2014, 6(2): 261-278. doi: 10.3934/jgm.2014.6.261

Periodic orbits in the Kepler-Heisenberg problem

1. 

Department of Mathematics and Statistics, California State University, Sacramento, 6000 J St., Sacramento, CA, United States

Received  November 2013 Revised  March 2014 Published  June 2014

One can formulate the classical Kepler problem on the Heisenberg group, the simplest sub-Riemannian manifold. We take the sub-Riemannian Hamiltonian as our kinetic energy, and our potential is the fundamental solution to the Heisenberg sub-Laplacian. The resulting dynamical system is known to contain a fundamental integrable subsystem. Here we use variational methods to prove that the Kepler-Heisenberg system admits periodic orbits with $k$-fold rotational symmetry for any odd integer $k\geq 3$. Approximations are shown for $k=3$.
Citation: Corey Shanbrom. Periodic orbits in the Kepler-Heisenberg problem. Journal of Geometric Mechanics, 2014, 6 (2) : 261-278. doi: 10.3934/jgm.2014.6.261
References:
[1]

R. Abraham and J. Marsden, Foundations of Mechanics,, Benjamin-Cummings, (1978).

[2]

A. Albouy, Projective dynamics and classical gravitation,, Regul. Chaotic Dyn., 13 (2008), 525. doi: 10.1134/S156035470806004X.

[3]

V. I. Arnold, Mathematical Methods of Classical Mechanics,, Springer-Verlag, (1978).

[4]

D. Barilari, U. Boscain and R. Neel, Small time heat kernel asymptotics at the sub-Riemannian cut locus,, Journal of Differential Geometry, 92 (2012), 373.

[5]

G. Bliss, The problem of Lagrange in the calculus of variations,, American J. Math., 52 (1930), 673. doi: 10.2307/2370714.

[6]

O. Bolza, Calculus of Variations,, $2^{nd}$ edition, (1960).

[7]

R. W. Brockett, Control theory and singular Riemannian geometry,, in New Directions in Appl. Math., (1982), 11.

[8]

A. Chenciner and R. Montgomery, A remarkable periodic solution of the three body problem in the case of equal masses,, Annals of Math., 152 (2000), 881. doi: 10.2307/2661357.

[9]

F. Diacu, E. Perez-Chavela and M. Santoprete, The n-body problem in spaces of constant curvature. Part I: Relative equilibria,, Journal of Nonlinear Science, 22 (2012), 247. doi: 10.1007/s00332-011-9116-z.

[10]

G. Folland, A fundamental solution for a subelliptic operator,, Bulletin of the AMS, 79 (1973), 373. doi: 10.1090/S0002-9904-1973-13171-4.

[11]

I. M. Gelfand and S. V. Fomin, Calculus of Variations,, Dover, (1963).

[12]

W. B. Gordon, A minimizing property of Keplerian orbits,, American J. Math., 99 (1977), 961. doi: 10.2307/2373993.

[13]

P. Griffiths, Exterior Differential Systems and the Calculus of Variations,, Birkhäuser, (1983).

[14]

N. I. Lobachevsky, The new foundations of geometry with full theory of parallels,, in Collected Works, II (1949), 1835.

[15]

R. Montgomery, A Tour of Subriemannian Geometries,, AMS, (2002).

[16]

R. Montgomery and C. Shanbrom, Keplerian dynamics on the Heisenberg group and elsewhere,, to appear in Geometry, ().

[17]

R. Palais, The principle of symmetric criticality,, Commun. Math. Phys, 69 (1979), 19. doi: 10.1007/BF01941322.

[18]

L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes,, Wiley, (1962).

[19]

C. Shanbrom, Two Problems in Sub-Riemannian Geometry,, Ph.D thesis, (2013).

[20]

L. C. Young, Lectures on the Calculus of Variations and Optimal Control Theory,, $2^{nd}$ edition, (1980).

show all references

References:
[1]

R. Abraham and J. Marsden, Foundations of Mechanics,, Benjamin-Cummings, (1978).

[2]

A. Albouy, Projective dynamics and classical gravitation,, Regul. Chaotic Dyn., 13 (2008), 525. doi: 10.1134/S156035470806004X.

[3]

V. I. Arnold, Mathematical Methods of Classical Mechanics,, Springer-Verlag, (1978).

[4]

D. Barilari, U. Boscain and R. Neel, Small time heat kernel asymptotics at the sub-Riemannian cut locus,, Journal of Differential Geometry, 92 (2012), 373.

[5]

G. Bliss, The problem of Lagrange in the calculus of variations,, American J. Math., 52 (1930), 673. doi: 10.2307/2370714.

[6]

O. Bolza, Calculus of Variations,, $2^{nd}$ edition, (1960).

[7]

R. W. Brockett, Control theory and singular Riemannian geometry,, in New Directions in Appl. Math., (1982), 11.

[8]

A. Chenciner and R. Montgomery, A remarkable periodic solution of the three body problem in the case of equal masses,, Annals of Math., 152 (2000), 881. doi: 10.2307/2661357.

[9]

F. Diacu, E. Perez-Chavela and M. Santoprete, The n-body problem in spaces of constant curvature. Part I: Relative equilibria,, Journal of Nonlinear Science, 22 (2012), 247. doi: 10.1007/s00332-011-9116-z.

[10]

G. Folland, A fundamental solution for a subelliptic operator,, Bulletin of the AMS, 79 (1973), 373. doi: 10.1090/S0002-9904-1973-13171-4.

[11]

I. M. Gelfand and S. V. Fomin, Calculus of Variations,, Dover, (1963).

[12]

W. B. Gordon, A minimizing property of Keplerian orbits,, American J. Math., 99 (1977), 961. doi: 10.2307/2373993.

[13]

P. Griffiths, Exterior Differential Systems and the Calculus of Variations,, Birkhäuser, (1983).

[14]

N. I. Lobachevsky, The new foundations of geometry with full theory of parallels,, in Collected Works, II (1949), 1835.

[15]

R. Montgomery, A Tour of Subriemannian Geometries,, AMS, (2002).

[16]

R. Montgomery and C. Shanbrom, Keplerian dynamics on the Heisenberg group and elsewhere,, to appear in Geometry, ().

[17]

R. Palais, The principle of symmetric criticality,, Commun. Math. Phys, 69 (1979), 19. doi: 10.1007/BF01941322.

[18]

L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes,, Wiley, (1962).

[19]

C. Shanbrom, Two Problems in Sub-Riemannian Geometry,, Ph.D thesis, (2013).

[20]

L. C. Young, Lectures on the Calculus of Variations and Optimal Control Theory,, $2^{nd}$ edition, (1980).

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