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Journal of Geometric Mechanics

March 2021 , Volume 13 , Issue 1

Special issue dedicated to Professor Tony Bloch on the occasion of his 65th birthday

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Preface to the special issue dedicated to Anthony Bloch
Leonardo Colombo, Manuel de León and Tomoki Ohsawa
2021, 13(1): i-iii doi: 10.3934/jgm.2021004 +[Abstract](247) +[HTML](112) +[PDF](2156.14KB)
Geometric optimal techniques to control the muscular force response to functional electrical stimulation using a non-isometric force-fatigue model
Bernard Bonnard and Jérémy Rouot
2021, 13(1): 1-23 doi: 10.3934/jgm.2020032 +[Abstract](467) +[HTML](185) +[PDF](4559.97KB)

A recent force-fatigue parameterized mathematical model, based on the seminal contributions of V. Hill to describe muscular activity, allows to predict the muscular force response to external electrical stimulation (FES) and it opens the road to optimize the FES-input to maximize the force response to a pulse train, to track a reference force while minimizing the fatigue for a sequence of pulse trains or to follow a reference joint angle trajectory to produce motion in the non-isometric case. In this article, we introduce the geometric frame to analyze the dynamics and we present Pontryagin types necessary optimality conditions adapted to digital controls, used in the experiments, vs permanent control and which fits in the optimal sampled-data control frame. This leads to Hamiltonian differential variational inequalities, which can be numerically implemented vs direct optimization schemes.

Contact Hamiltonian and Lagrangian systems with nonholonomic constraints
Manuel de León, Víctor M. Jiménez and Manuel Lainz
2021, 13(1): 25-53 doi: 10.3934/jgm.2021001 +[Abstract](511) +[HTML](172) +[PDF](542.75KB)

In this article we develop a theory of contact systems with nonholonomic constraints. We obtain the dynamics from Herglotz's variational principle, by restricting the variations so that they satisfy the nonholonomic constraints. We prove that the nonholonomic dynamics can be obtained as a projection of the unconstrained Hamiltonian vector field. Finally, we construct the nonholonomic bracket, which is an almost Jacobi bracket on the space of observables and provides the nonholonomic dynamics.

A Lagrangian approach to extremal curves on Stiefel manifolds
Knut Hüper, Irina Markina and Fátima Silva Leite
2021, 13(1): 55-72 doi: 10.3934/jgm.2020031 +[Abstract](484) +[HTML](222) +[PDF](378.67KB)

A unified framework for studying extremal curves on real Stiefel manifolds is presented. We start with a smooth one-parameter family of pseudo-Riemannian metrics on a product of orthogonal groups acting transitively on Stiefel manifolds. In the next step Euler-Langrange equations for a whole class of extremal curves on Stiefel manifolds are derived. This includes not only geodesics with respect to different Riemannian metrics, but so-called quasi-geodesics and smooth curves of constant geodesic curvature, as well. It is shown that they all can be written in closed form. Our results are put into perspective to recent related work where a Hamiltonian rather than a Lagrangian approach was used. For some specific values of the parameter we recover certain well-known results.

On nomalized differentials on spectral curves associated with the sinh-Gordon equation
Thomas Kappeler and Yannick Widmer
2021, 13(1): 73-143 doi: 10.3934/jgm.2020023 +[Abstract](549) +[HTML](313) +[PDF](1680.37KB)

The spectral curve associated with the sinh-Gordon equation on the torus is defined in terms of the spectrum of the Lax operator appearing in the Lax pair formulation of the equation. If the spectrum is simple, it is an open Riemann surface of infinite genus. In this paper we construct normalized differentials on this curve and derive estimates for the location of their zeroes, needed for the construction of angle variables.

A unifying approach for rolling symmetric spaces
Krzysztof A. Krakowski, Luís Machado and Fátima Silva Leite
2021, 13(1): 145-166 doi: 10.3934/jgm.2020016 +[Abstract](887) +[HTML](426) +[PDF](1302.18KB)

The main goal of this paper is to present a unifying theory to describe the pure rolling motions of Riemannian symmetric spaces, which are submanifolds of Euclidean or pseudo-Euclidean spaces. Rolling motions provide interesting examples of nonholonomic systems and symmetric spaces appear associated to important applications. We make a connection between the structure of the kinematic equations of rolling and the natural decomposition of the Lie algebra associated to the symmetric space. This emphasises the relevance of Lie theory in the geometry of rolling manifolds and explains why many particular examples scattered through the existing literature always show a common pattern.

2019  Impact Factor: 0.649



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