Projection methods and discrete gradient methods for preserving first integrals of ODEs
Richard A. Norton David I. McLaren G. R. W. Quispel Ari Stern Antonella Zanna
Discrete & Continuous Dynamical Systems - A 2015, 35(5): 2079-2098 doi: 10.3934/dcds.2015.35.2079
In this paper we study linear projection methods for approximating the solution and simultaneously preserving first integrals of autonomous ordinary differential equations. We show that each (linear) projection method is equivalent to a class of discrete gradient methods, in both single and multiple first integral cases, and known results for discrete gradient methods also apply to projection methods. Thus we prove that in the single first integral case, under certain mild conditions, the numerical solution for a projection method exists and is locally unique, and preserves the order of accuracy of the underlying method. Our results allow considerable freedom for the choice of projection direction and do not have a time step restriction close to critical points.
keywords: discrete gradients Hamiltonian systems. projection energy preserving integrators Geometric integration
Symplectic groupoids and discrete constrained Lagrangian mechanics
Juan Carlos Marrero David Martín de Diego Ari Stern
Discrete & Continuous Dynamical Systems - A 2015, 35(1): 367-397 doi: 10.3934/dcds.2015.35.367
In this article, we generalize the theory of discrete Lagrangian mechanics and variational integrators in two principal directions. First, we show that Lagrangian submanifolds of symplectic groupoids give rise to discrete dynamical systems, and we study the properties of these systems, including their regularity and reversibility, from the perspective of symplectic and Poisson geometry. Next, we use this framework---along with a generalized notion of generating function due to Śniatycki and Tulczyjew [18]---to develop a theory of discrete constrained Lagrangian mechanics. This allows for systems with arbitrary constraints, including those which are non-integrable (in an appropriate discrete, variational sense). In addition to characterizing the dynamics of these constrained systems, we also develop a theory of reduction and Noether symmetries, and study the relationship between the dynamics and variational principles. Finally, we apply this theory to discretize several concrete examples of constrained systems in mechanics and optimal control.
keywords: non-integrable constraints Discrete Lagrangian mechanics Lagrangian submanifolds symplectic groupoids generating functions Lie algebroids. Lie groupoids

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