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David Martin de Diego

Publications and source records attributed to David Martin de Diego.

At least 19 recordsLinked to original sources
arXiv2026

Learning Forced Multibody Dynamics on Lie Groups

We propose an architecture for learning the dynamics of mechanical systems based on discrete forced Euler-Lagrange equations on Lie groups using only position data. By formulating the dynamics directly on manifold-valued configuration spaces, the method naturally respects the geometric structure of the systems and preserves geometric invariants and conservation laws. The reliance on position measurements alone makes the framework applicable in settings where velocity data are unavailable or noisy. The approach extends naturally to multibody systems, accommodates external control inputs, and demonstrates strong performance on both synthetic and real-world datasets.

cs.LG↗
arXiv2026

Discrete-time generalized canonical transformations for non-autonomous systems

A dynamical system is said to be \emph{non-autonomous} when the differential equations describing its evolution depends explicitly on time. Among the various geometric approaches to investigate such systems, the cosymplectic formulation provides a natural framework that extends symplectic geometry to time-dependent Hamiltonians systems. However, preserving the associated geometric structures under numerical discretization remains a challenging problem: standard integrators, such as explicit Euler schemes, generally fail to conserve the cosymplectic volume or the underlying Poisson structure. In this work we propose a geometric method for the discretization of non-autonomous Hamiltonian systems based on \emph{generalized canonical transformations}. The approach constructs a symplectomorphism on the extended phase space $T^*(Q \times \mathbb{R})$ whose projection onto $T^*Q \times \mathbb{R}$ defines a structure-preserving discrete flow. We show that this formulation guarantees the preservation of key invariants, including the volume form, the Poisson bracket, and the symplectic structure on each time fiber.

math.DS↗
arXiv2026

Geometric Integrators for Nonholonomic Systems on Lie Groups

We present a general framework for constructing structure-preserving numerical integrators for nonholonomically constrained mechanical systems evolving on Lie groups using retraction maps. Retraction maps generalize the exponential map and provide a convenient tool for performing numerical integration on manifolds. In nonholonomic mechanics, the constraints restrict the dynamics to a nonintegrable distribution rather than the entire tangent bundle. Using the Hamel formulation, the equations of motion can be expressed in local coordinates adapted to this constraint distribution. We then specialize the framework to the case of Lie groups, where both the dynamics and the constraints exhibit symmetries, allowing a simplified formulation of the numerical scheme. The resulting integrator respects the constraint distribution and enforces the nonholonomic constraints at each discrete time step. The approach is illustrated using the Suslov problem.

math.NA↗
arXiv2026

A Geometric Approach to Structure-Preserving Integrators for Mechanical Systems

We develop a geometric framework for the numerical integration of mechanical systems evolving on manifolds. After briefly reviewing classical numerical methods and highlighting their limitations and shortcomings in non-flat (non-Euclidean) spaces, we include a short interlude reviewing the differential geometric notions required in the sequel. We then introduce retraction maps as intrinsic generalizations of the Riemannian exponential, which induce discretization maps tailored to manifold-valued dynamics. Adopting the Tulczyjew unified viewpoint, mechanical systems are formulated as Lagrangian submanifolds, providing a natural and coordinate-free foundation for the construction of structure-preserving integrators for both Hamiltonian and Lagrangian systems. The framework is specialized to Lie groups, where parallelizability allows for the global trivialization of tangent and cotangent bundles and the systematic derivation of integrators for Euler-Poincare and Lie-Poisson equations. The effectiveness of the proposed approach is illustrated through the rigid body and heavy top, and is further extended to the construction of a geometric integrator for an underactuated mechanical system-a quadrotor-demonstrating the applicability of the framework beyond fully symmetric systems and toward problems relevant in robotics and control.

math.GM↗
arXiv2025

Numerical Integrators for Mechanical Systems on Lie Groups

Retraction maps are known to be the seed for all numerical integrators. These retraction maps-based integrators can be further lifted to tangent and cotangent bundles, giving rise to structure-preserving integrators for mechanical systems. We explore the particular case where the configuration space of our mechanical system is a Lie group with certain symmetries. Here, the integrator simplifies based on the property that the tangent and cotangent bundles of Lie groups are trivializable. Finally, we present a framework for designing numerical integrators for Euler- Poincare and Lie-Poisson type equations.

math.NA↗
arXiv2020

Radial kinetic nonholonomic trajectories are Riemannian geodesics!

