Fialkow, The conformal theory of curves, Trans. Amer.Math.Soc., 51 (1942), 435–501.
[5] M. H. Freedman, Z-X. He, Z. Wang, Mœbius energy of knots and unknots Annals of Mathematics, 139
(1994), 1–50.
[6] V. Guillemin, S. Sternberg, Variations on a Theme by Kepler, AMS Colloquium Publications 285, Providence
R. I. (1990).
[7] H. Liebmann, Beitra¨ge zur inversionsgeometrie der Kurven, Mœnchener Berichte (1923).
[8] R. Langevin, J. O’Hara, Conformally invariant energies of knots, J. Inst. Math. Jussieu, 4 (2005), 219-280.
[9] C-M. Marle, A property of conformally invariant Hamiltonian vector fields; application to the Kepler problem,
Journal of Geometric Mechanics, 4 (2012), 181–206.
[10] G. Mari-Beffa, Relative and Absolute Differential Invariants for Conformal Curves, Journal of Lie Theory,
12 (2003), 213–245.
[11] G. Marih-Beffa, Poisson brackets associated to the Conformal geometry of curves, Transaction of the AMS,
357 (2005), 2799–2827.
[12] E. Musso, The conformal arclength functional, Math. Nachr., 165 (1994), 107–131.
[13] M. Magliaro, L. Mari, M. Rigoli, On the geometry of curves and conformal geodesics in the Moebius space,
Ann. Global Anal. Geom., 40 (2011), 133–165.
[14] J.-M. Soriau, Sur la variete´ de Kepler, Sympos. Math. 14 (1974), 343–360.
[15] R. Sulanke, Submanifolds of the Mœbius space II, Frenet Formulas and Curves of Constant Curvatures,
Math. Nachr., 100 (1981), 235–257.
[16] T. Takasu, Differentialgeometrie in den Kugelra¨ume, Vol. 1, Konforme Differentialgeometrie von Liouville
and Mœbius, Tokyo (1938).
[17] E. Vessiot, Enveloppes de sphe´res et courbes gauches, J, Ecole Polytechnique, 25 (1925), 43–91.
IC-MSQUARE 2012: International Conference on Mathematical Modelling in Physical Sciences IOP Publishing
Journal of Physics: Conference Series 410 (2013) 012031 doi:10.1088/1742-6596/410/1/012031
4


Closed Trajectories of the conformal arclength functional

Emilio Musso

Crossref

Journal of Physics Conference Series

Closed trajectories of the conformal arclength functional

MUSSO, EMILIO

PORTO@iris (Publications Open Repository TOrino - Politecnico di Torino)

