Reflection symmetric formulation of generalized fractional variational calculus

2013 ◽  
Vol 16 (1) ◽  
Author(s):  
Małgorzata Klimek ◽  
Maria Lupa
Keyword(s):  
Fractional Derivatives ◽  
Finite Interval ◽  
Equations Of Motion ◽  
Variational Calculus ◽  
Reflection Symmetry ◽  
Lagrange Equations ◽  
Representation Formulas ◽  
Symmetry Properties ◽  
Symmetric Formulation ◽  
Generalized Fractional Derivatives

AbstractWe define generalized fractional derivatives (GFDs) symmetric and anti-symmetric w.r.t. the reflection symmetry in a finite interval. Arbitrary functions are split into parts with well defined reflection symmetry properties in a hierarchy of intervals [0, b/2m], m ∈ ℕ0. For these parts — [J]-projections of function, we derive the representation formulas for generalized fractional operators (GFOs) and examine integration properties. It appears that GFOs can be reduced to operators determined in subintervals [0, b/2m]. The results are applied in the derivation of Euler-Lagrange equations for action dependent on Riemann-Liouville type GFDs. We show that for Lagrangian being a sum (finite or not) of monomials, the obtained equations of motion can be localized in arbitrary short subinterval [0, b/2m].

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.

Hamiltonian Formulation for Continuous Third-order Systems Using Fractional Derivatives

2021 ◽  
Vol 14 (1) ◽  
pp. 35-47
Keyword(s):  
Variational Principle ◽  
Fractional Derivatives ◽  
Equations Of Motion ◽  
Hamiltonian Formulation ◽  
Lagrange Equations ◽  
Third Order ◽  
Derivative Operator ◽  
Hamilton Equations ◽  
Liouville Fractional Derivative ◽  
Continuous Field

Abstract: We constructed the Hamiltonian formulation of continuous field systems with third order. A combined Riemann–Liouville fractional derivative operator is defined and a fractional variational principle under this definition is established. The fractional Euler equations and the fractional Hamilton equations are obtained from the fractional variational principle. Besides, it is shown that the Hamilton equations of motion are in agreement with the Euler-Lagrange equations for these systems. We have examined one example to illustrate the formalism. Keywords: Fractional derivatives, Lagrangian formulation, Hamiltonian formulation, Euler-lagrange equations, Third-order lagrangian.

scholarly journals Generalized Multiparameters Fractional Variational Calculus

2012 ◽  
Vol 2012 ◽  
pp. 1-38 ◽  
Author(s):  
Om Prakash Agrawal
Keyword(s):  
Fractional Derivatives ◽  
Variational Calculus ◽  
The Past ◽  
Special Cases ◽  
Generalized Fractional Derivatives ◽  
The One

This paper builds upon our recent paper on generalized fractional variational calculus (FVC). Here, we briefly review some of the fractional derivatives (FDs) that we considered in the past to develop FVC. We first introduce new one parameter generalized fractional derivatives (GFDs) which depend on two functions, and show that many of the one-parameter FDs considered in the past are special cases of the proposed GFDs. We develop several parts of FVC in terms of one parameter GFDs. We point out how many other parts could be developed using the properties of the one-parameter GFDs. Subsequently, we introduce two new two- and three-parameter GFDs. We introduce some of their properties, and discuss how they can be used to develop FVC. In addition, we indicate how these formulations could be used in various fields, and how the generalizations presented here can be further extended.

scholarly journals Usual Approximations to the Equations of Atmospheric Motion: A Variational Perspective

2014 ◽  
Vol 71 (7) ◽  
pp. 2452-2466 ◽  
Author(s):  
Marine Tort ◽  
Thomas Dubos
Keyword(s):  
Equations Of Motion ◽  
Curvilinear Coordinates ◽  
Hamilton’S Principle ◽  
Lagrange Equations ◽  
Hamilton's Principle ◽  
Energy Potential ◽  
Conservation Of Energy ◽  
Asymptotic Consistency ◽  
Variational Framework ◽  
Symmetry Properties