Nonholonomic mechanics describes the motion of systems constrained by nonintegrable constraints. One of its most remarkable properties is that the derivation of the nonholonomic equations is not variational in nature. {However, in} this paper, we prove (Theorem 1.1) that for kinetic nonholonomic {systems}, the solutions starting from a fixed point $q$ are true geodesics for a family of Riemannian metrics on the image submanifold ${\mathcal M}^{nh}_q$ of the nonholonomic exponential map. This implies a surprising result: the kinetic nonholonomic trajectories with starting point $q$, for sufficiently small times, minimize length in ${\mathcal M}^{nh}_q$!

math-ph↗
arXiv2020

Jacobi fields in nonholonomic mechanics

In this paper, we define Jacobi fields for nonholonomic mechanics using a similar characterization than in Riemannian geometry. We give explicit conditions to find Jacobi fields and finally we find the nonholonomic Jacobi equations in two equivalent ways: the first one, using an appropriate complete lift of the nonholonomic system and, in the second one, using the curvature and torsion of the associated nonholonomic connection.

math.DG↗
arXiv2020

Exact discrete Lagrangian mechanics for nonholonomic mechanics

We construct the exponential map associated to a nonholonomic system that allows us to define an exact discrete nonholonomic constraint submanifold. We reproduce the continuous nonholonomic flow as a discrete flow on this discrete constraint submanifold deriving an exact discrete version of the nonholonomic equations. Finally, we derive a general family of nonholonomic integrators.

math-ph↗
arXiv2018

Lie-Poisson integrators

In this paper, we discuss the geometric integration of hamiltonian systems on Poisson manifolds, in particular, in the case, when the Poisson structure is induced by a Lie algebra, that is, it is a Lie-Poisson structure. A Hamiltonian system on a Poisson manifold $(P, Π)$ is a smooth manifold $P$ equipped with a bivector field $Π$ satisfying $[Π, Π]=0$ (Jacobi identity), inducing the Poisson bracket on $C^{\infty}(P)$, $\{f, g\}\equiv Π(df, dg)$ where $f, g\in C^{\infty}(P)$. For any $f\in C^{\infty}(P)$ the Hamiltonian vector field is defined by $X_f(g)=\{g, f\}$. The Hamiltonian vector fields $X_f$ generate an integrable generalized distribution on $P$ and the leaves of this foliation are symplectic. The flow of any hamiltonian vector field preserves the Poisson structure, it fixes each leaf and the hamiltonian itself is a first integral. It is important to characterize numerical methods preserving some of these fundamental properties of the hamiltonian flow on Poisson manifolds (geometric integrators). We discuss the difficulties of deriving these Poisson methods using standard techniques and we present some modern approaches to the problem.

math.NA↗
arXiv2015

Second-order variational problems on Lie groupoids and optimal control applications

In this paper we study, from a variational and geometrical point of view, second-order variational problems on Lie groupoids and the construction of variational integrators for optimal control problems. First, we develop variational techniques for second-order variational problems on Lie groupoids and their applications to the construction of variational integrators for optimal control problems of mechanical systems. Next, we show how Lagrangian submanifolds of a symplectic groupoid gives intrinsically the discrete dynamics for second-order systems, both unconstrained and constrained, and we study the geometric properties of the implicit flow which defines the dynamics in the Lagrangian submanifold. We also study the theory of reduction by symmetries and the corresponding Noether theorem.