04 August 2020POLITECNICO DI TORINORepository ISTITUZIONALEClosed Trajectories of the conformal arclength functional / E. Musso. - In: JOURNAL OF PHYSICS. CONFERENCESERIES. - ISSN 1742-6588. - STAMPA. - 410:012031(2013), pp. 1-4.OriginalClosed Trajectories of the conformal arclength functionalPublisher:PublishedDOI:10.1088/1742-6596/410/1/012031Terms of use:openAccessPublisher copyright(Article begins on next page)This article is made available under terms and conditions as specified in the  corresponding bibliographic description inthe repositoryAvailability:This version is available at: 11583/2506109 since:JOP PublishingClosed trajectories of the conformal arclength functionalThis article has been downloaded from IOPscience. Please scroll down to see the full text article.2013 J. Phys.: Conf. Ser. 410 012031(http://iopscience.iop.org/1742-6596/410/1/012031)Download details:IP Address: 2.34.64.155The article was downloaded on 10/02/2013 at 09:20Please note that terms and conditions apply.View the table of contents for this issue, or go to the journal homepage for moreHome Search Collections Journals About Contact us My IOPscienceClosed trajectories of the conformal arclengthfunctionalEmilio MussoDipartimento di Scienze Matematiche, Politecnico di Torino, Corso Duca degli Abruzzi 24,I-10129 Torino, ItalyE-mail: emilio.musso@polito.itAbstract. The purpose of this report is to give a brief overview of some unpublished resultsabout the geometry of closed critical curves of a conformally invariant functional for spacecurves.1. IntroductionConformal geometry of curves is a well studied subject in classical differential geometry[4, 15, 16, 17]. In more recent times, the topic has been considered in connection with theregularization of the Kepler problem [6, 9, 14], within the theory of integrable systems andin the topology of knots [3, 1, 2, 5, 8, 10, 11]. In a previous paper, published several yearsago [12], I studied the variational problem defined by the conformal density of a space curve,improperly called conformal arc-element. In that paper I computed the critical curves in termsof elliptic integrals. This was not much surprising because the trajectories of the variationalproblem can be obtained as projections of the integral curves of a collective completely integrablecontact Hamiltonian system, defined on a 13-dimensional momentum space. Recently, my workwas taken up and its results have been generalized to curves immersed in Euclidean spaces ofarbitrary dimension [13]. The problem that I will consider in this report is the existence andthe explicit determination of the closed trajectories of the variational problem. A more completeand exhaustive analysis will be the subject of a forthcoming paper.2. Conformal geometry of space curves and the period mapConsider an oriented, complete bi-regular curve γ : R → R3, parameterized by the Euclideanarc-length. A vertex is a point γ(s) such that κ˙(s)2 + κ2(s)τ(s)2 = 0, where κ and τ are thecurvature and the torsion respectively. This notion is invariant by conformal transformations ofR3. The conformal arc element (or more appropriately the conformal density) of γ is the 1-formζ = 4√κ˙2 + κ2τ2ds. Assuming that the curve is generic (i.e., bi-regular and without vertices),then the conformal density is nowhere vanishing and we can define the two conformal curvaturesk1 = r5(κ2τ3 + κκ˙τ˙ + τ(2κ˙2 − κκ¨)) ,k2 =12(r˙2 − 2rr¨ − r2κ2) . (1)IC-MSQUARE 2012: International Conference on Mathematical Modelling in Physical Sciences IOP PublishingJournal of Physics: Conference Series 410 (2013) 012031 doi:10.1088/1742-6596/410/1/012031Published under licence by IOP Publishing Ltd 1where r = (κ˙2 + κ2τ2)−1/4. The conformal density and the two curvatures determine the shapeof the curve, up to conformal transformations. This means that if two curves γ and γ˜ have thesame invariants, then there is a conformal transformation of R3 that sends one trajectory ontothe other. The conformal symmetry group of a curve γ is the group Gγ made up of all conformaltransformations which leave unchanged its trajectory. From now on we will assume that thecurves are bi-regular and without vertices.Definiton. The