scholarly journals Fractional Euler Lagrange Equations for Irregular Lagrangian with Holonomic Constraints

2018 ◽  
Vol 09 (08) ◽  
pp. 1690-1696
Author(s):  
Ola A. Jarab’ah
Keyword(s):  
Lagrange Equations ◽  
Holonomic Constraints

Constrained Lagrangian Mechanics

2018 ◽  
Author(s):  
Peter Mann
Keyword(s):  
Lagrange Equation ◽  
Constrained Systems ◽  
Lagrangian Mechanics ◽  
Lagrange Equations ◽  
Minimal Set ◽  
Holonomic Constraints ◽  
Constrained Optimisation ◽  
Constraint Equations ◽  
Linearly Independent ◽  
Set Up

This chapter builds on the previous two chapters to tackle constrained systems, using Lagrangian mechanics and constrained variations. The first section deals with holonomic constraint equations using Lagrange multipliers; these can be used to reduce the number of coordinates until a linearly independent minimal set is obtained that describes a constraint surface within configuration space, so that Lagrange equations can be set up and solved. Motion is understood to be confined to a constraint submanifold. The variational formulation of non-holonomic constraints is then discussed to derive the vakonomic formulation. These erroneous equations are then compared to the central Lagrange equation, and the precise nature of the variations used in each formulation is investigated. The vakonomic equations are then presented in their Suslov form (Suslov–vakonomic form) in an attempt to reconcile the two approaches. In addition, the structure of biological membranes is framed as a constrained optimisation problem.

scholarly journals Power and beauty of the Lagrange equations

2020 ◽  
Vol 17 (1 Jan-Jun) ◽  
pp. 47
Author(s):  
Luis De la Peña ◽  
Ana María Cetto ◽  
Andrea Valdés-Hernández
Keyword(s):  
Variational Principle ◽  
Equations Of Motion ◽  
Classical Mechanics ◽  
Integrals Of Motion ◽  
Lagrange Equations ◽  
Holonomic Constraints ◽  
Point Particles ◽  
Lagrange Formulation ◽  
Classical Particles ◽  
Advanced Treatments

The Lagrangian formulation of the equations of motion for point particles isusually presented in classical mechanics as the outcome of a series ofinsightful algebraic transformations or, in more advanced treatments, as theresult of applying a variational principle. In this paper we stress two mainreasons for considering the Lagrange equations as a fundamental descriptionof the dynamics of classical particles. Firstly, their structure can benaturally disclosed from the existence of integrals of motion, in a waythat, though elementary and easy to prove, seems to be less popular--or less frequently made explicit-- than others insupport of the Lagrange formulation. The second reason is that the Lagrangeequations preserve their form in \emph{any} coordinate system --even in moving ones, if required. Their covariant nature makes themparticularly suited to deal with dynamical problems in curved spaces orinvolving (holonomic) constraints. We develop the above and related ideas inclear and simple terms, keeping them throughout at the level of intermediatecourses in classical mechanics. This has the advantage of introducing sometools and concepts that are useful at this stage, while they may also serveas a bridge to more advanced courses.

Deterministic Mechanism of Irreversibility

2018 ◽  
Vol 14 (3) ◽  
pp. 5708-5733 ◽  
Author(s):  
Vyacheslav Michailovich Somsikov
Keyword(s):  
Internal Energy ◽  
Hamiltonian Systems ◽  
Classical Mechanics ◽  
Motion Equation ◽  
Holonomic Constraints ◽  
Dissipative Processes ◽  
Motion Energy ◽  
Analytical Review ◽  
History Of ◽  
Material Points

The analytical review of the papers devoted to the deterministic mechanism of irreversibility (DMI) is presented. The history of solving of the irreversibility problem is briefly described. It is shown, how the DMI was found basing on the motion equation for a structured body. The structured body was given by a set of potentially interacting material points. The taking into account of the body’s structure led to the possibility of describing dissipative processes. This possibility caused by the transformation of the body’s motion energy into internal energy. It is shown, that the condition of holonomic constraints, which used for obtaining of the canonical formalisms of classical mechanics, is excluding the DMI in Hamiltonian systems. The concepts of D-entropy and evolutionary non-linearity are discussed. The connection between thermodynamics and the laws of classical mechanics is shown. Extended forms of the Lagrange, Hamilton, Liouville, and Schrödinger equations, which describe dissipative processes, are presented.

