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We extend known constructions of almost-Poisson brackets and their gauge transformations to nonholonomic systems whose Lagrangian is not mechanical but possesses a gyroscopic term linear in the velocities. The new feature introduced by such a term is that the Legendre transformation is an affine, instead of linear, bundle isomorphism between the tangent and cotangent bundles of the configuration space and some care is needed in the development of the geometric formalism. At the end of the day, the affine nature of the Legendre transform is reflected in the affine dependence of the brackets that we construct on the momentum variables. Our study is motivated by a wide class of nonholonomic systems involving rigid bodies with internal rotors which are of interest in control. Our construction provides a natural geometric framework for the (known) Hamiltonisations of the gyrostatic generalisations of the Suslov and Chaplygin sphere problems. [ABSTRACT FROM AUTHOR]

For a given material, controllable deformations are those deformations that can be maintained in the absence of body forces and by applying only boundary tractions. For a given class of materials, universal deformations are those deformations that are controllable for any material within the class. In this paper, we characterize the universal deformations in compressible isotropic implicit elasticity defined by solids whose constitutive equations, in terms of the Cauchy stress σ and the left Cauchy-Green strain b , have the implicit form f (σ , b) = 0 . We prove that universal deformations are homogeneous. However, an important observation is that, unlike Cauchy (and Green) elasticity, not every homogeneous deformation is constitutively admissible for a given implicit-elastic solid. In other words, the set of universal deformations is material-dependent, yet it remains a subset of homogeneous deformations. [ABSTRACT FROM AUTHOR]

We extend the method of controlled Lagrangians to nonholonomic Euler–Poincaré equations with advected parameters, specifically to those mechanical systems on Lie groups whose symmetry is broken not only by a potential force but also by nonholonomic constraints. We introduce advected-parameter-dependent quasivelocities in order to systematically eliminate the Lagrange multipliers in the nonholonomic Euler–Poincaré equations. The quasivelocities facilitate the method of controlled Lagrangians for these systems, and lead to matching conditions that are similar to those by Bloch, Leonard, and Marsden for the standard holonomic Euler–Poincaré equation. Our motivating example is what we call the pendulum skate, a simple model of a figure skater developed by Gzenda and Putkaradze. We show that the upright spinning of the pendulum skate is stable under certain conditions, whereas the upright sliding equilibrium is always unstable. Using the matching condition, we derive a control law to stabilize the sliding equilibrium. [ABSTRACT FROM AUTHOR]

In this article, we focus on the formulation of dissipative mechanical systems through contact Hamiltonian systems. Different forms of symmetry of a contact dynamical system (geometric, dynamic, and gage) are defined to, in the realm of Noether, find their corresponding dissipated quantities. We also address the existence of dissipated quantities associated with a general vector field X on T Q × R , focusing on the case where its contact Hamiltonian function is dissipative. [ABSTRACT FROM AUTHOR]

Let X be a compact connected Riemann surface and (V , ϕ) a holomorphic Lie algebroid on X such that the holomorphic vector bundle V is stable. We give a necessary and sufficient condition on holomorphic vector bundles E on X to admit a Lie algebroid connection. [ABSTRACT FROM AUTHOR]

We show that contact reductions can be described in terms of symplectic reductions in the traditional Marsden–Weinstein–Meyer as well as the constant rank picture. The point is that we view contact structures as particular (homogeneous) symplectic structures. A group action by contactomorphisms is lifted to a Hamiltonian action on the corresponding symplectic manifold, called the symplectic cover of the contact manifold. In contrast to the majority of the literature in the subject, our approach includes general contact structures (not only co-oriented) and changes the traditional view point: contact Hamiltonians and contact moment maps for contactomorphism groups are no longer defined on the contact manifold itself, but on its symplectic cover. Actually, the developed framework for reductions is slightly more general than purely contact, and includes a precontact and presymplectic setting which is based on the observation that there is a one-to-one correspondence between isomorphism classes of precontact manifolds and certain homogeneous presymplectic manifolds. [ABSTRACT FROM AUTHOR]

The classical notion of retraction map used to approximate geodesics is extended and rigorously defined to become a powerful tool to construct geometric integrators and it is called discretization map. Using the geometry of the tangent and cotangent bundles, we are able to tangently and cotangent lift such a map so that these lifts inherit the same properties as the original one and they continue to be discretization maps. In particular, the cotangent lift of a discretization map is a natural symplectomorphism, what plays a key role for constructing geometric integrators and symplectic methods. As a result, a wide range of (higher-order) numerical methods are recovered and canonically constructed by using different discretization maps, as well as some operations with Lagrangian submanifolds. [ABSTRACT FROM AUTHOR]

We perform a systematic study and classification of the possibility to decouple a class of second-order differential equations which have quadratic terms, either in the velocities or in the coordinates, plus linear damping terms. Our basic approach is the existing coordinate-free theory for the characterization of separable systems. But the focus on this specific class of second-order equations spontaneously leads to interesting side issues, such as the question of linearizability of the equations, and a potential role for other, less geometrical ideas of decoupling, such as the so-called technique of phase synchronization for linear systems. [ABSTRACT FROM AUTHOR]

