Solutions of Euler-Lagrange equations in fractional mechanics

2007 ◽  
Author(s):  
M. Klimek ◽  
Piotr Kielanowski ◽  
Anatol Odzijewicz ◽  
Martin Schlichenmeier ◽  
Theodore Voronov
Keyword(s):  
Lagrange Equations ◽  
Fractional Mechanics

scholarly journals On reflection symmetry and its application to the Euler–Lagrange equations in fractional mechanics

2013 ◽  
Vol 371 (1990) ◽  
pp. 20120145 ◽  
Author(s):  
Małgorzata Klimek
Keyword(s):  
Fractional Derivatives ◽  
Finite Interval ◽  
Reflection Symmetry ◽  
Symmetric Part ◽  
Fractional Differentiation ◽  
Lagrange Equations ◽  
Caputo Derivatives ◽  
Lagrange System ◽  
Fractional Mechanics ◽  
Derivatives Of

We study the properties of fractional differentiation with respect to the reflection symmetry in a finite interval. The representation and integration formulae are derived for symmetric and anti-symmetric fractional derivatives, both of the Riemann–Liouville and Caputo type. The action dependent on the left-sided Caputo derivatives of orders in the range (1,2) is considered and we derive the Euler–Lagrange equations for the symmetric and anti-symmetric part of the trajectory. The procedure is illustrated with an example of the action dependent linearly on fractional velocities. For the obtained Euler–Lagrange system, we discuss its localization resulting from the subsequent symmetrization of the action.

On Reflection Symmetry and Its Application to the Euler-Lagrange Equations in Fractional Mechanics

2011 ◽  
Author(s):  
Malgorzata Klimek
Keyword(s):  
Fractional Derivatives ◽  
Finite Interval ◽  
Reflection Symmetry ◽  
Symmetric Part ◽  
Fractional Differentiation ◽  
Lagrange Equations ◽  
Integration Formulas ◽  
Lagrange System ◽  
Fractional Mechanics ◽  
Derivatives Of

We study the properties of fractional differentiation with respect to the reflection symmetry in a finite interval. The representation and integration formulas are derived for the symmetric and anti-symmetric fractional derivatives, both of the Riemann -Liouville and Caputo type. The action dependent on the left -sided Caputo derivatives of orders in range (1.2) is considered and we derive the Euler-Lagrange equations for the symmetric and anti-symmetric part of the trajectory. The procedure is illustrated with an example of the action dependent linearly on fractional velocities. For the obtained Euler-Lagrange system we discuss its localization resulting from the subsequent sym-metrization of the action.

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.

The Geometry of d2y1/dt2 = f (y, ẏ, t) and d2y2/dt2 = g(y, ẏ, t), and Euclidean Spaces

2006 ◽  
Vol 49 (2) ◽  
pp. 170-184
Author(s):  
Richard Atkins
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Shortest Paths ◽  
Equivalence Problem ◽  
Second Order ◽  
System Of Differential Equations ◽  
Lagrange Equations ◽  
Invariant Functions ◽  
Underlying Geometry ◽  
The Relationship

AbstractThis paper investigates the relationship between a system of differential equations and the underlying geometry associated with it. The geometry of a surface determines shortest paths, or geodesics connecting nearby points, which are defined as the solutions to a pair of second-order differential equations: the Euler–Lagrange equations of the metric. We ask when the converse holds, that is, when solutions to a system of differential equations reveals an underlying geometry. Specifically, when may the solutions to a given pair of second order ordinary differential equations d2y1/dt2 = f (y, ẏ, t) and d2y2/dt2 = g(y, ẏ, t) be reparameterized by t → T(y, t) so as to give locally the geodesics of a Euclidean space? Our approach is based upon Cartan's method of equivalence. In the second part of the paper, the equivalence problem is solved for a generic pair of second order ordinary differential equations of the above form revealing the existence of 24 invariant functions.

scholarly journals Energy Minimization Scheme for Split Potential Systems Using Exponential Variational Integrators

2021 ◽  
Vol 2 (3) ◽  
pp. 431-441
Author(s):  
Odysseas Kosmas
Keyword(s):  
Numerical Solutions ◽  
Oscillatory Behavior ◽  
Lagrangian Function ◽  
Computational Time ◽  
Variational Integrators ◽  
Weighted Sum ◽  
Lagrange Equations ◽  
Exponential Functions ◽  
Intermediate Time ◽  
Definition Of

In previous works we developed a methodology of deriving variational integrators to provide numerical solutions of systems having oscillatory behavior. These schemes use exponential functions to approximate the intermediate configurations and velocities, which are then placed into the discrete Lagrangian function characterizing the physical system. We afterwards proved that, higher order schemes can be obtained through the corresponding discrete Euler–Lagrange equations and the definition of a weighted sum of “continuous intermediate Lagrangians” each of them evaluated at an intermediate time node. In the present article, we extend these methods so as to include Lagrangians of split potential systems, namely, to address cases when the potential function can be decomposed into several components. Rather than using many intermediate points for the complete Lagrangian, in this work we introduce different numbers of intermediate points, resulting within the context of various reliable quadrature rules, for the various potentials. Finally, we assess the accuracy, convergence and computational time of the proposed technique by testing and comparing them with well known standards.

