## Journals

- Advances in Mathematics of Communications
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### Open Access Journals

DCDS

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.

DCDS

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.## Year of publication

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