critical curves of the functional defined by the integral of the conformal densityare called conformal geodesics. For simplicity, a closed conformal geodesic with non-constantconformal curvatures is referred to as a conformal string.The critical curves with periodic curvature functions are characterized by the Euler-Lagrangeequations [12]dk1 + (k21 − a)(k21 − b)ζ = 0, k2 = −32k1 +a+ ba, (2)where a and b are two real constants, the natural parameters, such that a > 0, b 6= 0 and a > b.An analysis of the momentum map shows that the natural parameters of a string belong to theadmissible regionΣ = {(a, b) : a > 1, a−1 < b < a}.Using the conservation laws it can be checked (see [12]) that, up to conformal transformations,the parametrization of a critical curve with natural parameters (a, b) ∈ Σ is given byx(t) =√2r(t)µ√k(t)2 − υ2 cos(Θ2(t)),y(t) =√2r(t)µ√k(t)2 − υ2 sin(Θ2(t)),z(t) =√2r(t)υ√µ2 − k(t)2 sin(Θ1(t)),(3)where k(t) is the Jacobi elliptic function√a ·dn(√at, a−1(a− b)), µ and υ are the real constantsµ =√a+ b+√4 + (a− b)22, υ =√a+ b−√4 + (a− b)22, (4)Θ1(t) and Θ2(t) are the incomplete elliptic integrals of the third kindΘ1(t) =∫ t0µµ2 − k(u)2du, Θ2(t) =∫ t0υυ2 − k(u)2du.and r(t) is the radial functionr(t) =√µ2 − υ2k(t) + υ√µ2 − k(t)2 cos(Θ1(t)).The curve defined by the formula (3) is said to be the symmetrical configuration of the criticalcurve with natural parameters (a, b). We denote by Π(m,n), −1 < n,m < 1, the completeelliptic integral of the third kindΠ(m,n) =∫ pi/20dt(1− n sin2(t))√1−m sin2(t),IC-MSQUARE 2012: International Conference on Mathematical Modelling in Physical Sciences IOP PublishingJournal of Physics: Conference Series 410 (2013) 012031 doi:10.1088/1742-6596/410/1/0120312and, for every (a, b) ∈ Σ we setΦ1(a, b) = − µ(a, b)pi√a(a− µ(a, b)2)Π(a− ba− µ(a, b)2 ,a− ba),Φ2(a, b) =υ(a, b)pi√a(a− υ(a, b)2)Π(a− ba− υ(a, b)2 ,a− ba).(5)where µ(a, b) and υ(a, b) are defined as in (4). The real-analytic functionΦ = (a, b) ∈ Σ→ (Φ1(a, b),Φ2(a, b)) ∈ R2is called the period map of the conformal arc-length functional. With a thorough analysis of thebehavior of the period map it can be proved that Φ is a real-analytic diffeomorphism of Σ ontothe circular domainΩ = {(x, y) ∈ R2 : 1/2 < x < 1/√2, y > 0, x2 + y2 < 1/2}.3. Closed trajectories and their phenomenological invariantsFrom the previous discussion it follows that a critical curve with natural parameters (a, b) ∈ Σis closed if and only if Φ(a, b) is a rational point of the domain Ω. This shows that, up toconformal transformations, the conformal strings are in one to one correspondence with thepairs (q1, q2) of rational numbers belonging to Ω. More precisely, for every rational point q ofthe domain Ω there is a unique conformal string in its symmetrical configuration, with naturalparameters (a, b) ∈ Σ such that q = Φ(a, b). The rational points of Ω are called the modulesof the conformal strings. To clarify the geometric meaning of the modules we must say a fewwords about the maximal tori of the conformal group. The conformal group of the Euclideanspace is isomorphic to the pseudo-orthogonal group SO(4, 1). Its maximal tori are conjugatesto the subgroup SO(2) × SO(2) ⊂ SO(4, 1). The infinitesimal generators for the action ofSO(2)× SO(2) on R3 areξ1 = −y∂x + x∂y, ξ2 = − xz√2∂x − yz√2∂y +x2 + y2 − z2 − 22√2∂z.The first vector field generates the rotations around the Oz-axis, while the second one generatesthe toroidal rotations around the Clifford circle, that is the circle of the Oxy-plane, centered atthe origin, with radius√2. The geometric meaning of the modules is clarified by the followingTheorem.Theorem. Let γ : R → R3 be a conformal string in its symmetrical configuration. Denote byq1 = m1/n1 and q2 = m2/n2 the modules of γ and assume that gcd(m1, n1) = gcd(m2, n2) = 1.Let n be the least common multiple of n1 and n2. Then, the conformal symmetry group of γ isthe cyclic subgroup Gγ ⊂ SO(2)× SO(2) of order n generated by the composition of a rotationR1(2piq1) of an angle 2piq1 around the Oz-axis and a toroidal rotation R2(2piq2) of an angle2piq2 around the Clifford circle. Moreover, if we denote by h1 and h2 the coprime integers n/n1and n/n2, then m1h1 is the linking