scholarly journals Determination of Load-Carrying Capacity of C-Profile Glued Ti-Al Column under Temperature Environment

Materials ◽  
2021 ◽  
Vol 14 (11) ◽  
pp. 3013
Author(s):  
Leszek Czechowski
Keyword(s):  
Carrying Capacity ◽  
Tensile Tests ◽  
Discrete Models ◽  
Image Correlation ◽  
Load Carrying Capacity ◽  
Lagrange Equations ◽  
Load Carrying ◽  
Different Temperatures ◽  
Influence Of Temperature On

The paper deals with an examination of the behaviour of glued Ti-Al column under compression at elevated temperature. The tests of compressed columns with initial load were performed at different temperatures to obtain their characteristics and the load-carrying capacity. The deformations of columns during tests were registered by employing non-contact Digital Image Correlation Aramis® System. The numerical computations based on finite element method by using two different discrete models were carried out to validate the empirical results. To solve the problems, true stress-logarithmic strain curves of one-directional tensile tests dependent on temperature both for considered metals and glue were implemented to software. Numerical estimations based on Green–Lagrange equations for large deflections and strains were conducted. The paper reveals the influence of temperature on the behaviour of compressed C-profile Ti-Al columns. It was verified how the load-carrying capacity of glued bi-metal column decreases with an increase in the temperature increment. The achieved maximum loads at temperature 200 °C dropped by 2.5 times related to maximum loads at ambient temperature.

The Determination of Maximum Range for an Unpowered Atmospheric Vehicle

1964 ◽  
Vol 68 (638) ◽  
pp. 111-116 ◽  
Author(s):  
D. J. Bell
Keyword(s):  
Calculus Of Variations ◽  
Lagrange Multipliers ◽  
Digital Computer ◽  
Necessary Conditions ◽  
Lagrange Equations ◽  
Initial Portion ◽  
End Conditions ◽  
Maximum Range ◽  
Isothermal Atmosphere

SummaryThe problem of maximising the range of a given unpowered, air-launched vehicle is formed as one of Mayer type in the calculus of variations. Eulers’ necessary conditions for the existence of an extremal are stated together with the natural end conditions. The problem reduces to finding the incidence programme which will give the greatest range.The vehicle is assumed to be an air-to-ground, winged unpowered vehicle flying in an isothermal atmosphere above a flat earth. It is also assumed to be a point mass acted upon by the forces of lift, drag and weight. The acceleration due to gravity is assumed constant.The fundamental constraints of the problem and the Euler-Lagrange equations are programmed for an automatic digital computer. By considering the Lagrange multipliers involved in the problem a method of search is devised based on finding flight paths with maximum range for specified final velocities. It is shown that this method leads to trajectories which are sufficiently close to the “best” trajectory for most practical purposes.It is concluded that such a method is practical and is particularly useful in obtaining the optimum incidence programme during the initial portion of the flight path.

scholarly journals Design of Tunable Holographic Liquid Crystalline Diffraction Gratings

Sensors ◽  
2020 ◽  
Vol 20 (23) ◽  
pp. 6789
Author(s):  
Katarzyna A. Rutkowska ◽  
Anna Kozanecka-Szmigiel
Keyword(s):  
Liquid Crystal ◽  
Communication Networks ◽  
Optical Communication ◽  
Liquid Crystalline ◽  
Incident Light ◽  
Diffraction Gratings ◽  
Lagrange Equations ◽  
Experimental Conditions ◽  
Molecular Distribution ◽  
Polarization Gratings

Tunable diffraction gratings and phase filters are important functional devices in optical communication and sensing systems. Polarization gratings, in particular, capable of redirecting an incident light beam completely into the first diffraction orders may be successfully fabricated in liquid crystalline cells assembled from substrates coated with uniform transparent electrodes and orienting layers that force a specific molecular distribution. In this work, the diffraction properties of liquid crystal (LC) cells characterized by a continually rotating cycloidal director pattern at the cell substrates and in the bulk, are studied theoretically by solving a relevant set of the Euler-Lagrange equations. The electric tunability of the gratings is analyzed by estimating the changes in liquid crystalline molecular distribution and thus in effective birefringence, as a function of external voltage. To the best of our knowledge, such detailed numerical calculations have not been presented so far for liquid crystal polarization gratings showing a cycloidal director pattern. Our theoretical predictions may be easily achieved in experimental conditions when exploiting, for example, photo-orienting material, to induce a permanent LC alignment with high spatial resolution. The proposed design may be for example, used as a tunable passband filter with adjustable bandwidths, thus allowing for potential applications in optical spectroscopy, optical communication networks, remote sensing and beyond.

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