
Previous Article
Threedimensional discrete systems of HirotaKimura type and deformed LiePoisson algebras
 JGM Home
 This Issue
 Next Article
Dirac cotangent bundle reduction
1.  Applied Mechanics and Aerospace Engineering, Waseda University, Okubo, Shinjuku, Tokyo 1698555, Japan 
2.  Control and Dynamical Systems 10781, California Institute of Technology, Pasadena, CA 91125, United States 
First of all, we establish a reduction theory starting with the HamiltonPontryagin variational principle, which enables one to formulate an implicit analogue of the LagrangePoincaré equations. To do this, we assume that a Lie group acts freely and properly on a configuration manifold, in which case there is an associated principal bundle and we choose a principal connection. Then, we develop a reduction theory for the canonical Dirac structure on the cotangent bundle to induce a gauged Dirac structure . Second, it is shown that by making use of the gauged Dirac structure, one obtains a reduction procedure for standard implicit Lagrangian systems, which is called LagrangePoincaréDirac reduction . This procedure naturally induces the horizontal and vertical implicit LagrangePoincaré equations , which are consistent with those derived from the reduced HamiltonPontryagin principle. Further, we develop the case in which a Hamiltonian is given (perhaps, but not necessarily, coming from a regular Lagrangian); namely, HamiltonPoincaréDirac reduction for the horizontal and vertical HamiltonPoincaré equations . We illustrate the reduction procedures by an example of a satellite with a rotor.
The present work is done in a way that is consistent with, and may be viewed as a specialization of the larger context of Dirac reduction, which allows for Dirac reduction by stages . This is explored in a paper in preparation by Cendra, Marsden, Ratiu and Yoshimura.
[1] 
Marco Castrillón López, Pablo M. Chacón, Pedro L. García. LagrangePoincaré reduction in affine principal bundles. Journal of Geometric Mechanics, 2013, 5 (4) : 399414. doi: 10.3934/jgm.2013.5.399 
[2] 
Anthony Bloch, Leonardo Colombo, Fernando Jiménez. The variational discretization of the constrained higherorder LagrangePoincaré equations. Discrete & Continuous Dynamical Systems  A, 2019, 39 (1) : 309344. doi: 10.3934/dcds.2019013 
[3] 
Henry Jacobs, Joris Vankerschaver. Fluidstructure interaction in the LagrangePoincaré formalism: The NavierStokes and inviscid regimes. Journal of Geometric Mechanics, 2014, 6 (1) : 3966. doi: 10.3934/jgm.2014.6.39 
[4] 
Hernán Cendra, Viviana A. Díaz. Lagranged'alembertpoincaré equations by several stages. Journal of Geometric Mechanics, 2018, 10 (1) : 141. doi: 10.3934/jgm.2018001 
[5] 
Melvin Leok, Diana Sosa. Dirac structures and HamiltonJacobi theory for Lagrangian mechanics on Lie algebroids. Journal of Geometric Mechanics, 2012, 4 (4) : 421442. doi: 10.3934/jgm.2012.4.421 
[6] 
Jeffrey K. Lawson, Tanya Schmah, Cristina Stoica. EulerPoincaré reduction for systems with configuration space isotropy. Journal of Geometric Mechanics, 2011, 3 (2) : 261275. doi: 10.3934/jgm.2011.3.261 
[7] 
Maria J. Esteban, Eric Séré. An overview on linear and nonlinear Dirac equations. Discrete & Continuous Dynamical Systems  A, 2002, 8 (2) : 381397. doi: 10.3934/dcds.2002.8.381 
[8] 
Yvette KosmannSchwarzbach. Dirac pairs. Journal of Geometric Mechanics, 2012, 4 (2) : 165180. doi: 10.3934/jgm.2012.4.165 
[9] 
Xu Zhang. On the concentration of semiclassical states for nonlinear Dirac equations. Discrete & Continuous Dynamical Systems  A, 2018, 38 (11) : 53895413. doi: 10.3934/dcds.2018238 
[10] 
Xiaoyan Lin, Xianhua Tang. Solutions of nonlinear periodic Dirac equations with periodic potentials. Discrete & Continuous Dynamical Systems  S, 2019, 12 (7) : 20512061. doi: 10.3934/dcdss.2019132 
[11] 
Sebastián Ferrer, Martin Lara. Families of canonical transformations by HamiltonJacobiPoincaré equation. Application to rotational and orbital motion. Journal of Geometric Mechanics, 2010, 2 (3) : 223241. doi: 10.3934/jgm.2010.2.223 
[12] 
Henry O. Jacobs, Hiroaki Yoshimura. Tensor products of Dirac structures and interconnection in Lagrangian mechanics. Journal of Geometric Mechanics, 2014, 6 (1) : 6798. doi: 10.3934/jgm.2014.6.67 
[13] 
Ünver Çiftçi. LeibnizDirac structures and nonconservative systems with constraints. Journal of Geometric Mechanics, 2013, 5 (2) : 167183. doi: 10.3934/jgm.2013.5.167 
[14] 
EmanuelCiprian Cismas. EulerPoincaréArnold equations on semidirect products II. Discrete & Continuous Dynamical Systems  A, 2016, 36 (11) : 59936022. doi: 10.3934/dcds.2016063 
[15] 
V. Afraimovich, J. Schmeling, Edgardo Ugalde, Jesús Urías. Spectra of dimensions for Poincaré recurrences. Discrete & Continuous Dynamical Systems  A, 2000, 6 (4) : 901914. doi: 10.3934/dcds.2000.6.901 
[16] 
Eva Miranda, Romero Solha. A Poincaré lemma in geometric quantisation. Journal of Geometric Mechanics, 2013, 5 (4) : 473491. doi: 10.3934/jgm.2013.5.473 
[17] 
Nikolaos Bournaveas, Timothy Candy, Shuji Machihara. A note on the ChernSimonsDirac equations in the Coulomb gauge. Discrete & Continuous Dynamical Systems  A, 2014, 34 (7) : 26932701. doi: 10.3934/dcds.2014.34.2693 
[18] 
Noboru Okazawa, Kentarou Yoshii. Linear evolution equations with strongly measurable families and application to the Dirac equation. Discrete & Continuous Dynamical Systems  S, 2011, 4 (3) : 723744. doi: 10.3934/dcdss.2011.4.723 
[19] 
Hyungjin Huh. Selfsimilar solutions to nonlinear Dirac equations and an application to nonuniqueness. Evolution Equations & Control Theory, 2018, 7 (1) : 5360. doi: 10.3934/eect.2018003 
[20] 
Changhun Yang. Scattering results for Dirac Hartreetype equations with small initial data. Communications on Pure & Applied Analysis, 2019, 18 (4) : 17111734. doi: 10.3934/cpaa.2019081 
2017 Impact Factor: 0.561
Tools
Metrics
Other articles
by authors
[Back to Top]