Bloch, AnthonyColombo, LeonardoJiménez Alburquerque, Fernando2026-01-192026-01-192019-01-01Bloch, A., Colombo, L., & Jiménez, F. (2019). The variational discretization of the constrained higher-order lagrange-poincaré equations. Discrete and Continuous Dynamical Systems- Series A, 39(1), 309-344. https://doi.org/10.3934/DCDS.20190131078-0947https://doi.org/10.3934/dcds.2019013https://hdl.handle.net/20.500.14468/31475The registered version of this article, first published in Discrete and Continuous Dynamical Systems- Series A, is available online at the publisher's website: American Institute of Mathematical Sciences, https://doi.org/10.3934/dcds.2019013La versión registrada de este artículo, publicado por primera vez en Discrete and Continuous Dynamical Systems- Series A, está disponible en línea en el sitio web del editor: American Institute of Mathematical Sciences, https://doi.org/10.3934/dcds.2019013In this paper we investigate a variational discretization for the class of mechanical systems in presence of symmetries described by the action of a Lie group which reduces the phase space to a (non-trivial) principal bundle. By introducing a discrete connection we are able to obtain the discrete constrained higher-order Lagrange-Poincaré equations. These equations describe the dynamics of a constrained Lagrangian system when the Lagrangian function and the constraints depend on higher-order derivatives such as the acceleration, jerk or jounces. The equations, under some mild regularity conditions, determine a well defined (local) flow which can be used to define a numerical scheme to integrate the constrained higher-order Lagrange-Poincaré equations. Optimal control problems for underactuated mechanical systems can be viewed as higher-order constrained variational problems. We study how a variational discretization can be used in the construction of variational integrators for optimal control of underactuated mechanical systems where control inputs act soley on the base manifold of a principal bundle (the shape space). Examples include the energy minimum control of an electron in a magnetic field and two coupled rigid bodies attached at a common center of mass.eninfo:eu-repo/semantics/openAccess12 MatemáticasartículoVariational integratorsdiscrete mechanical systemsdiscrete mechanical systemsLagrangePoincar´e equationsgeometric integrationdiscrete variational calculusordinary differential equationscontrol of mechanical systemsreduction by symmetries1553-5231