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We present a qualitative analysis of the dynamics of a rolling and sliding disk in a horizontal plane. It is based on using three classes of asymptotic solutions: straight-line rolling, spinning about a vertical diameter and tumbling solutions. Their linear stability analysis is given and it is complemented with computer simulations of solutions starting in the vicinity of the asymptotic solutions. The results on asymptotic solutions and their linear stability apply also to an annulus and to a hoop. [ABSTRACT FROM AUTHOR]

The relationship of equations of motion of a Lagrangian $$\phi (L)$$ to those of L is studied, and the question of the existence of a function $$\phi $$ such that $$\phi (L)$$ is dynamically equivalent to L is answered. [ABSTRACT FROM AUTHOR]

In this work we investigate Ricci flows of almost Kähler structures on Lie algebroids when the fundamental geometric objects are completely determined by (semi) Riemannian metrics, or (effective) regular generating Lagrange/Finsler functions. There are constructed canonical almost symplectic connections for which the geometric flows can be represented as gradient ones and characterized by nonholonomic deformations of Grigory Perelman's functionals. The first goal of this paper is to define such thermodynamical type values and derive almost Kähler-Ricci geometric evolution equations. The second goal is to study how fixed Lie algebroid, i.e. Ricci soliton, configurations can be constructed for Riemannian manifolds and/or (co) tangent bundles endowed with nonholonomic distributions modelling (generalized) Einstein or Finsler-Cartan spaces. Finally, some examples of generic off-diagonal solutions for Lie algebroid-type Ricci solitons and (effective) Einstein and Lagrange-Finsler algebroids are provided. [ABSTRACT FROM AUTHOR]

We show that the Siegel upper half space $${\Sigma_{d}}$$ is identified with the Marsden-Weinstein quotient obtained by symplectic reduction of the cotangent bundle $${T^{*} \mathbb{R}^{2d^{2}}}$$ with O(2 d)-symmetry. The reduced symplectic form on $${\Sigma_{d}}$$ corresponding to the standard symplectic form on $${T^{*} \mathbb{R}^{2d^{2}}}$$ turns out to be a constant multiple of the symplectic form on $${\Sigma_{d}}$$ obtained by Siegel. Our motivation is to understand the geometry behind two different formulations of the Gaussian wave packet dynamics commonly used in semiclassical mechanics. Specifically, we show that the two formulations are related via the symplectic reduction. [ABSTRACT FROM AUTHOR]

We derive the reduced equations of motion for an articulated n-trailer vehicle that moves under its own inertia on the plane. We show that the energy level surfaces in the reduced space are ( n + 1)-tori and we classify the equilibria within them, determining their stability. A thorough description of the dynamics is given in the case n = 1. .. [ABSTRACT FROM AUTHOR]

This note introduces the construction of relational symplectic groupoids as a way to integrate every Poisson manifold. Examples are provided and the equivalence, in the integrable case, with the usual notion of symplectic groupoid is discussed. [ABSTRACT FROM AUTHOR]

The equations of motion of a mechanical system subjected to nonholonomic linear constraints can be formulated in terms of a linear almost Poisson structure in a vector bundle. We study the existence of invariant measures for the system in terms of the unimodularity of this structure. In the presence of symmetries, our approach allows us to give necessary and sufficient conditions for the existence of an invariant volume, which unify and improve results existing in the literature. We present an algorithm to study the existence of a smooth invariant volume for nonholonomic mechanical systems with symmetry and we apply it to several concrete mechanical examples. [ABSTRACT FROM AUTHOR]

We introduce a method in differential geometry to study the derivative operators of Siegel modular forms. By determining the coefficients of the invariant Levi-Civita connection on a Siegel upper half plane, and further by calculating the expressions of the differential forms under this connection, we get a non-holomorphic derivative operator of the Siegel modular forms. In order to get a holomorphic derivative operator, we introduce a weaker notion, called modular connection, on the Siegel upper half plane. Then we show that on a Siegel upper half plane there exists at most one holomorphic $${{\rm Sp}(2g, {\mathbb {Z}})}$$ -modular connection in some sense, and get a possible holomorphic derivative operator of Siegel modular forms. [ABSTRACT FROM AUTHOR]

In this note we construct a canonical lifting of arbitrary Poisson structures on a manifold to its algbera of densities. Using this construction we proceed to classify all extensions of a fixed structure on the original manifold to its algebra of densities. The question is analogous to the problem studied by H. M. Khudaverdian and Th. Voronov for odd Poisson structures and differential operators. Although the questions are similar the results are distinctly marked, namely in the case of even Poisson structures there always exists a lift which is naturally defined. The proof of this result bears a remarkable resemblance to the construction of the Frolicher - Nijenhuis bracket. In the final section we find a cocycle for the odd Lie algbera $${C^\infty (\Pi T* M)}$$ originating from a singular point in the above lifitng. [ABSTRACT FROM AUTHOR]