The aim of this work is the study of magnetic trajectories on 2-step nilmanifolds. The magnetic equation is written and the corresponding solutions are found for a family of invariant Lorentz forces on a 2-step nilpotent Lie group equipped with a left-invariant metric. Some explicit examples are computed in the Heisenberg Lie groups H n for n = 3 , 5 , showing differences with the case of exact forms. Interesting magnetic trajectories related to elliptic integrals appear in H 3 . The question of existence of closed or periodic magnetic trajectories for every energy level on Lie groups or on compact quotients is treated. Conditions for the closeness of magnetic trajectories are proved. [ABSTRACT FROM AUTHOR]

This article introduces a new map on the symplectic Stiefel manifold. The operation that requires the highest computational cost to compute the novel retraction is a inversion of size 2p-by-2p, which is much less expensive than those required for the available retractions in the literature. Later, with the new retraction, we design a constraint preserving gradient method to minimize smooth functions defined on the symplectic Stiefel manifold. To improve the numerical performance of our approach, we use the non-monotone line-search of Zhang and Hager with an adaptive Barzilai–Borwein type step-size. Our numerical studies show that the proposed procedure is computationally promising and is a very good alternative to solve large-scale optimization problems over the symplectic Stiefel manifold. [ABSTRACT FROM AUTHOR]

For the Newtonian 4-body problem in space we prove that any zero angularmomentum bounded solution suffers infinitely many coplanar instants, that is,times at which all 4 bodies lie in the same plane. This result generalizes aknown result for collinear instants ("syzygies") in the zero angular momentumplanar 3-body problem, and extends to the $d+1$ body problem in $d$-space. Theproof, for $d=3$, starts by identifying the center-of-mass zero configurationspace with real $3 \times 3$ matrices, the coplanar configurations withmatrices whose determinant is zero, and the mass metric with the Frobenius(standard Euclidean) norm. Let $S$ denote the signed distance from a matrix tothe hypersurface of matrices with determinant zero. The proof hinges onestablishing a harmonic oscillator type ODE for $S$ along solutions. Bounds oninter-body distances then yield an explicit lower bound $\omega$ for thefrequency of this oscillator, guaranteeing a degeneration within every timeinterval of length $\pi/\omega$. The non-negativity of the curvature oforiented shape space (the quotient of configuration space by the rotationgroup) plays a crucial role in the proof.

In this paper, we present a relation between Jacobi–Reeb dynamics and the dynamics associated with a mechanical Hamiltonian system with respect to a linear Poisson structure on a vector bundle. For this purpose, we will use the so-called Jacobi bundle metrics induced by the mechanical Hamiltonian system. These constructions extend classical results on the relation between standard mechanical Hamiltonian systems on cotangent bundles and Reeb dynamics. [ABSTRACT FROM AUTHOR]

We recall and prove some basic properties of both the Schouten and Wagner curvature tensors. We consider—in some detail—the construction of the Wagner tensor, and then discuss how the vanishing of this tensor characterizes the flat structures, i.e., those for which the parallel translation is path-independent. (As a special case, we revisit the flatness of three-dimensional nonholonomic Riemannian manifolds.) In order to facilitate our presentation, we employ an extension of the notion of a connection to a "restricted connection." Such connections are equivalently viewed as horizontal lifts (or horizontal distributions); hence we revisit the Schouten and Wagner tensors from this perspective. In particular, it turns out that the construction of the Wagner tensor may be equivalently formulated as a flag of horizontal distributions. [ABSTRACT FROM AUTHOR]

We study the class of nonholonomic mechanical systems formed by a heavy symmetric ball that rolls without sliding on a surface of revolution, which is either at rest or rotates about its (vertical) figure axis with uniform angular velocity Ω . The first studies of these systems go back over a century, but a comprehensive understanding of their dynamics is still missing. The system has an SO (3) × SO (2) symmetry and reduces to four dimensions. We extend in various directions, particularly from the case Ω = 0 to the case Ω ≠ 0 , a number of previous results and give new results. In particular, we prove that the reduced system is Hamiltonizable even if Ω ≠ 0 and, exploiting the recently introduced “moving energy,” we give sufficient conditions on the profile of the surface that ensure the periodicity of the reduced dynamics and hence the quasiperiodicity of the unreduced dynamics on tori of dimension up to three. Furthermore, we determine all the equilibria of the reduced system, which are classified in three distinct families, and determine their stability properties. In addition to this, we give a new form of the equations of motion of nonholonomic systems in quasi-velocities which, at variance from the well-known Hamel equations, use any set of quasi-velocities and explicitly contain the reaction forces. [ABSTRACT FROM AUTHOR]

