On the geometry of Lagrangian one-forms

This work is primarily concerned with various aspects of Lagrangian and Hamiltonian mechanics. These different aspects are related in a somewhat complicated way, which is clarified by the introduction of the unifying concept of a jet bundle. Here, bundles are first considered in some generality and then jet bundles are introduced and their bundle structure investigated especially with a view to formulating Lagrangian and Hamiltonian mechanics invariantly. The second chapter is devoted to a discussion of the two Schouten brackets and it is explained why each is of importance in mechanics. The symmetric Schouten bracket enables the notion of a killing tensor to be defined on a Riemannian or pseudo-Riemannian manifold. A theorem is proved which gives an upper bound on the dimension of the space of killing tensors of a fixed rank. In chapter 3 Hamilton-Jacobi theory and a version of Noether's theorem are presented from a m o d e m viewpoint. Then conditions are obtained which entail the existence of a constant of motion which is polynomial in momenta. These conditions are used to construct several classical Hamiltonians with hidden symmetries - the usage of the latter term is briefly justified. Most importantly, all systems with two degrees of freedom which have a quadratic integral independent of the Hamiltonian are characterized. In chapter 4 the geometrical properties of TM and the closely related space J1 (IR,M) are investigated. It is shown that J1(IR,M) has an intrinsically defined 1-1 tensor in analogy to TM. The behaviour of these tensors under diffeomorphisms is investigated. The chapter concludes with a discussion of the various notions of symmetry in Lagrangian theory. Chapter 5 begins with a review of symplectic geometry and continues with the definition and examination of contact manifolds in the same spirit as symplectic geometry. In particular, contact diffeomorphisms are described on the space J1(IR,M). Finally, two theorems are given which endeavour to explain the general principle enabling the Lie algebra of a subalgebra of vector fields to be transferred to a collection of forms.

Characterisation of metal combustion with DUST code

Published in two volumes, this is the first book to provide a thorough and systematic explanation of symplectic topology, and the analytical details and techniques used in applying the machinery arising from Floer theory as a whole. Volume 2 provides a comprehensive introduction to both Hamiltonian Floer theory and Lagrangian Floer theory, including many examples of their applications to various problems in symplectic topology. The first volume covered the basic materials of Hamiltonian dynamics and symplectic geometry and the analytic foundations of Gromov's pseudoholomorphic curve theory. Symplectic Topology and Floer Homology is a comprehensive resource suitable for experts and newcomers alike.

We consider the system of nonlinear differential equations governing shallow water waves over a uniform or sloping bottom. By using the hodograph method we construct solutions, conservation laws, and Böcklund transformations for these equations. We show that these constructions are canonical relative to a symplectic form introduced by Manin.

Phase field-elasticity analysis of austenite–martensite phase transformation at the nanoscale: Finite element modeling

Newton–Cotes formulae for long-time integration

This study explores the multistage differential transformation method (MsDTM) as an efficient approach for solving non-autonomous differential equations. The proposed method demonstrates wide-ranging applicability in fields such as computer graphics, quantum optics, biomathematics, and image processing. Specifically, the Bloch equations, which describe the interaction of a spin-1/2 system (or two-level atom) with a mono- or bichromatic laser field in the presence of an off-resonant broad-band squeezed vacuum (SV), are examined. As a system of non-autonomous ordinary differential equations, the Bloch model captures the quantum dynamics between matter and electromagnetic fields, offering insights into more complex and experimentally relevant models. The MsDTM is employed to obtain numerical solutions with high precision and computational efficiency, outperforming the classical 4th-order Runge-Kutta (RK4) method. A key advantage of the MsDTM is its adaptability; its accuracy can be further enhanced by either increasing the number of iterations or refining the time-step in the numerical scheme. Consequently, MsDTM emerges as a robust tool for computing solutions to a broad class of non-autonomous equations.

In this second paper of a series of papers, we explore the difference discrete versions for the Euler–Lagrange cohomology and apply them to the symplectic or multisymplectic geometry and their preserving properties in both the Lagrangian and Hamiltonian formalisms for discrete mechanics and field theory in the framework of multi-parameter differential approach. In terms of the difference discrete Euler–Lagrange cohomological concepts, we show that the symplectic or multisymplectic geometry and their difference discrete structure-preserving properties can always be established not only in the solution spaces of the discrete Euler–Lagrange or canonical equations derived by the difference discrete variational principle but also in the function space in each case if and only if the relevant closed Euler–Lagrange cohomological conditions are satisfied.

