# American Institute of Mathematical Sciences

December  2018, 10(4): 373-410. doi: 10.3934/jgm.2018014

## Generalized variational calculus for continuous and discrete mechanical systems

 1 Departamento de Matemática, Universidad Nacional del Sur (UNS), Avenida Alem 1253, 8000 Bahía Blanca, Argentina 2 Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM), Nicolás Cabrera, 13-15, Campus Cantoblanco, UAM, 28049 Madrid, Spain

Received  October 2014 Revised  September 2018 Published  November 2018

Fund Project: This work has been partially supported by UNS, Argentina (project PGI 24/ZL06); FONCYT, Argentina (project PICT 2010-2746); CONICET, Argentina (project PIP 2010–2012 11220090101018); MEC (Spain) Grants MTM2013-42870-P, MTM2009-08166-E, and IRSES-project "Geomech-246981".

In this paper, we consider a generalization of variational calculus which allows us to consider in the same framework different cases of mechanical systems, for instance, Lagrangian mechanics, Hamiltonian mechanics, systems subjected to constraints, optimal control theory... This generalized variational calculus is based on two main notions: the tangent lift of curves and the notion of complete lift of a vector field. Both concepts are also adapted for the case of skew-symmetric algebroids, therefore, our formalism easily extends to the case of Lie algebroids and nonholonomic systems (see also [20]). Hence, this framework automatically includes reduced mechanical systems subjected or not to constraints. Finally, we show that our formalism can be used to tackle the case of discrete mechanics, including reduced systems, systems subjected to constraints and discrete optimal control theory.

Citation: Viviana Alejandra Díaz, David Martín de Diego. Generalized variational calculus for continuous and discrete mechanical systems. Journal of Geometric Mechanics, 2018, 10 (4) : 373-410. doi: 10.3934/jgm.2018014
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