
SIGMA 6 (2010), 020, 17 pages arXiv:1002.4156
http://dx.doi.org/10.3842/SIGMA.2010.020
Geodesic Reduction via Frame Bundle Geometry
Ajit Bhand
Department of Mathematics, University of Oklahoma, Norman, OK, USA
Received October 12, 2009, in final form February 18, 2010; Published online February 22, 2010
Abstract
A manifold with an arbitrary affine connection is considered and the geodesic spray associated with the connection is studied in the presence of a Lie group action. In particular, results are obtained that provide insight into the structure of the reduced dynamics associated with the given invariant affine connection. The geometry of the frame bundle of the given manifold is used to provide an intrinsic description of the geodesic spray. A fundamental relationship between the geodesic spray, the tangent lift and the vertical lift of the symmetric product is obtained, which provides a key to understanding reduction in this formulation.
Key words:
affine connection; geodesic spray; reduction; linear frame bundle.
pdf (320 kb)
ps (196 kb)
tex (19 kb)
References
 Abraham R., Marsden J.E.,
Foundations of mechanics, 2nd ed.,
Benjamin/Cummings Publishing Co., Inc., Advanced Book Program, Reading, Mass., 1978.
 Bates L.,
Problems and progress in nonholonomic reduction,
Rep. Math. Phys. 49 (2002), 143149.
 Bhand A.,
Geodesic reduction via frame bundle geometry,
PhD Thesis, Queen's University, Kingston, ON, Canada, 2007, available at https://qspace.library.queensu.ca/handle/1974/456.
 Bloch A.M., Krishnaprasad P.S., Marsden J.E., Murray R.M.,
Nonholonomic mechanical systems with symmetry,
Arch. Rational Mech. Anal. 136 (1996), 2199.
 Bullo F., Lewis A.D.,
Geometric control of mechanical systems. Modeling, analysis, and design for simple mechanical control systems,
Texts in Applied Mathematics, Vol. 49,
SpringerVerlag, New York, 2005.
 Bullo F., Lewis A.D.,
Reduction, linearization, and stability of relative equilibria for mechanical systems on Riemannian manifolds,
Acta Appl. Math. 99 (2007), 5395.
 Cantrijn F., de León M., Marrero J.C., Martín de Diego D.,
Reduction of nonholonomic mechanical systems with symmetries,
Rep. Math. Phys. 42 (1998), 2545.
 Cantrijn F., Langerock B.,
Generalised connections over a vector bundle map,
Differential Geom. Appl. 18 (2003), 295317,
math.DG/0201274.
 Cendra H., Marsden J.E., Ratiu T.S.,
Geometric mechanics, Lagrangian reduction, and nonholonomic systems,
in Mathematics Unlimited2001 and Beyond, Springer, Berlin, 2001, 221273.
 Cendra H., Marsden J.E., Ratiu T.S.,
Lagrangian reduction by stages,
Mem. Amer. Math. Soc. 152 (2001), no. 722, 108 pages.
 Cortés J., Martínez E.,
Mechanical control systems on Lie algebroids,
IMA J. Math. Control Inform. 21 (2004), 457492.
 Cortés Monforte J.,
Geometric, control and numerical aspects of nonholonomic systems,
Lecture Notes in Mathematics, Vol. 1793, SpringerVerlag, Berlin, 2002.
 Crampin M., Mestdag T.,
Reduction and reconstruction aspects of secondorder dynamical systems with symmetry,
Acta Appl. Math. 105 (2009), 241266,
arXiv:0807.0156.
 Ehlers K., Koiller J., Montgomery R., Rios P.M.,
Nonholonomic systems via moving frames: Cartan equivalence and Chaplygin Hamiltonization,
in The Breadth of Symplectic and Poisson Geometry,
Progr. Math., Vol. 232, Birkhäuser Boston, Boston, MA, 2005, 75120,
mathph/0408005.
 Kobayashi S., Nomizu K.,
Foundations of differential geometry,
Tracts in Pure and Applied Mathematics, Vol. 1, Interscience Publishers, New York  London, 1964.
 Koiller J.,
Reduction of some classical nonholonomic systems with symmetry,
Arch. Rational Mech. Anal. 118 (1992), 113148.
 Lewis A.D.,
Affine connections and distributions with applications to nonholonomic mechanics,
Rep. Math. Phys. 42 (1998), 135164.
 Lewis A.D., Murray R.M.,
Decompositions for control systems on manifolds with an affine connection,
Systems Control Lett. 31 (1997), 199205.
 Marsden J.E., Ratiu T.S.,
Introduction to mechanics and symmetry. A basic exposition of classical mechanical systems, 2nd ed.,
Texts in Applied Mathematics, Vol. 17,
SpringerVerlag, New York, 1999.
 Marsden J.E., Weinstein A.,
Reduction of symplectic manifolds with symmetry,
Rep. Math. Phys. 5 (1974), 121130.
 Smale S.,
Topology and mechanics,
Invent. Math. 10 (1970), 305331.
 Sniatycki J.,
Nonholonomic Noether theorem and reduction of symmetries,
Rep. Math. Phys. 42 (1998), 523.