We introduce and study a notion of 'Sasaki with torsion structure' ( st ) as an odd-dimensional analogue of Kähler with torsion geometry ( kt ). These are normal almost contact metric manifolds that admit a unique compatible connection with $$ 3 $$-form torsion. Any odd-dimensional compact Lie group is shown to admit such a structure; in this case, the structure is left-invariant and has closed torsion form. We illustrate the relation between st structures and other generalisations of Sasaki geometry, and we explain how some standard constructions in Sasaki geometry can be adapted to this setting. In particular, we relate the st structure to a kt structure on the space of leaves and show that both the cylinder and the cone over an st manifold are kt , although only the cylinder behaves well with respect to closedness of the torsion form. Finally, we introduce a notion of ' $$ G $$-moment map'. We provide criteria based on equivariant cohomology ensuring the existence of these maps and then apply them as a tool for reducing st structures. [ABSTRACT FROM AUTHOR]

In this paper we introduce poly-Poisson structures as a higher-order extension of Poisson structures. It is shown that any poly-Poisson structure is endowed with a polysymplectic foliation. It is also proved that if a Lie group acts polysymplectically on a polysymplectic manifold then, under certain regularity conditions, the reduced space is a poly-Poisson manifold. In addition, some interesting examples of poly-Poisson manifolds are discussed. [ABSTRACT FROM AUTHOR]

We extend the notion of multi-moment map to geometries defined by closed forms of arbitrary degree. We give fundamental existence and uniqueness results and discuss a number of essential examples, including geometries related to special holonomy. For forms of degree four, multi-moment maps are guaranteed to exist and are unique when the symmetry group is (3,4)-trivial, meaning that the group is connected and the third and fourth Lie algebra Betti numbers vanish. We give a structural description of some classes of (3,4)-trivial algebras and provide a number of examples. [ABSTRACT FROM AUTHOR]

It is known that the reduced equations for an axially symmetric homogeneous ellipsoid that rolls without slipping on the plane possess a smooth invariant measure. We show that such an invariant measure does not exist in the case when all of the semi-axes of the ellipsoid have different length. [ABSTRACT FROM AUTHOR]

We see how the first jet bundle of curves into affine space can be realized as a homogeneous space of the Galilean group. Cartan connections with this model are precisely the geometric structure of second-order ordinary differential equations under time-preserving transformations - sometimes called KCC-theory. With certain regularity conditions, we show that any such Cartan connection induces 'laboratory' coordinate systems, and the geodesic equations in this coordinates form a system of second-order ordinary differential equations. We then show the converse - the 'fundamental theorem' - that given such a coordinate system, and a system of second order ordinary differential equations, there exists regular Cartan connections yielding these, and such connections are completely determined by their torsion. [ABSTRACT FROM AUTHOR]

The curvature and the reduced curvature are basic differential invariants of the pair (Hamiltonian system, Lagrangian distribution) on a symplectic manifold. We consider the Hamiltonian flows of the curve of least action of natural mechanical systems in sub-Riemannian structures with symmetries. We give sufficient conditions for the reduced flows (after reduction of the first integrals induced from the symmetries) to be hyperbolic in terms of the reduced curvature and show new examples of Anosov flows. [ABSTRACT FROM AUTHOR]

We consider a Dirac structure D on a manifold Q invariant under a proper action of a Lie group G on Q. Our aim is to describe the structure of the orbit space D/ G in terms of the structure of Q/ G. If the action of G on Q is free, then Q is a left principal fibre bundle with structure group G and base manifold Q/ G. We denote by $${Q[\mathfrak{g}]}$$ the adjoint bundle and by $${Q[\mathfrak{g}^{*}]}$$ the coadjoint bundle of Q. We show that for a free and proper action, D/ G is a maximal isotropic subbundle of the direct sum $${Q[\mathfrak{g}] \oplus Q[\mathfrak{g}^{*}] \oplus T(Q/G) \oplus T^{*}(Q/G)}$$. If the action of G on Q is proper but not free, then the orbit space Q/ G is stratified. For each stratum of Q/ G, the restriction of D/ G to the stratum can be described in the same way as for a free and proper action. [ABSTRACT FROM AUTHOR]

Identities for the curvature tensor of the Levi-Cività connection on an almost para-cosymplectic manifold are proved. Elements of harmonic theory for almost product structures are given and a Bochner-type formula for the leaves of the canonical foliation is established. [ABSTRACT FROM AUTHOR]

