arXiv subjects

Marcela Zuccalli

Publications and source records attributed to Marcela Zuccalli.

At least 19 recordsLinked to original sources

arXiv2026

Lagrangian reduction of symmetric discrete mechanical systems: a survey

In this note we survey some of our results on the Lagrangian reduction of discrete-time mechanical systems (DMSs). It is intended as an introduction to the general ideas that we used in the reduction of DMSs with nonholonomic constraints, DMSs with external forcing, as well as a theory of reduction by stages for such systems. This line of work was inspired by the paper and the monograph written by H. Cendra, J. Marsden and T. Ratiu in 2001.

math.DG↗

arXiv2026

Remarks on structures and preservation in forced discrete mechanical systems of Routh type

We study a type of forced discrete mechanical system $(Q,L_d,f_d)$ -- that we name of Routh type -- whose (discrete) time-flow preserves a symplectic structure on $Q\times Q$. That structure arises as the pullback via the forced discrete Legendre transform of the canonical symplectic structure on $T^*Q$ modified by a "magnetic term". One example of this type of system is provided by the Lagrangian reduction of a symmetric (unforced) discrete mechanical system in the Routh style. In this particular case, we do not reduce by the full symmetry group but, rather, by an appropriate isotropy subgroup. In this context, the preserved symplectic structure can be alternatively seen as the Marsden-Weinstein reduction of the canonical symplectic structure $ω_{L_d}$ on $Q\times Q$.

math.DG↗

arXiv2025

Lagrangian neural networks for nonholonomic mechanics

Lagrangian Neural Networks (LNNs) are a powerful tool for addressing physical systems, particularly those governed by conservation laws. LNNs can parametrize the Lagrangian of a system to predict trajectories with nearly conserved energy. These techniques have proven effective in unconstrained systems as well as those with holonomic constraints. In this work, we adapt LNN techniques to mechanical systems with nonholonomic constraints. We test our approach on some well-known examples with nonholonomic constraints, showing that incorporating these restrictions into the neural network's learning improves not only trajectory estimation accuracy but also ensures adherence to constraints and exhibits better energy behavior compared to the unconstrained counterpart.

cs.LG↗

arXiv2025

Homogeneous bi-Hamiltonian structures and integrable contact systems

Bi-Hamiltonian structures can be utilised to compute a maximal set of functions in involution for certain integrable systems, given by the eigenvalues of the recursion operator relating both Poisson structures. We show that the recursion operator relating two compatible Jacobi structures cannot produce a maximal set of functions in involution. However, as we illustrate with an example, bi-Hamiltonian structures can still be used to obtain a maximal set of functions in involution on a contact manifold, at the cost of symplectisation.

math-ph↗

arXiv2024

Discrete connections on principal bundles: the Discrete Atiyah Sequence

In this work we study discrete analogues of an exact sequence of vector bundles introduced by M. Atiyah in 1957, associated to any smooth principal $G$-bundle $π:Q\rightarrow Q/G$. In the original setting, the splittings of the exact sequence correspond to connections on the principal bundle $π$. The discrete analogues that we consider here can be studied in two different categories: the category of fiber bundles with a (chosen) section, FBS, and the category of local Lie groupoids, lLgpdC. In FBS we find a correspondence between a) (semi-local) splittings of the discrete Atiyah sequence (DAS) of $π$, b) discrete connections on the same bundle $π$, and c) isomorphisms of the DAS with certain fiber product extensions in FBS. We see that the right splittings of the DAS (in FBS) are not necessarily right splittings in lLgpdC: we use this obstruction to define the discrete curvature of a discrete connection. Then, there is a correspondence between the splittings of the DAS in lLgpdC and discrete connections with trivial discrete curvature. We also introduce a semidirect product between (some) local Lie groupoids and prove that there is a correspondence between semidirect product extensions and splittings of the DAS in lLgpdC.

math.DG↗

arXiv2023

Lagrangian reduction of forced discrete mechanical systems

In this paper we propose a process of Lagrangian reduction and reconstruction for symmetric discrete-time mechanical systems acted on by external forces, where the symmetry group action on the configuration manifold turns it into a principal bundle. We analyze the evolution of momentum maps and Poisson structures under different conditions.

math.DG↗

arXiv2022

Momentum map reduction for nonholonomic systems

This paper presents a reduction procedure for nonholonomic systems admitting suitable types of symmetries and conserved quantities. The full procedure contains two steps. The first (simple) step results in a Chaplygin system, described by an almost symplectic structure, carrying additional symmetries. The focus of this paper is on the second step, which consists of a Marsden-Weinstein--type reduction that generalizes constructions in [4,17]. The almost symplectic manifolds obtained in the second step are proven to coincide with the leaves of the reduced nonholonomic brackets defined in [7]. We illustrate our construction with several classical examples.

math-ph↗

arXiv2022

Discrete Mechanical Systems in a Dirac Setting: a Proposal

In these notes, we present an alternative version of discrete Dirac mechanics using Dirac structures. We first establish a notion of 'continuous Dirac system' and then propose a definition of discrete Dirac system, proving that it is possible to recover discrete Lagrangian and Hamiltonian systems as particular cases. We also note that this approach allows for kinematic as well as variational constraints.

math.DG↗

arXiv2021

Discrete connections on principal bundles: abelian group case

In this note we consider a few interesting properties of discrete connections on principal bundles when the structure group of the bundle is an abelian Lie group. In particular, we show that the discrete connection form and its curvature can be interpreted as singular $1$ and $2$ cochains respectively, with the curvature being the coboundary of the connection form. Using this formalism we prove a discrete analogue of a formula for the holonomy around a loop given by Marsden, Montgomery and Ratiu for (continuous) connections in a similar setting.

