Kepler Problem in Lagrangian Formulation Discussed from Topological Viewpoint

2005 ◽  
Vol 22 (6) ◽  
pp. 1303-1304 ◽  
Author(s):  
Xu Gong-Ou ◽  
Xu Ming-Jie
Keyword(s):  
Kepler Problem ◽  
Lagrangian Formulation

scholarly journals Hamiltonian structures for integrable hierarchies of Lagrangian PDEs

2021 ◽  
Vol Volume 1 ◽  
Author(s):  
Mats Vermeeren
Keyword(s):  
Differential Equations ◽  
Integrable Hierarchies ◽  
Kepler Problem ◽  
Lagrange Function ◽  
Lagrangian Formulation ◽  
Integrable Hierarchy ◽  
Poisson Brackets ◽  
Exterior Derivative ◽  
Hamiltonian Structures ◽  
Fundamental Object

Many integrable hierarchies of differential equations allow a variational description, called a Lagrangian multiform or a pluri-Lagrangian structure. The fundamental object in this theory is not a Lagrange function but a differential $d$-form that is integrated over arbitrary $d$-dimensional submanifolds. All such action integrals must be stationary for a field to be a solution to the pluri-Lagrangian problem. In this paper we present a procedure to obtain Hamiltonian structures from the pluri-Lagrangian formulation of an integrable hierarchy of PDEs. As a prelude, we review a similar procedure for integrable ODEs. We show that exterior derivative of the Lagrangian $d$-form is closely related to the Poisson brackets between the corresponding Hamilton functions. In the ODE (Lagrangian 1-form) case we discuss as examples the Toda hierarchy and the Kepler problem. As examples for the PDE (Lagrangian 2-form) case we present the potential and Schwarzian Korteweg-de Vries hierarchies, as well as the Boussinesq hierarchy.

scholarly journals Approaching the self-dual point of the sinh-Gordon model

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Robert Konik ◽  
Márton Lájer ◽  
Giuseppe Mussardo
Keyword(s):  
Small Volume ◽  
Zero Mode ◽  
Form Factors ◽  
Lagrangian Formulation ◽  
The Self ◽  
Quantum Mechanical ◽  
Exact Form ◽  
Coupling Constant ◽  
Self Duality ◽  
Hyperbolic Cosine

Abstract One of the most striking but mysterious properties of the sinh-Gordon model (ShG) is the b → 1/b self-duality of its S-matrix, of which there is no trace in its Lagrangian formulation. Here b is the coupling appearing in the model’s eponymous hyperbolic cosine present in its Lagrangian, cosh(bϕ). In this paper we develop truncated spectrum methods (TSMs) for studying the sinh-Gordon model at a finite volume as we vary the coupling constant. We obtain the expected results for b ≪ 1 and intermediate values of b, but as the self-dual point b = 1 is approached, the basic application of the TSM to the ShG breaks down. We find that the TSM gives results with a strong cutoff Ec dependence, which disappears according only to a very slow power law in Ec. Standard renormalization group strategies — whether they be numerical or analytic — also fail to improve upon matters here. We thus explore three strategies to address the basic limitations of the TSM in the vicinity of b = 1. In the first, we focus on the small-volume spectrum. We attempt to understand how much of the physics of the ShG is encoded in the zero mode part of its Hamiltonian, in essence how ‘quantum mechanical’ vs ‘quantum field theoretic’ the problem is. In the second, we identify the divergencies present in perturbation theory and perform their resummation using a supra-Borel approximate. In the third approach, we use the exact form factors of the model to treat the ShG at one value of b as a perturbation of a ShG at a different coupling. In the light of this work, we argue that the strong coupling phase b > 1 of the Lagrangian formulation of model may be different from what is naïvely inferred from its S-matrix. In particular, we present an argument that the theory is massless for b > 1.

Multibody Dynamic Model of Web Guiding System With Moving Web

2010 ◽  
Vol 132 (5) ◽  
Author(s):  
Lei Yu ◽  
Zhihua Zhao ◽  
Gexue Ren
Keyword(s):  
Dynamic Model ◽  
Lateral Displacement ◽  
Dynamical Behavior ◽  
Contact Forces ◽  
Lagrangian Formulation ◽  
Pi Control ◽  
Control Law ◽  
Reference Surface ◽  
Multibody Dynamic ◽  
Guiding System

In this paper, a multibody dynamic model is established to simulate the dynamics and control of moving web with its guiding system, where the term moving web is used to describe thin materials, which are manufactured and processed in a continuous, flexible strip form. In contrast with available researches based on Eulerian description and beam assumption, webs are described by Lagrangian formulation with the absolute nodal coordinate formulation (ANCF) plate element, which is based on Kirchhoff’s assumptions that material normals to the original reference surface remain straight and normal to the deformed reference surface, and the nonlinear elasticity theory that accounts for large displacement, large rotation, and large deformation. The rollers and guiding mechanism are modeled as rigid bodies. The distributed frictional contact forces between rollers and web are considered by Hertz contact model and are evaluated by Gauss quadrature. The proportional integral (PI) control law for web guiding is also embedded in the multibody model. A series of simulations on a typical web-guide system is carried out using the multibody dynamics approach for web guiding system presented in this study. System dynamical information, for example, lateral displacement, stress distribution, and driving moment for web guiding, are obtained from simulations. Parameter sensitivity analysis illustrates the effect of influence variables and effectiveness of the PI control law for lateral movement control of web that are verified under different gains. The present Lagrangian formulation of web element, i.e., ANCF element, is not only capable of describing the large movement and deformation but also easily adapted to capture the distributed contact forces between web and rollers. The dynamical behavior of the moving web can be accurately described by a small number of ANCF thin plate elements. Simulations carried out in this paper show that the present approach is an effective method to assess the design of web guiding system with easily available desktop computers.

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