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This article explores some geometric and algebraic properties of the dynamical system which is represented by matrix differential equations arising from inertial navigation problems, such as the symplecticity and the orthogonality.…
We analyze a variational time discretization of geodesic calculus on finite- and certain classes of infinite-dimensional Riemannian manifolds. We investigate the fundamental properties of discrete geodesics, the associated discrete…
In this work we study the dynamic behaviour of compound shells of revolution partially filled with an ideal incompressible fluid based on boundary-value problems. New analytical mathematical model with corresponding discrete scheme for the…
This paper is a review of results which have been recently obtained by applying mathematical concepts drawn, in particular, from differential geometry and topology, to the physics of Hamiltonian dynamical systems with many degrees of…
Attitude control systems naturally evolve on nonlinear configuration spaces, such as S^2 and SO(3). The nontrivial topological properties of these configuration spaces result in interesting and complicated nonlinear dynamics when studying…
In this manuscript, an original numerical procedure for the nonlinear peridynamics on arbitrarily--shaped two-dimensional (2D) closed manifolds is proposed. When dealing with non parameterized 2D manifolds at the discrete scale, the problem…
We study the dynamical symmetries of a class of two-dimensional superintegrable systems on a 2-sphere, obtained by a procedure based on the Marsden-Weinstein reduction, by considering its shape-invariant intertwining operators. These are…
We investigate the periodically driven dynamics of many-body systems, either classical or quantum, finite-dimensional or mean-field, displaying an unbounded phase-space. Using the lattice $\phi^4$ model and the $p$-spin spherical model as…
This paper presents theoretical analysis and software implementation for real harmonics analysis on the special orthogonal group. Noncommutative harmonic analysis for complex-valued functions on the special orthogonal group has been studied…
Stochastic geometric mechanics (SGM) is known for its potential utility in quantifying uncertainty in global climate modelling of the Earth's ocean and atmosphere while also preserving the fundamental advective transport properties of ideal…
Discussed are quantized dynamical systems on orthogonal and affine groups. The special stress is laid on geodetic systems with affinely-invariant kinetic energy operators. The resulting formulas show that such models may be useful in…
Optimal control problems are formulated and efficient computational procedures are proposed for combined orbital and rotational maneuvers of a rigid body in three dimensions. The rigid body is assumed to act under the influence of forces…
We present a comprehensive study on discrete morphological symmetries of dynamical systems, which are commonly observed in biological and artificial locomoting systems, such as legged, swimming, and flying animals/robots/virtual characters.…
A new class of exclusion type processes acting in continuum with synchronous updating is introduced and studied. Ergodic averages of particle velocities are obtained and their connections to other statistical quantities, in particular to…
We consider discrete dynamical systems and lattice models in statistical mechanics from the point of view of their symmetry groups. We describe a C program for symmetry analysis of discrete systems. Among other features, the program…
This paper reports a novel result: with proper robot models on matrix Lie groups, one can formulate the kinodynamic motion planning problem for rigid body systems as \emph{exact} polynomial optimization problems that can be relaxed as…
These lectures demonstrate the development of a PID control framework for mechanical systems. Based on the observation that mechanical systems are essentially double integrator systems, we generalize the linear PID controller to mechanical…
This paper introduces a set of numerical methods for Riemannian shape analysis of 3D surfaces within the setting of invariant (elastic) second-order Sobolev metrics. More specifically, we address the computation of geodesics and geodesic…
In this article, we present an extension of the formulation recently developed by the authors (A Framework for Data-Driven Computational Mechanics Based on Nonlinear Optimization, arXiv:1910.12736 [math.NA]) to the structural dynamics…
Understanding the global dynamics of a robot controller, such as identifying attractors and their regions of attraction (RoA), is important for safe deployment and synthesizing more effective hybrid controllers. This paper proposes a…