Стаття описує каркасну установку паралельної конструкції. Наведено формули механіки та руху каркасної установки. За допомогою залежностей визначені положення центру платформи, кут відхилення норми від вертикальної осі. Описано також співвідношення координат структури. Проаналізовано рівняння динаміки для многопріводних систем. За допомогою рівнянь Лагранжа отримана система диференціальних рівнянь, що описують оптимальне по відхиленню від заданої траєкторії рух маніпулятора.

We lay down the foundations of the theory of Poisson vertex algebras aimed at its applications to integrability of Hamiltonian partial differential equations. Such an equation is called integrable if it can be included in an infinite hierarchy of compatible Hamiltonian equations, which admit an infinite sequence of linearly independent integrals of motion in involution. The construction of a hierarchy and its integrals of motion is achieved by making use of the so called Lenard scheme. We find simple conditions which guarantee that the scheme produces an infinite sequence of closed 1-forms $$\omega_j, j \in {\mathbb {Z}}_+$$ , of the variational complex Ω. If these forms are exact, i.e., ω j are variational derivatives of some local functionals ∫ h j , then the latter are integrals of motion in involution of the hierarchy formed by the corresponding Hamiltonian vector fields. We show that the complex Ω is exact, provided that the algebra of functions is $$\fancyscript {V}$$ is “normal”; in particular, for arbitrary $$\fancyscript {V}$$ , any closed form in Ω becomes exact if we add to $$\fancyscript {V}$$ a finite number of antiderivatives. We demonstrate on the examples of the KdV, HD and CNW hierarchies how the Lenard scheme works. We also discover a new integrable hierarchy, which we call the CNW hierarchy of HD type. Developing the ideas of Dorfman, we extend the Lenard scheme to arbitrary Dirac structures, and demonstrate its applicability on the examples of the NLS, pKdV and KN hierarchies.

Necessary and sufficient conditions for the existence of a Lagrangian in field theory. I. Variational approach to self-adjointness for tensorial field equations

The representations of Galilean generators are constructed on a space where both position and momentum coordinates are noncommutating operators. A dynamical model invariant under noncommutative phase space transformations is constructed. The Dirac brackets of this model reproduce the original noncommutative algebra. Also, the generators in terms of noncommutative phase-space variables are abstracted from this model in a consistent manner. Finally, the role of Jacobi identities is emphasized to produce the noncommuting structure that occurs when an electron is subjected to a constant magnetic field and Berry curvature.

In this paper, a modeling and simulating method is presented for the planar rigid multi-body system with friction-affected translational joints. The dynamical equations of the non-smooth system are formulated by the first kind of Lagrange's equations in Cartesian coordinate system. The normal forces are expressed by the Lagrange multipliers, and the frictional forces are characterized by Coulomb's friction law. The state transition problem of stick-slip and normal forces of bilateral constraints is solved by an improved trial-and-error method. Finally, the planar rigid multi-body system with two translational joints is considered as a illustrative application example.

It is well known that the traditional modeling theories adopt the Cartesian coordinates to establish the dynamic equation of solar array system. The order of this equation is often very high, which is inconvenient for controller design. The Cartesian coordinates are difficult to be measured in practice. In this paper, the joint coordinates are used as generalized variables to establish the dynamic equation of the solar array system. The single direction recursive construction method and the Jourdain’s velocity variation principle are used in modeling the system. The order of the established equation is lower, and the joint coordinates are easily measured in practice. Besides the dynamics modeling, joint friction and control design are extensively studied in this paper too. A three-dimensional revolute joint model is introduced, and the contribution of joint friction to the dynamic equation of the system is deduced. A fuzzy PD controller is designed to eliminate the drift of spacecraft caused by the deployment of solar array. In the end of this paper, numerical simulations are carried out to analyze the deployment dynamics and control effect of the solar array system with friction.

We present the arising of the Fick cross-diffusion system of equations for fluid mixtures from the multi-species Boltzmann equation in a rigorous manner in Sobolev spaces. To this end, we formally show that, in a diffusive scaling, the hydrodynamical limit of the kinetic system is the Fick model supplemented with a closure relation and we give explicit formulae for the macroscopic diffusion coefficients from the Boltzmann collision operator. Then, we provide a perturbative Cauchy theory in Sobolev spaces for the constructed Fick system, which turns out to be a dilated parabolic equation. We finally prove the stability of the system in the Boltzmann equation, ensuring a rigorous derivation between the two models.