506 research outputs found
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
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