Pawel Urbanski
An affine Cartan calculus is developed. The concepts of special affine bundles and special affine duality are introduced. The canonical isomorphisms, fundamental for Lagrangian and Hamiltonian formulations of the dynamics in the affine setting are proved.
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Main ideas of the differential geometry on affine bundles are presented. Affine counterparts of Lie algebroid and Poisson structures are introduced and discussed. The developed concepts are applied in a frame-independent formulation of the time-dependent and the Newtonian mechanics.
A frame independent formulation of analytical mechanics in the Newtonian space-time is presented. The differential geometry of affine values i.e., the differential geometry in which affine bundles replace vector bundles and sections of one dimensional affine bundles replace functions on manifolds, is used. Lagrangian and hamiltonian generating objects, together with the Legendre transformation independent on inertial frame are constructed.
Classical relativistic nonholonomic mechanics and time-dependent $G$-Chaplygin systems with affine constraints
该研究从活动标架视角研究经典含时非完整拉格朗日力学的相对论表述，引入带仿射约束的含时$G$-Chaplygin系统，通过修正Chaplygin乘子法，证明平衡圆盘在变半径竖直圆上无滑动滚动的约化系统在新时间下为常规拉格朗日系统，为含时约束哈密顿化研究迈出第一步。
Invariance of subbundles and nonholonomic trajectories
本文研究非完整力学与约束变分力学比较中产生的不变仿射子丛，通过Spencer上同调判定形式可积性，给出求最大不变子丛的迭代公式，并以斜面滚动圆盘为例验证。
A linear connection is associated to a nonlinear connection on a vector bundle by a linearization procedure. Our definition is intrinsic in terms of vector fields on the bundle. For a connection on an affine bundle our procedure can be applied after homogenization and restriction. Several applications in Classical Mechanics are provided.
Based on ideas of W. M. Tulczyjew, a geometric framework for a frame-independent formulation of different problems in analytical mechanics is developed. In this approach affine bundles replace vector bundles of the standard description and functions are replaced by sections of certain affine line bundles called AV-bundles. Categorial constructions for affine and special affine bundles as well as natural analogs of Lie algebroid structures on affine bundles (Lie affgebroids) are investigated. One discovers certain Lie algebroids and Lie affgebroids canonically associated with an AV-bundle which…
A theory of double affine and special double affine bundles, i.e. differential manifolds with two compatible (special) affine bundle structures, is developed as an affine counterpart of the theory of double vector bundles. The motivation and basic examples come from Analytical Mechanics, where double affine bundles have been recognized as a proper geometrical tool in a frame-independent description of many important systems. Different approaches to the (special) double affine bundles are compared and carefully studied together with the problems of double vector bundle models and hulls, duality…
A general, consistent and complete framework for geometrical formulation of mechanical systems is proposed, based on certain structures on affine bundles (affgebroids) that generalize Lie algebras and Lie algebroids. This scheme covers and unifies various geometrical approaches to mechanics in the Lagrangian and Hamiltonian pictures, including time-dependent lagrangians and hamiltonians. In our approach, lagrangians and hamiltonians are, in general, sections of certain $\R$-principal bundles, and the solutions of analogs of Euler-Lagrange equations are curves in certain affine bundles. The cor…
This paper presents a geometric description of Lagrangian and Hamiltonian systems on Lie affgebroids subject to affine nonholonomic constraints. We define the notion of nonholonomically constrained system, and characterize regularity conditions that guarantee that the dynamics of the system can be obtained as a suitable projection of the unconstrained dynamics. It is shown that one can define an almost aff-Poisson bracket on the constraint AV-bundle, which plays a prominent role in the description of nonholonomic dynamics. Moreover, these developments give a general description of nonholonomic…
The aim of the present text is twofold: to provide a compendium of Lagrangian and Hamiltonian geometries and to introduce and investigate new analytical Mechanics: Finslerian, Lagrangian and Hamiltonian. The fundamental equations (or evolution equations) of these Mechanics are derived from the variational calculus applied to the integral of action and these can be studied by using the methods of Lagrangian or Hamiltonian geometries. More general, the notions of higher order Lagrange or Hamilton spaces have been introduced and developed by the present author. The applications led to the notions…
Geometry in global coordinates in mechanics and optimal transport
该研究提出嵌入内积空间流形的几何量全局坐标表示方法，推导向量丛仿射联络的高斯-科达齐方程，可结合数值框架应用，并证明了Kim-McCann度量交叉曲率的相关性质。
Chern-Simons field theory on the general affine group, $3d$-gravity and the extension of Cartan connections
该研究从几何力学角度探究结构群为一般仿射群时3d引力与Chern-Simons场论的对应，通过推广Cartan联络下的联络延拓约化概念，建立了主纤维丛上Chern-Simons型变分问题与常规理论的关联。
Natural analogs of Lie brackets on affine bundles are studied, based on natural examples from differential geometry and analytical mechanics. In particular, a close relation to Lie algebroids and, by a sort of duality, to affine analogs of Poisson structures is established as well as affine versions of the complete lift of a section and the Cartan exterior calculus.
The book contains a collection of works on Riemann-Cartan and metric-affine manifolds provided with nonlinear connection structure and on generalized Finsler-Lagrange and Cartan-Hamilton geometries and Clifford structures modelled on such manifolds. The choice of material presented has evolved from various applications in modern gravity and geometric mechanics and certain generalizations to noncommutative Riemann-Finsler geometry. The authors develop and use the method of anholonomic frames with associated nonlinear connection structure and apply it to a number of concrete problems: constructi…
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