The application of exterior differential forms in variational problems on manifolds

Exterior and evolutionary skew-symmetric differential forms and their role in mathematical physics

At present the theory of skew-symmetric exterior differential forms has been developed. The closed exterior forms possess the invariant properties that are of great importance. The operators of the exterior form theory lie at the basis of the differential and integral operators of the field theory. However, the theory of exterior forms, being invariant one, does not answer the questions related to the evolutionary processes. In the work the readers are introduced to the skew-symmetric differential forms that possess evolutionary properties. They were called evolutionary ones. The radical disti…

Geometric Structures and Differential Operators on Manifolds Having Super tangent Bundle

In this paper, we introduce the notion of a super tangent bundle of a manifold, and extend the basic notions of differential geometry such as differential forms, exterior derivation, connection, metric and divergence on manifolds that equipped with a super tangent bundle.

传递李代数胚上外微分系统的Cartan-Kähler定理

The Cartan-Kähler theorem for exterior differential systems on transitive Lie algebroids

将外微分系统推广到李代数胚，在锚映射满射条件下给出两个Cartan-Kähler定理版本，并应用于变分学的不变逆问题。

Dirac constraints in field theory and exterior differential systems

The usual treatment of a (first order) classical field theory such as electromagnetism has a little drawback: It has a primary constraint submanifold that arise from the fact that the dynamics is governed by the antisymmetric part of the jet variables. So it is natural to ask if there exists a formulation of this kind of field theories which avoids this problem, retaining the versatility of the known approach. The following paper deals with a family of variational problems, namely, the so called non standard variational problems, which intends to capture the data necessary to set up such a for…

Reduction for constrained variational problems on 3D null curves

We consider the optimal control problem for null curves in de Sitter 3-space defined by a functional which is linear in the curvature of the trajectory. We show how techniques based on the method of moving frames and exterior differential systems, coupled with the reduction procedure for systems with a Lie group of symmetries lead to the integration by quadratures of the extremals. Explicit solutions are found in terms of elliptic functions and integrals.

Differential complexes and exterior calculus

In this paper we present a new theory of calculus over $k$-dimensional domains in a smooth $n$-manifold, unifying the discrete, exterior, and continuum theories. The calculus begins at a single point and is extended to chains of finitely many points by linearity, or superposition. It converges to the smooth continuum with respect to a norm on the space of ``pointed chains,'' culminating in the chainlet complex. Through this complex, we discover a broad theory of coordinate free, multivector analysis in smooth manifolds for which both the classical Newtonian calculus and the Cartan exterior cal…

Variational Problems in Elastic Theory of Biomembranes, Smectic-a Liquid Crystals, and Carbon Related Structures

After a brief introduction to several variational problems in the study of shapes of thin thickness structures, we deal with variational problems on 2-dimensional surface in 3-dimensional Euclidian space by using exterior differential forms. The morphological problems of lipid bilayers and stabilities of cell membranes are also discussed. The key point is that the first and the second order variations of the free energy determine equilibrium shapes and mechanical stabilities of structures.

Stocastic variational principle on riemannian manifolds

Some little considerations concerning the application of the Theory of Dirichlet Forms to stocastic variational principle on riemannian manifolds are performed

Discrete Exterior Geometry Approach to Structure-Preserving Discretization of Distributed-Parameter Port-Hamiltonian Systems

This paper addresses the issue of structure-preserving discretization of open distributed-parameter systems with Hamiltonian dynamics. Employing the formalism of discrete exterior calculus, we introduce a simplicial Dirac structure as a discrete analogue of the Stokes-Dirac structure and demonstrate that it provides a natural framework for deriving finite-dimensional port-Hamiltonian systems that emulate their infinite-dimensional counterparts. The spatial domain, in the continuous theory represented by a finite-dimensional smooth manifold with boundary, is replaced by a homological manifold-l…

混沌拱门，一阶平均运动共振的异宿连接以及太阳-木星系统中小天体的混沌输运

Arches of chaos, heteroclinic connections of first-order MMRs and the chaotic transport of small bodies in the Sun-Jupiter system