scholarly journals Second-order integrable Lagrangians and WDVV equations

2021 ◽  
Vol 111 (2) ◽  
Author(s):  
E. V. Ferapontov ◽  
M. V. Pavlov ◽  
Lingling Xue
Keyword(s):  
Lagrangian Density ◽  
Theta Functions ◽  
Second Order ◽  
Elementary Functions ◽  
Lagrange Equations ◽  
Wdvv Equations ◽  
Integrability Conditions ◽  
Jacobi Theta Functions

AbstractWe investigate the integrability of Euler–Lagrange equations associated with 2D second-order Lagrangians of the form $$\begin{aligned} \int f(u_{xx},u_{xy},u_{yy})\ \mathrm{d}x\mathrm{d}y. \end{aligned}$$ ∫ f ( u xx , u xy , u yy ) d x d y . By deriving integrability conditions for the Lagrangian density f, examples of integrable Lagrangians expressible via elementary functions, Jacobi theta functions and dilogarithms are constructed. A link of second-order integrable Lagrangians to WDVV equations is established. Generalisations to 3D second-order integrable Lagrangians are also discussed.

scholarly journals Introduction to Macroscopic Optimal Design in the Mechanics of Composite Materials and Structures

2021 ◽  
Vol 5 (2) ◽  
pp. 36
Author(s):  
Aleksander Muc
Keyword(s):  
Composite Materials ◽  
Composite Structures ◽  
Constitutive Relations ◽  
A Priori ◽  
Chemical Properties ◽  
Composite Plates ◽  
Failure Criteria ◽  
Lagrange Equations ◽  
Homogenization Methods ◽  
Mechanics Of Composite Materials

The main goal of building composite materials and structures is to provide appropriate a priori controlled physico-chemical properties. For this purpose, a strengthening is introduced that can bear loads higher than those borne by isotropic materials, improve creep resistance, etc. Composite materials can be designed in a different fashion to meet specific properties requirements.Nevertheless, it is necessary to be careful about the orientation, placement and sizes of different types of reinforcement. These issues should be solved by optimization, which, however, requires the construction of appropriate models. In the present paper we intend to discuss formulations of kinematic and constitutive relations and the possible application of homogenization methods. Then, 2D relations for multilayered composite plates and cylindrical shells are derived with the use of the Euler–Lagrange equations, through the application of the symbolic package Mathematica. The introduced form of the First-Ply-Failure criteria demonstrates the non-uniqueness in solutions and complications in searching for the global macroscopic optimal solutions. The information presented to readers is enriched by adding selected review papers, surveys and monographs in the area of composite structures.

scholarly journals A least action principle for interceptive walking

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Soon Ho Kim ◽  
Jong Won Kim ◽  
Hyun Chae Chung ◽  
MooYoung Choi
Keyword(s):  
Social Systems ◽  
Equations Of Motion ◽  
Classical Mechanics ◽  
Action Principle ◽  
Lagrange Equations ◽  
Least Action Principle ◽  
Least Action ◽  
The Least Action Principle ◽  
Principle Of Least Action ◽  
Human Participants

AbstractThe principle of least effort has been widely used to explain phenomena related to human behavior ranging from topics in language to those in social systems. It has precedence in the principle of least action from the Lagrangian formulation of classical mechanics. In this study, we present a model for interceptive human walking based on the least action principle. Taking inspiration from Lagrangian mechanics, a Lagrangian is defined as effort minus security, with two different specific mathematical forms. The resulting Euler–Lagrange equations are then solved to obtain the equations of motion. The model is validated using experimental data from a virtual reality crossing simulation with human participants. We thus conclude that the least action principle provides a useful tool in the study of interceptive walking.

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