math.DS↗
arXiv2014

A geometric approach to the optimal control of nonholnomic mechanical systems

In this paper, we describe a constrained Lagrangian and Hamiltonian formalism for the optimal control of nonholonomic mechanical systems. In particular, we aim to minimize a cost functional, given initial and final conditions where the controlled dynamics is given by nonholonomic mechanical system. In our paper, the controlled equations are derived using a basis of vector fields adapted to the nonholonomic distribution and the Riemannian metric determined by the kinetic energy. Given a cost function, the optimal control problem is understood as a constrained problem or equivalently, under some mild regularity conditions, as a Hamiltonian problem on the cotangent bundle of the nonholonomic distribution. A suitable Lagrangian submanifold is also shown to lead to the correct dynamics. We demonstrate our techniques in several examples including a continuously variable transmission problem and motion planning for obstacle avoidance problems.

math.OC↗
arXiv2012

Discrete Variational Optimal Control

This paper develops numerical methods for optimal control of mechanical systems in the Lagrangian setting. It extends the theory of discrete mechanics to enable the solutions of optimal control problems through the discretization of variational principles. The key point is to solve the optimal control problem as a variational integrator of a specially constructed higher-dimensional system. The developed framework applies to systems on tangent bundles, Lie groups, underactuated and nonholonomic systems with symmetries, and can approximate either smooth or discontinuous control inputs. The resulting methods inherit the preservation properties of variational integrators and result in numerically robust and easily implementable algorithms. Several theoretical and a practical examples, e.g. the control of an underwater vehicle, will illustrate the application of the proposed approach.

math.OC↗
arXiv2012

Hamiltonian dynamics and constrained variational calculus: continuous and discrete settings

The aim of this paper is to study the relationship between Hamiltonian dynamics and constrained variational calculus. We describe both using the notion of Lagrangian submanifolds of convenient symplectic manifolds and using the so-called Tulczyjew's triples. The results are also extended to the case of discrete dynamics and nonholonomic mechanics. Interesting applications to geometrical integration of Hamiltonian systems are obtained.

math-ph↗
arXiv2011

Discrete second-order Euler-Poincaré equations. Applications to optimal control

In this paper we will discuss some new developments in the design of numerical methods for optimal control problems of Lagrangian systems on Lie groups. We will construct these geometric integrators using discrete variational calculus on Lie groups, deriving a discrete version of the second-order Euler-Lagrange equations. Interesting applications as, for instance, a discrete derivation of the Euler-Poincaré equations for second-order Lagrangians and its application to optimal control of a rigid body, and of a Cosserat rod are shown at the end of the paper.

math-ph↗
arXiv2011

On the geometry of higher-order variational problems on Lie groups

In this paper, we describe a geometric setting for higher-order lagrangian problems on Lie groups. Using left-trivialization of the higher-order tangent bundle of a Lie group and an adaptation of the classical Skinner-Rusk formalism, we deduce an intrinsic framework for this type of dynamical systems. Interesting applications as, for instance, a geometric derivation of the higher-order Euler-Poincaré equations, optimal control of underactuated control systems whose configuration space is a Lie group are shown, among others, along the paper.

math-ph↗
arXiv2010

Constrained Variational Calculus for Higher Order Classical Field Theories

We develop an intrinsic geometrical setting for higher order constrained field theories. As a main tool we use an appropriate generalization of the classical Skinner-Rusk formalism. Some examples of application are studied, in particular, applications to the geometrical description of optimal control theory for partial differential equations.

math-ph↗
arXiv2010

Unambiguous Formalism for Higher-Order Lagrangian Field Theories

The aim of this paper is to propose an unambiguous intrinsic formalism for higher-order field theories which avoids the arbitrariness in the generalization of the conventional description of field theories, which implies the existence of different Cartan forms and Legendre transformations. We propose a differential-geometric setting for the dynamics of a higher-order field theory, based on the Skinner and Rusk formalism for mechanics. This approach incorporates aspects of both, the Lagrangian and the Hamiltonian description, since the field equations are formulated using the Lagrangian on a higher-order jet bundle and the canonical multisymplectic form on its dual. As both of these objects are uniquely defined, the Skinner-Rusk approach has the advantage that it does not suffer from the arbitrariness in conventional descriptions. The result is that we obtain a unique and global intrinsic version of the Euler-Lagrange equations for higher-order field theories. Several examples illustrate our construction.

math.DG↗