研究太阳-木星平面限制三体问题中主要平均运动共振的不稳定周期轨道稳定和不稳定流形之间的异宿连接，计算一阶内共振2:1、3:2和外共振2:3的流形，并通过FLI图展示混沌鞍点结构，解释小行星轨道平面的混沌拱门现象。

Symmetries of Helmholtz forms and globally variational dynamical forms

Invariance properties of classes in the variational sequence suggested to Krupka et al. the idea that there should exist a close correspondence between the notions of variationality of a differential form and invariance of its exterior derivative. It was shown by them that the invariance of a closed Helmholtz form of a dynamical form is equivalent with local variationality of the Lie derivative of the dynamical form, so that the latter is locally the Euler--Lagrange form of a Lagrangian. We show that the corresponding local system of Euler--Lagrange forms is variationally equivalent to a globa…

闵可夫斯基流形上旋量变量的p-微分形式导论

An Introduction to the $p$-Differential Forms in Spinor Variables on the Minkowskian Manifold

本文研究闵可夫斯基流形上旋量变量的p-微分形式，基于旋量空间与克利福德代数建立时空对应，构造旋量形式并表述狄拉克算子，推广至非单演函数，验证其一致性及应用潜力。

The Canonical Differential Complex of Poisson Manifold and Distributions

We study variuos homological structures associated with Poisson algebra, the canonical differential complex for singular Poisson structure and the analogue of the star operator for such manifolds. Give the interpretation of the classical Koszul differential of exterior forms, as the supercommutator with some second order element. Describe the space of invariant distributions on manifold with singular Poisson structure.

Role of exterior and evolutionary skew-symmetric differential forms in mathematical physics

A role of skew-symmetric differential forms in mathematical physics relates to the fact that they reflect the properties of conservation laws. The closed exterior forms correspond to the conservation laws for physical fields, whereas the evolutionary forms correspond to the conservation laws for material media. Skew-symmetric differential forms can describe a conjugacy of any objects (that correspond to the conservation laws). The closed exterior forms describe conjugated objects. And the evolutionary forms, whose basis are deforming manifolds, describe the process of conjugating objects and o…

Variational problems on Riemannian manifolds with constrained accelerations

We introduce variational problems on Riemannian manifolds with constrained acceleration and derive necessary conditions for normal extremals in the constrained variational problem. The problem consists on minimizing a higher-order energy functional, among a set of admissible curves defined by a constraint on the covariant acceleration. In addition, we use this framework to address the elastic splines problem with obstacle avoidance in the presence of this type of contraints.

A variational principle for computing slow invariant manifolds in dissipative dynamical systems

A key issue in dimension reduction of dissipative dynamical systems with spectral gaps is the identification of slow invariant manifolds. We present theoretical and numerical results for a variational approach to the problem of computing such manifolds for kinetic models using trajectory optimization. The corresponding objective functional reflects a variational principle that characterizes trajectories on, respectively near, slow invariant manifolds. For a two-dimensional linear system and a common nonlinear test problem we show analytically that the variational approach asymptotically identi…

Z_3-graded exterior differential calculus and gauge theories of higher order

We present a possible generalization of the exterior differential calculus, based on the operator d such that d^3=0, but d^2\not=0. The first and second order differentials generate an associative algebra; we shall suppose that there are no binary relations between first order differentials, while the ternary products will satisfy the cyclic relations based on the representation of cyclic group Z_3 by cubic roots of unity. We shall attribute grade 1 to the first order differentials and grade 2 to the second order differentials; under the associative multiplication law the grades add up modulo …

Cohomology of the variational complex in BRST theory

We show that cohomology of the variational complex in field-antifield BRST theory on an arbitrary manifold is equal to the de Rham cohomology of this manifold.

Variational problem for Hamiltonian system on so(k, m) Lie-Poisson manifold and dynamics of semiclassical spin

We describe the procedure for obtaining Hamiltonian equations on a manifold with $so(k, m)$ Lie-Poisson bracket from a variational problem. This implies identification of the manifold with base of a properly constructed fiber bundle embedded as a surface into the phase space with canonical Poisson bracket. Our geometric construction underlies the formalism used for construction of spinning particles in [24-27], and gives precise mathematical formulation of the oldest idea about spin as the "inner angular momentum".