High-order geometric methods for nonholonomic mechanical systems

In the last two decades, significant effort has been put in understanding and designing so-called structure-preserving numerical methods for the simulation of mechanical systems. Geometric integrators attempt to preserve the geometry associated to the original system as much as possible, such as the structure of the configuration space, the energy behaviour, preservation of constants of the motion and of constraints or other structures associated to the continuous system (symplecticity, Poisson structure...). In this article, we develop high-order geometric (or pseudo-variational) integrators for nonholonomic systems, i.e., mechanical systems subjected to constraint functions which are, roughly speaking, functions on velocities that are not derivable from position constraints. These systems realize rolling or certain kinds of sliding contact and are important for describing different classes of vehicles.Comment: 59 pages, 26 figure

Generalized variational calculus for continuous and discrete mechanical systems

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 and so on. 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. 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

Hamilton-Jacobi theory, Symmetries and Coisotropic Reduction

Reduction theory has played a major role in the study of Hamiltonian systems. On the other hand, the Hamilton-Jacobi theory is one of the main tools to integrate the dynamics of certain Hamiltonian problems and a topic of research on its own. Moreover, the construction of several symplectic integrators rely on approximations of a complete solution of the Hamilton-Jacobi equation. The natural question that we address in this paper is how these two topics (reduction and Hamilton-Jacobi theory) fit together. We obtain a reduction and reconstruction procedure for the Hamilton-Jacobi equation with symmetries, even in a generalized sense to be clarified below. Several applications and relations to other reductions of the Hamilton-Jacobi theory are shown in the last section of the paper. It is remarkable that as a by-product we obtain a generalization of the Ge-Marsden reduction procedure. Quite surprinsingly, the classical ansatzs available in the literature to solve the Hamilton-Jacobi equation are also particular instances of our framework.Comment: 30 page

A Universal Hamilton-Jacobi Theory

In this paper we develop a Hamilton-Jacobi theory in the setting of almost Poisson manifolds. The theory extends the classical Hamilton-Jacobi theory and can be also applied to very general situations including nonholonomic mechanical systems and time dependent systems with external forces.Comment: 19 page

Higher-order discrete variational problems with constraints

An interesting family of geometric integrators for Lagrangian systems can be defined using discretizations of the Hamilton's principle of critical action. This family of geometric integrators is called variational integrators. In this paper, we derive new variational integrators for higher-order lagrangian mechanical system subjected to higher-order constraints. From the discretization of the variational principles, we show that our methods are automatically symplectic and, in consequence, with a very good energy behavior. Additionally, the symmetries of the discrete Lagrangian imply that momenta is conserved by the integrator. Moreover, we extend our construction to variational integrators where the lagrangian is explicitly time-dependent. Finally, some motivating applications of higher-order problems are considered; in particular, optimal control problems for explicitly time-dependent underactuated systems and an interpolation problem on Riemannian manifolds.Comment: Comments Welcome

New developments on the Geometric Nonholonomic Integrator

In this paper, we will discuss new developments regarding the Geometric Nonholonomic Integrator (GNI) [23, 24]. GNI is a discretization scheme adapted to nonholonomic mechanical systems through a discrete geometric approach. This method was designed to account for some of the special geometric structures associated to a nonholonomic motion, like preservation of energy, preservation of constraints or the nonholonomic momentum equation. First, we study the GNI versions of the symplectic-Euler methods, paying special attention to their convergence behavior. Then, we construct an extension of the GNI in the case of affine constraints. Finally, we generalize the proposed method to nonholonomic reduced systems, an important subclass of examples in nonholonomic dynamics. We illustrate the behavior of the proposed method with the example of the Chaplygin sphere, which accounts for the last two features, namely it is both a reduced and an affine system.Comment: 28 pages. v2: Added references and the example of the Chaplygin spher

Variational integrators for underactuated mechanical control systems with symmetries

Optimal control problems for underactuated mechanical systems can be seen as a higher-order variational problem subject to higher-order constraints (that is, when the Lagrangian function and the constraints depend on higher-order derivatives such as the acceleration, jerk or jounces). In this paper we discuss the variational formalism for the class of underactuated mechanical control systems when the configuration space is a trivial principal bundle and the construction of variational integrators for such mechanical control systems. An interesting family of geometric integrators can be defined using discretizations of the Hamilton's principle of critical action. This family of geometric integrators is called variational integrators, being one of their main properties the preservation of geometric features as the symplecticity, momentum preservation and good behavior of the energy. We construct variational integrators for higher-order mechanical systems on trivial principal bundles and their extension for higher-order constrained systems and we devote special attention to the particular case of underactuated mechanical system

Morse families in optimal control problems

We geometrically describe optimal control problems in terms of Morse families in the Hamiltonian framework. These geometric structures allow us to recover the classical first order necessary conditions for optimality and the starting point to run an integrability algorithm. Moreover the integrability algorithm is adapted to optimal control problems in such a way that the trajectories originated by discontinuous controls are also obtained. From the Hamiltonian viewpoint we obtain the equations of motion for optimal control problems in the Lagrangian formalism by means of a proper Lagrangian submanifold. Singular optimal control problems and overdetermined ones are also studied along the paper.Comment: 22 page

The local description of discrete Mechanics

In this paper, we introduce local expressions for discrete Mechanics. To apply our results simultaneously to several interesting cases, we derive these local expressions in the framework of Lie groupoids, following the program proposed by Alan Weinstein in [19]. To do this, we will need some results on the geometry of Lie groupoids, as, for instance, the construction of symmetric neighborhoods or the existence of local bisections. These local descriptions will be particular useful for the explicit construction of geometric integrators for mechanical systems (reduced or not), in particular, discrete Euler-Lagrange equations, discrete Euler-Poincar\'e equations, discrete Lagrange-Poincar\'e equations... The results contained in this paper can be considered as a local version of the study that we have started in [13], on the geometry of discrete Mechanics on Lie groupoids.Comment: 25 page

Symplectic groupoids and discrete constrained Lagrangian mechanics

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 Sniatycki and Tulczyjew -- 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.Comment: 36 pages; v2: minor revision