number of γ with the Clifford circle and m2h2 is the linkingnumber of γ with the Oz-axis.This result implies that, up to the action of the conformal group, the conformal strings are areuniquely determined by three numerical invariants: the order of the conformal symmetry groupand the linking numbers with the Oz-axis and the Clifford circle.The inverse of the period map can be calculated by numerical methods. Therefore, once thatwe know the three phenomenological invariants (order of symmetry and linking numbers), theIC-MSQUARE 2012: International Conference on Mathematical Modelling in Physical Sciences IOP PublishingJournal of Physics: Conference Series 410 (2013) 012031 doi:10.1088/1742-6596/410/1/0120313symmetrical configuration of the string can be determined explicitly. The numerical experimentssuggest that the strings are simple curves. However, we do not have a mathematically rigorousproof of this fact. Another interesting problem is to find an asymptotic estimate of the cardinalityρ(n) of the set of the equivalence classes of conformal strings with order of symmetry n. Ourexpectation is that the growth of ρ(n) is quadratic.4. ProspectivesIt is likely that the techniques and the results obtained so far can be extended to time-like curvesin the conformal completion of the Minkowski space (and more generally, in any conformallyflat Lorenzian 4-manifold). In this framework, it is important to analyze the physical meaningof the scattering data of the critical curves with non-periodic conformal curvatures.References[1] S. Bryson, M. H. Freedman, Z-X. He, Z. Wang, Mœbius invariance of knot energy Bull. of Am.Math.Soc.,28 (1993), pp 99–103.[2] K. S. Chou, C. Qu, Integrable equations arising from motions of plane curves. II, J. Nonlinear Sci. 13 (2003),487–517.[3] M. Eastwood, G. Mari-Beffa, Geometric Poisson brackets on Grassmannians and Conformal Spheres, toappear in Proceedings Royal Society of Edimburg, (2012), arXiv:1006.5753, 30 June 2010.[4] A. Fialkow, The conformal theory of curves, Trans. Amer.Math.Soc., 51 (1942), 435–501.[5] M. H. Freedman, Z-X. He, Z. Wang, Mœbius energy of knots and unknots Annals of Mathematics, 139(1994), 1–50.[6] V. Guillemin, S. Sternberg, Variations on a Theme by Kepler, AMS Colloquium Publications 285, ProvidenceR. I. (1990).[7] H. Liebmann, Beitra¨ge zur inversionsgeometrie der Kurven, Mœnchener Berichte (1923).[8] R. Langevin, J. O’Hara, Conformally invariant energies of knots, J. Inst. Math. Jussieu, 4 (2005), 219-280.[9] C-M. Marle, A property of conformally invariant Hamiltonian vector fields; application to the Kepler problem,Journal of Geometric Mechanics, 4 (2012), 181–206.[10] G. Mari-Beffa, Relative and Absolute Differential Invariants for Conformal Curves, Journal of Lie Theory,12 (2003), 213–245.[11] G. Marih-Beffa, Poisson brackets associated to the Conformal geometry of curves, Transaction of the AMS,357 (2005), 2799–2827.[12] E. Musso, The conformal arclength functional, Math. Nachr., 165 (1994), 107–131.[13] M. Magliaro, L. Mari, M. Rigoli, On the geometry of curves and conformal geodesics in the Moebius space,Ann. Global Anal. Geom., 40 (2011), 133–165.[14] J.-M. Soriau, Sur la variete´ de Kepler, Sympos. Math. 14 (1974), 343–360.[15] R. Sulanke, Submanifolds of the Mœbius space II, Frenet Formulas and Curves of Constant Curvatures,Math. Nachr., 100 (1981), 235–257.[16] T. Takasu, Differentialgeometrie in den Kugelra¨ume, Vol. 1, Konforme Differentialgeometrie von Liouvilleand Mœbius, Tokyo (1938).[17] E. Vessiot, Enveloppes de sphe´res et courbes gauches, J, Ecole Polytechnique, 25 (1925), 43–91.IC-MSQUARE 2012: International Conference on Mathematical Modelling in Physical Sciences IOP PublishingJournal of Physics: Conference Series 410 (2013) 012031 doi:10.1088/1742-6596/410/1/0120314

English

https://iris.polito.it/retrieve/handle/11583/2506109/59271/Closed%20Trajectories%20Conformal%20-%20Published%201742-6596_410_1_012031.pdf

Closed Trajectories of the conformal arclength functional

Abstract

Similar works

Full text

Available Versions

PORTO Publications Open Repository TOrino

Crossref

PORTO@iris (Publications Open Repository TOrino - Politecnico di Torino)