On the Euler–Lagrange Equation for Functionals of the Calculus of Variations without Upper Growth Conditions

2009 ◽  
Vol 48 (4) ◽  
pp. 2857-2870 ◽  
Author(s):  
Marco Degiovanni ◽  
Marco Marzocchi
Keyword(s):  
Calculus Of Variations ◽  
Lagrange Equation ◽  
Growth Conditions

On conservation laws and necessary conditions in the calculus of variations

2002 ◽  
Vol 132 (6) ◽  
pp. 1361-1371 ◽  
Author(s):  
G. Francfort ◽  
J. Sivaloganathan
Keyword(s):  
Conservation Laws ◽  
Calculus Of Variations ◽  
Conservation Law ◽  
Necessary Conditions ◽  
Lagrange Equation ◽  
Integral Functional ◽  
Variational Symmetry

It is well known from the work of Noether that every variational symmetry of an integral functional gives rise to a corresponding conservation law. In this paper, we prove that each such conservation law arises directly as the Euler-Lagrange equation for the functional on taking suitable variations around a minimizer.

Differential Equations

2018 ◽  
Author(s):  
Peter Mann
Keyword(s):  
Calculus Of Variations ◽  
Lagrange Equation ◽  
Shortest Paths ◽  
Classical Mechanics ◽  
Classical Field Theory ◽  
Variational Problems ◽  
Functional Derivative ◽  
Mathematical Framework ◽  
Analytical Mechanics ◽  
Mathematical Technique

This chapter presents the general formulation of the calculus of variations as applied to mechanics, relativity and field theories. The calculus of variations is a common mathematical technique used throughout classical mechanics. First developed by Euler to determine the shortest paths between fixed points along a surface, it was applied by Lagrange to mechanical problems in analytical mechanics. The variational problems in the chapter have been simplified for ease of understanding upon first introduction, in order to give a general mathematical framework. This chapter takes a relaxed approach to explain how the Euler–Lagrange equation is derived using this method. It also discusses first integrals. The chapter closes by defining the functional derivative, which is used in classical field theory.

Optimality Principles for Fuzzy Dual Uncertain Systems

2018 ◽  
pp. 47-72
Author(s):  
Félix Mora-Camino ◽  
Hakim Bouadi ◽  
Roger Marcelin Faye ◽  
Lunlong Zhong
Keyword(s):  
Optimality Conditions ◽  
Calculus Of Variations ◽  
Optimization Problems ◽  
Lagrange Equation ◽  
Necessary Optimality Conditions ◽  
Continuous System ◽  
Propagation Of Uncertainty ◽  
Performance Indexes ◽  
Definition Of ◽  
Optimality Principles

This chapter considers the extension of the calculus of variations to the optimization of a class of fuzzy systems where the uncertainty of variables and parameters is represented by symmetrical triangular membership functions. The concept of fuzzy dual numbers is introduced, and the consideration of the necessary differentiability conditions for functions of dual variables leads to the definition of fuzzy dual functions. It is shown that when this formalism is adopted to represent performance indexes for uncertain optimization problems, the calculus of variations can be used to establish necessary optimality conditions as an extension to this case of the Euler-Lagrange equation. Then the chapter discusses the propagation of uncertainty when the fuzzy dual formalism is adopted for the state representation of a time continuous system. This leads to the formulation of a fuzzy dual optimization problem for which necessary optimality conditions, corresponding to an extension of Pontryagine's optimality principle, are established.

scholarly journals Special Functions of Mathematical Physics: A Unified Lagrangian Formalism

Mathematics ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 379
Author(s):  
Zdzislaw E. Musielak ◽  
Niyousha Davachi ◽  
Marialis Rosario-Franco
Keyword(s):  
Differential Equations ◽  
Calculus Of Variations ◽  
Special Functions ◽  
Lagrange Equation ◽  
Mathematical Physics ◽  
Lagrangian Formalism ◽  
Lagrange Equations ◽  
Gauge Functions ◽  
Helmholtz Conditions

Lagrangian formalism is established for differential equations with special functions of mathematical physics as solutions. Formalism is based on either standard or non-standard Lagrangians. This work shows that the procedure of deriving the standard Lagrangians leads to Lagrangians for which the Euler–Lagrange equation vanishes identically, and that only some of these Lagrangians become the null Lagrangians with the well-defined gauge functions. It is also demonstrated that the non-standard Lagrangians require that the Euler–Lagrange equations are amended by the auxiliary conditions, which is a new phenomenon in the calculus of variations. The existence of the auxiliary conditions has profound implications on the validity of the Helmholtz conditions. The obtained results are used to derive the Lagrangians for the Airy, Bessel, Legendre and Hermite equations. The presented examples clearly demonstrate that the developed Lagrangian formalism is applicable to all considered differential equations, including the Airy (and other similar) equations, and that the regular and modified Bessel equations are the only ones with the gauge functions. Possible implications of the existence of the gauge functions for these equations are discussed.

scholarly journals Euler-Lagrange equation for a delay variational problem

2017 ◽  
Vol 4 (1) ◽  
pp. 52-61
Author(s):  
Joël Blot ◽  
Mamadou I. Koné
Keyword(s):  
Variational Problem ◽  
Calculus Of Variations ◽  
Lagrange Equation ◽  
Lagrange Equations ◽  
Unknown Variable

Abstract We establish Euler-Lagrange equations for a problem of Calculus of Variations where the unknown variable contains a term of delay on a segment

Generalized Euler–Lagrange equation for nonsmooth calculus of variations

2013 ◽  
Vol 75 (1-2) ◽  
pp. 85-100 ◽  
Author(s):  
M. H. Noori Skandari ◽  
A. V. Kamyad ◽  
S. Effati
Keyword(s):  
Calculus Of Variations ◽  
Lagrange Equation ◽  
Nonsmooth Calculus

Partial regularity of strong local minimizers of quasiconvex integrals with (p, q)-growth

2009 ◽  
Vol 139 (3) ◽  
pp. 595-621 ◽  
Author(s):  
Sabine Schemm ◽  
Thomas Schmidt
Keyword(s):  
Calculus Of Variations ◽  
Partial Regularity ◽  
Growth Conditions ◽  
Local Minimizers ◽  
Image Position

We consider strictly quasiconvex integralsin the multi-dimensional calculus of variations. For the C2-integrand f : ℝNn → ℝ we impose (p, q)-growth conditionswith γ, Γ > 0 and 1 < p ≤ q < min {p + 1/n, p(2n − 1)/(2n − 2)}. Under these assumptions we prove partial C1, αloc-regularity for strong local minimizers of F and the associated relaxed functional F.

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