# American Institute of Mathematical Sciences

December  2019, 11(4): 539-552. doi: 10.3934/jgm.2019026

## The problem of Lagrange on principal bundles under a subgroup of symmetries

We dedicate this work to our friend Darryl D. Holm on the occasion of his 70th birthday.

Received  April 2018 Revised  May 2019 Published  November 2019

Fund Project: This work has been partially supported by MICINN (Spain) under projects MTM2015-63612-P and PGC2018-098321-B-I00, as well as Consejería de Educación, Junta de Castilla-León (Spain) under project SA090G19.

Given a Lagrangian density $L{\bf{v}}$ defined on the $1$-jet extension $J^1P$ of a principal $G$-bundle $\pi \colon P\to M$ invariant under the action of a closed subgroup $H\subset G$, its Euler-Poincaré reduction in $J^1P/H = C(P)\times_M P/H$ ($C(P)\to M$ being the bundle of connections of $P$ and $P/H\to M$ being the bundle of $H$-structures) induces a Lagrange problem defined in $J^1(C(P)\times_M P/H)$ by a reduced Lagrangian density $l{\bf{v}}$ together with the constraints ${\rm{Curv}}\sigma = 0, \nabla ^\sigma \bar{s} = 0$, for $\sigma$ and $\bar{s}$ sections of $C(P)$ and $P/H$ respectively. We prove that the critical section of this problem are solutions of the Euler-Poincaré equations of the reduced problem. We also study the Hamilton-Cartan formulation of this Lagrange problem, where we find some common points with Pontryagin's approach to optimal control problems for $\sigma$ as control variables and $\bar{s}$ as dynamical variables. Finally, the theory is illustrated with the case of affine principal fiber bundles and its application to the modelisation of the molecular strands on a Lorentzian plane.

Citation: Marco Castrillón López, Pedro Luis García Pérez. The problem of Lagrange on principal bundles under a subgroup of symmetries. Journal of Geometric Mechanics, 2019, 11 (4) : 539-552. doi: 10.3934/jgm.2019026
##### References:

show all references

##### References:
 [1] Marco Castrillón López, Pablo M. Chacón, Pedro L. García. Lagrange-Poincaré reduction in affine principal bundles. Journal of Geometric Mechanics, 2013, 5 (4) : 399-414. doi: 10.3934/jgm.2013.5.399 [2] Takeshi Fukao, Nobuyuki Kenmochi. Abstract theory of variational inequalities and Lagrange multipliers. Conference Publications, 2013, 2013 (special) : 237-246. doi: 10.3934/proc.2013.2013.237 [3] Sergio Grillo, Marcela Zuccalli. Variational reduction of Lagrangian systems with general constraints. Journal of Geometric Mechanics, 2012, 4 (1) : 49-88. doi: 10.3934/jgm.2012.4.49 [4] Karla L. Cortez, Javier F. Rosenblueth. Normality and uniqueness of Lagrange multipliers. Discrete & Continuous Dynamical Systems - A, 2018, 38 (6) : 3169-3188. doi: 10.3934/dcds.2018138 [5] Bas Janssens. Infinitesimally natural principal bundles. Journal of Geometric Mechanics, 2016, 8 (2) : 199-220. doi: 10.3934/jgm.2016004 [6] V. Balaji, I. Biswas and D. S. Nagaraj. Principal bundles with parabolic structure. Electronic Research Announcements, 2001, 7: 37-44. [7] Maria do Rosário de Pinho, Ilya Shvartsman. Lipschitz continuity of optimal control and Lagrange multipliers in a problem with mixed and pure state constraints. Discrete & Continuous Dynamical Systems - A, 2011, 29 (2) : 505-522. doi: 10.3934/dcds.2011.29.505 [8] Marco Castrillón López, Mark J. Gotay. Covariantizing classical field theories. Journal of Geometric Mechanics, 2011, 3 (4) : 487-506. doi: 10.3934/jgm.2011.3.487 [9] Tao Jie, Gao Yan. Computing shadow prices with multiple Lagrange multipliers. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020070 [10] Javier Fernández, Marcela Zuccalli. A geometric approach to discrete connections on principal bundles. Journal of Geometric Mechanics, 2013, 5 (4) : 433-444. doi: 10.3934/jgm.2013.5.433 [11] Alberto Ibort, Amelia Spivak. Covariant Hamiltonian field theories on manifolds with boundary: Yang-Mills theories. Journal of Geometric Mechanics, 2017, 9 (1) : 47-82. doi: 10.3934/jgm.2017002 [12] Pedro L. García, Antonio Fernández, César Rodrigo. Variational integrators for discrete Lagrange problems. Journal of Geometric Mechanics, 2010, 2 (4) : 343-374. doi: 10.3934/jgm.2010.2.343 [13] Harald Markum, Rainer Pullirsch. Classical and quantum chaos in fundamental field theories. Conference Publications, 2003, 2003 (Special) : 596-603. doi: 10.3934/proc.2003.2003.596 [14] Chjan C. Lim, Da Zhu. Variational analysis of energy-enstrophy theories on the sphere. Conference Publications, 2005, 2005 (Special) : 611-620. doi: 10.3934/proc.2005.2005.611 [15] J. Húska, Peter Poláčik. Exponential separation and principal Floquet bundles for linear parabolic equations on $R^N$. Discrete & Continuous Dynamical Systems - A, 2008, 20 (1) : 81-113. doi: 10.3934/dcds.2008.20.81 [16] Agnieszka B. Malinowska, Delfim F. M. Torres. Euler-Lagrange equations for composition functionals in calculus of variations on time scales. Discrete & Continuous Dynamical Systems - A, 2011, 29 (2) : 577-593. doi: 10.3934/dcds.2011.29.577 [17] Stefano Bianchini. On the Euler-Lagrange equation for a variational problem. Discrete & Continuous Dynamical Systems - A, 2007, 17 (3) : 449-480. doi: 10.3934/dcds.2007.17.449 [18] Hernán Cendra, María Etchechoury, Sebastián J. Ferraro. An extension of the Dirac and Gotay-Nester theories of constraints for Dirac dynamical systems. Journal of Geometric Mechanics, 2014, 6 (2) : 167-236. doi: 10.3934/jgm.2014.6.167 [19] Artur M. C. Brito da Cruz, Natália Martins, Delfim F. M. Torres. Hahn's symmetric quantum variational calculus. Numerical Algebra, Control & Optimization, 2013, 3 (1) : 77-94. doi: 10.3934/naco.2013.3.77 [20] Eduardo Martínez. Higher-order variational calculus on Lie algebroids. Journal of Geometric Mechanics, 2015, 7 (1) : 81-108. doi: 10.3934/jgm.2015.7.81

2019 Impact Factor: 0.649