Abstract The usual geophysical approximations are reframed within a variational framework. Starting from the Lagrangian of the fully compressible Euler equations expressed in a general curvilinear coordinates system, Hamilton’s principle of least action yields Euler–Lagrange equations of motion. Instead of directly making approximations in these equations, the approach followed is that of Hamilton’s principle asymptotics; that is, all approximations are performed in the Lagrangian. Using a coordinate system where the geopotential is the third coordinate, diverse approximations are considered. The assumptions and approximations covered are 1) particular shapes of the geopotential; 2) shallowness of the atmosphere, which allows for the approximation of the relative and planetary kinetic energy; 3) small vertical velocities, implying quasi-hydrostatic systems; and 4) pseudoincompressibility, enforced by introducing a Lagangian multiplier. This variational approach greatly facilitates the derivation of the equations and systematically ensures their dynamical consistency. Indeed, the symmetry properties of the approximated Lagrangian imply the conservation of energy, potential vorticity, and momentum. Justification of the equations then relies, as usual, on a proper order-of-magnitude analysis. As an illustrative example, the asymptotic consistency of recently introduced shallow-atmosphere equations with a complete Coriolis force is discussed, suggesting additional corrections to the pressure gradient and gravity.

scholarly journals Variational Problems with Time Delay and Higher-Order Distributed-Order Fractional Derivatives with Arbitrary Kernels

Mathematics ◽  
2021 ◽  
Vol 9 (14) ◽  
pp. 1665
Author(s):  
Fátima Cruz ◽  
Ricardo Almeida ◽  
Natália Martins
Keyword(s):  
Time Delay ◽  
Fractional Derivatives ◽  
Variational Problems ◽  
Higher Order ◽  
Sufficient Optimality Conditions ◽  
Fractional Operator ◽  
Different Types ◽  
Generalized Fractional Derivatives ◽  
Necessary And Sufficient ◽  
Distributed Order

In this work, we study variational problems with time delay and higher-order distributed-order fractional derivatives dealing with a new fractional operator. This fractional derivative combines two known operators: distributed-order derivatives and derivatives with respect to another function. The main results of this paper are necessary and sufficient optimality conditions for different types of variational problems. Since we are dealing with generalized fractional derivatives, from this work, some well-known results can be obtained as particular cases.

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.

A Numerical Scheme for a Class of Parametric Problem of Fractional Variational Calculus

2012 ◽  
Vol 7 (2) ◽  
Author(s):  
Om P. Agrawal ◽  
M. Mehedi Hasan ◽  
X. W. Tangpong
Keyword(s):  
Analytical Solutions ◽  
Numerical Scheme ◽  
Fractional Derivatives ◽  
Optimization Problems ◽  
Practical Interest ◽  
Variational Calculus ◽  
Functional Minimization ◽  
Orthogonal Functions ◽  
Order Problem ◽  
Parametric Problem

Fractional derivatives (FDs) or derivatives of arbitrary order have been used in many applications, and it is envisioned that in the future they will appear in many functional minimization problems of practical interest. Since fractional derivatives have such properties as being non-local, it can be extremely challenging to find analytical solutions for fractional parametric optimization problems, and in many cases, analytical solutions may not exist. Therefore, it is of great importance to develop numerical methods for such problems. This paper presents a numerical scheme for a linear functional minimization problem that involves FD terms. The FD is defined in terms of the Riemann-Liouville definition; however, the scheme will also apply to Caputo derivatives, as well as other definitions of fractional derivatives. In this scheme, the spatial domain is discretized into several subdomains and 2-node one-dimensional linear elements are adopted to approximate the solution and its fractional derivative at point within the domain. The fractional optimization problem is converted to an eigenvalue problem, the solution of which leads to fractional orthogonal functions. Convergence study of the number of elements and error analysis of the results ensure that the algorithm yields stable results. Various fractional orders of derivative are considered, and as the order approaches the integer value of 1, the solution recovers the analytical result for the corresponding integer order problem.

A Method for Use of Cyclic Symmetry Properties in Analysis of Nonlinear Multiharmonic Vibrations of Bladed Disks

2004 ◽  
Vol 126 (1) ◽  
pp. 175-183 ◽  
Author(s):  
E. P. Petrov
Keyword(s):  
High Efficiency ◽  
Equations Of Motion ◽  
Linear Part ◽  
Forced Response ◽  
Mode Shapes ◽  
Solution Technique ◽  
Disk Model ◽  
Sector Model ◽  
Bladed Disk ◽  
Symmetry Properties

An effective method for analysis of periodic forced response of nonlinear cyclically symmetric structures has been developed. The method allows multiharmonic forced response to be calculated for a whole bladed disk using a periodic sector model without any loss of accuracy in calculations and modeling. A rigorous proof of the validity of the reduction of the whole nonlinear structure to a sector is provided. Types of bladed disk forcing for which the method may be applied are formulated. A multiharmonic formulation and a solution technique for equations of motion have been derived for two cases of description for a linear part of the bladed disk model: (i) using sector finite element matrices and (ii) using sector mode shapes and frequencies. Calculations validating the developed method and a numerical investigation of a realistic high-pressure turbine bladed disk with shrouds have demonstrated the high efficiency of the method.

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