Ricci-like solitons with arbitrary potential are introduced and studied on Sasaki-like almost contact B-metric manifolds. A manifold of this type can be considered as an almost contact complex Riemannian manifold which complex cone is a holomorphic complex Riemannian manifold. The soliton under study is characterized and proved that its Ricci tensor is equal to the vertical component of both B-metrics multiplied by a constant. Thus, the scalar curvatures with respect to both B-metrics are equal and constant. In the 3-dimensional case, it is found that the special sectional curvatures with respect to the structure are constant. Gradient almost Ricci-like solitons on Sasaki-like almost contact B-metric manifolds have been proved to have constant soliton coefficients. Explicit examples are provided of Lie groups as manifolds of dimensions 3 and 5 equipped with the structures under study. [ABSTRACT FROM AUTHOR]

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 that includes as a particular case the exact discrete nonholonomic trajectory. [ABSTRACT FROM AUTHOR]

We consider a natural mechanical system on a Finsler manifold and study its curvature using the intrinsic Jacobi equations (called Jacobi curves) along the extremals of the least action of the system. The curvature for such a system is expressed in terms of the Riemann curvature and the Chern curvature (involving the gradient of the potential) of the Finsler manifold and the Hessian of the potential w.r.t. a Riemannian metric induced from the Finsler metric. As an application, we give sufficient conditions for the Hamiltonian flows of the least action to be hyperbolic and show new examples of Anosov flows. [ABSTRACT FROM AUTHOR]

In this paper a numerical scheme for partitioned systems of index 2 DAEs, such as those arising from nonholonomic mechanical problems, is proposed and its order for a certain class of Runge–Kutta methods we call of Lobatto-type is proven. [ABSTRACT FROM AUTHOR]

We generalize Hashin's nonlinear isotropic hollow cylinder and sphere assemblages to nonlinear anisotropic solids. More specifically, we find the effective hydrostatic constitutive equation of nonlinear transversely isotropic hollow sphere assemblages with radial material preferred directions. We also derive the effective constitutive equations of finite and infinitely-long hollow cylinder assemblages made of incompressible orthotropic solids with axial, radial, and circumferential material preferred directions. In both sphere and cylinder assemblages the spherical and cylindrical shells can be radially inhomogeneous as long as Hashin's definition of similar shells is properly generalized. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the possibility of elastodynamic transformation cloaking in bodies made of non-centrosymmetric gradient solids. The goal of transformation cloaking is to hide a hole from elastic disturbances in the sense that the mechanical response of a homogeneous and isotropic body with a hole covered by a cloak would be identical to that of the corresponding homogeneous and isotropic body outside the cloak. It is known that in the case of centrosymmetric gradient solids exact transformation cloaking is not possible; the balance of angular momentum is the obstruction to transformation cloaking. We will show that this no-go theorem holds for non-centrosymmetric gradient solids as well. [ABSTRACT FROM AUTHOR]

We present two meaningful and effective non-ideal constitutive characterizations for a multiple impulsive constraints S comprising a finite number of non-ideal frictionless constraints of codimension 1, described in the geometric setup given by the space–time bundle M of a mechanical system in contact/impact with S . Thanks to the geometric structures associated to the elements of S , we introduce a symmetric characterization, that does not distinguish the elements forming S as regards mechanical behavior, and an asymmetric one that makes this distinction. Both the characterizations provide a generalization of the characterization of ideal multiple constraints presented in Pasquero (Q Appl Math 76(3):547–576, 2018). The iterative nature of these characterizations allows the introduction of two algorithms determining the right velocity of the system in case of single or multiple contact/impact with symmetric or asymmetric constraints S , once the elements forming S and the left velocity of the system are known. We show the effectiveness of the two possible choices with explicit implementations of these algorithms in two significant examples: a simplified Newton’s cradle system for the symmetric characterization and a disk in multiple contact/impact with two walls of a corner for the asymmetric one. [ABSTRACT FROM AUTHOR]

The Boltzmann–Hamel (BH) equations are central in the dynamics and control of nonholonomic systems described in terms of quasi-velocities. The rigid body is a classical example of such systems, and it is well-known that the BH-equations are the Newton–Euler (NE) equations when described in terms of rigid body twists as quasi-velocities. It is further known that the NE-equations are the Euler–Poincaré, respectively, the reduced Euler–Lagrange equations on SE(3) when using body-fixed or spatial representation of rigid body twists. The connection between these equations are the Hamel coefficients, which are immediately identified as the structure constants of SE(3). However, an explicit coordinate-free derivation has not been presented in the literature. In this paper the Hamel coefficients for the rigid body are derived in a coordinate-free way without resorting to local coordinates describing the rigid body motion. The three most relevant choices of quasi-velocities (body-fixed, spatial, and hybrid representation of rigid body twists) are considered. The corresponding BH-equations are derived explicitly for the rotating and free floating body. Further, the Hamel equations for nonholonomically constrained rigid bodies are discussed, and demonstrated for the inhomogenous ball rolling on a plane. [ABSTRACT FROM AUTHOR]