We demonstrate the usefulness of anholonomic frames in the contexts of nonholonomic and vakonomic systems. We take a consistently differential-geometric approach. As an application, we investigate the conditions under which the dynamics of the two systems will be consistent. A few illustrative examples confirm the results. [ABSTRACT FROM AUTHOR]

We study pairs of structures, such as the Poisson-Nijenhuis structures, on the tangent bundle of a manifold or, more generally, on a Lie algebroid or a Courant algebroid. These composite structures are defined by two of the following, a closed 2-form, a Poisson bivector or a Nijenhuis tensor, with suitable compatibility assumptions. We establish the relationships between PN-, P Ω- and Ω N-structures. We then show that the non-degenerate Monge-Ampère structures on 2-dimensional manifolds satisfying an integrability condition provide numerous examples of such structures, while in the case of 3-dimensional manifolds, such Monge-Ampère operators give rise to generalized complex structures or generalized product structures on the cotangent bundle of the manifold. [ABSTRACT FROM AUTHOR]

In this article the geometric structure of Hamiltonian systems with ports is clarified. Particularly, it is shown that the theory of Hamiltonian systems with ports fits into the paradigm that the state space of a mechanical system should be modeled by a manifold M endowed with a(n almost) Dirac structure in a Courant algebroid. Here, the Courant algebroid is TM ⊕ TM* ⊕ E ⊕ E*, where E is a vector bundle over M endowed with a flat covariant derivative ∇, and the bracket is Further, the integrability of almost Dirac structures D⊂ TM ⊕ TM* ⊕ E ⊕ E* given by the graph of $\bigl(\fontsize{7.6}{7.6}\selectfont \begin{array}{c@{\ }c}\omega &-(A\circ \omega)^{*}\\A\circ \omega &0\end{array}\bigr)$ for a two-form ω on M or the cograph of $\bigl(\fontsize{7.6}{7.6}\selectfont \begin{array}{c@{\ }c}\pi &-A^{*}\\A&0\end{array}\bigr)$ for a bivector field π on M is discussed, where the bundle map A: TM*→ E models the port. Moreover, conditions are obtained to guarantee the integrability of almost Dirac structures induced by closing ports or by power-conserving interconnections. These conditions are formulated in terms of the covariant Schouten bracket defined in the Appendix. [ABSTRACT FROM AUTHOR]

The purpose of this article is to describe Lagrangian mechanics for constrained systems on Lie algebroids, a natural framework which covers a wide range of situations (systems on Lie groups, quotients by the action of a Lie group, standard tangent bundles ...). In particular, we are interested in two cases: singular Lagrangian systems and vakonomic mechanics (variational constrained mechanics). Several examples illustrate the interest of these developments. [ABSTRACT FROM AUTHOR]

Generalized non-holonomic mechanical systems are analyzed from a geometric point of view. The existence and uniqueness of solutions, D’Alembert principle, Gauss principle of minimal constraint, the non-holonomic momentum and Gibbs–Appell equations for such systems are studied in an invariant Lagrangian framework. [ABSTRACT FROM AUTHOR]

Let Θ be a smooth compact oriented manifold without boundary, imbedded in a Euclidean space E s, and let γ be a smooth map of Θ into a Riemannian manifold Λ. An unknown state θ ∈ Θ is observed via X = θ + εξ, where ε > 0 is a small parameter and ξ is a white Gaussian noise. For a given smooth prior λ on Θ and smooth estimators g( X) of the map γ we derive a second-order asymptotic expansion for the related Bayesian risk. The calculation involves the geometry of the underlying spaces Θ and Λ, in particular, the integration-by-parts formula. Using this result, a second-order minimax estimator of γ is found based on the modern theory of harmonic maps and hypo-elliptic differential operators. [ABSTRACT FROM AUTHOR]

We introduce the notion of a symplectic Lie affgebroid and their Lagrangian submanifolds in order to describe the Lagrangian (Hamiltonian) dynamics on a Lie affgebroid in terms of this type of structures. Several examples are discussed. [Copyright &y& Elsevier]

Using the mechanical principle, the theory of modern geometry and advanced calculus, Hamiltonian mechanics was generalized to Kähler manifolds, and the Hamiltonian mechanics on Kähler manifolds was established. Then the complex mathematical aspect of Hamiltonian vector field and Hamilton’s equations was obtained, and so on. [ABSTRACT FROM AUTHOR]

Based on modern differential geometry, the symplectic structure of a Birkhoffian system which is an extension of conservative and nonconservative systems is analyzed. An integral invariant of Poincaré-Cartan's type is constructed for Birkhoffian systems. Finally, one-dimensional damped vibration is taken as an illustrative example and an integral invariant of Poincaré's type is found. [ABSTRACT FROM AUTHOR]