math.DG↗

arXiv2020

Lagrangian reduction of nonholonomic discrete mechanical systems by stages

In this work we introduce a category $LDP_d$ of discrete-time dynamical systems, that we call discrete Lagrange--D'Alembert--Poincaré systems, and study some of its elementary properties. Examples of objects of $LDP_d$ are nonholonomic discrete mechanical systems as well as their lagrangian reductions and, also, discrete Lagrange-Poincaré systems. We also introduce a notion of symmetry group for objects of $LDP_d$ and a process of reduction when symmetries are present. This reduction process extends the reduction process of discrete Lagrange--Poincaré systems as well as the one defined for nonholonomic discrete mechanical systems. In addition, we prove that, under some conditions, the two-stage reduction process (first by a closed and normal subgroup of the symmetry group and, then, by the residual symmetry group) produces a system that is isomorphic in $LDP_d$ to the system obtained by a one-stage reduction by the full symmetry group.

math.DG↗

arXiv2019

Explicit equations from orbit reduction: one and two stages

It is known that orbit reduction can be performed in one or two stages and it has been proven that the two processes are symplectically equivalent. In the context of orbit reduction by one stage we shall write an expression for the reduced two-form in the general case and obtain the equations of motion derived from this theory. Then we shall develop the same process in the case in which the symmetry group has a normal subgroup to get the reduced symplectic form by two stages and the consequent orbit reduced equations. In both cases we shall illustrate the method with the example of a rigid body with rotors and compare the obtained equations with the ones given by other authors in different frameworks.

math.SG↗

arXiv2019

Asymptotic Stabilizability of Underactuated Hamiltonian Systems With Two Degrees of Freedom

For an underactuated (simple) Hamiltonian system with two degrees of freedom and one degree of underactuation, a rather general condition that ensures its stabilizability, by means of the existence of a (simple) Lyapunov function, was found in a recent paper by D.E. Chang within the context of the energy shaping method. Also, in the same paper, some additional assumptions were presented in order to ensure also asymptotic stabilizability. In this paper we extend these results by showing that above mentioned condition is not only sufficient, but also a necessary one. And, more importantly, we show that no additional assumption is needed to ensure asymptotic stabilizability.

math.OC↗

arXiv2019

Variational reduction of Hamiltonian systems with general constraints

In the Hamiltonian formalism, and in the presence of a symmetry Lie group, a variational reduction procedure has already been developed for Hamiltonian systems without constraints. In this paper we present a procedure of the same kind, but for the entire class of the higher order constrained systems (HOCS), described in the Hamiltonian formalism. Last systems include the standard and generalized nonholonomic Hamiltonian systems as particular cases. When restricted to Hamiltonian systems without constraints, our procedure gives rise exactly to the so-called Hamilton-Poincaré equations, as expected. In order to illustrate the procedure, we study in detail the case in which both the configuration space of the system and the involved symmetry define a trivial principal bundle.

math-ph↗

arXiv2018

Explicit solutions of the kinetic and potential matching conditions of the energy shaping method

In this paper we present a procedure to integrate, up to quadratures, the matching conditions of the energy shaping method. We do that in the context of underactuated Hamiltonian systems defined by simple Hamiltonian functions. For such systems, the matching conditions split into two decoupled subsets of equations: the kinetic and potential equations. First, assuming that a solution of the kinetic equation is given, we find integrability and positivity conditions for the potential equation (because positive-definite solutions are the interesting ones), and we find an explicit solution of the latter. Then, in the case of systems with one degree of underactuation, we find in addition a concrete formula for the general solution of the kinetic equation. An example is included to illustrate our results.

math-ph↗

arXiv2016

On the relationship between the energy shaping and the Lyapunov constraint based methods

In this paper, we make a review of the controlled Hamiltonians (CH) method and its related matching conditions, focusing on an improved version recently developed by D.E. Chang. Also, we review the general ideas around the Lyapunov constraint based (LCB) method, whose related partial differential equations (PDEs) were originally studied for underactuated systems with only one actuator, and then we study its PDEs for an arbitrary number of actuators. We analyze and compare these methods within the framework of Differential Geometry, and from a purely theoretical point of view. We show, in the context of underactuated systems defined by simple Hamiltonian functions, that the LCB method and the Chang's version of the CH method are equivalent stabilization methods (i.e. they give rise to the same set of control laws). In other words, we show that the Chang's improvement of the energy shaping method is precisely the LCB method. As a by-product, coordinate-free and connection-free expressions of Chang's matching conditions are obtained.

math.OC↗

arXiv2015

Lagrangian reduction of discrete mechanical systems by stages

In this work we introduce a category of discrete Lagrange--Poincare systems LP_d and study some of its properties. In particular, we show that the discrete mechanical systems and the discrete mechanical systems obtained by the Lagrangian reduction of symmetric discrete mechanical systems are objects in LP_d. We introduce a notion of symmetry groups for objects of LP_d and introduce a reduction procedure that is closed in the category LP_d. Furthermore, under some conditions, we show that the reduction in two steps (first by a closed normal subgroup of the symmetry group and then by the residual symmetry group) is isomorphic in LP_d to the reduction by the full symmetry group.

math.DG↗

arXiv2013

A geometric approach to discrete connections on principal bundles

This work revisits, from a geometric perspective, the notion of discrete connection on a principal bundle, introduced by M. Leok, J. Marsden and A. Weinstein. It provides precise definitions of discrete connection, discrete connection form and discrete horizontal lift and studies some of their basic properties and relationships. An existence result for discrete connections on principal bundles equipped with appropriate Riemannian metrics is proved.

math.DG↗

arXiv2013

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.

math-ph↗