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A Discussion of the Quasi-Euler–Lagrange Equation
SIAM Journal on Applied Mathematics, 1992Summary: The quasi-Euler-Lagrange equation, \(a(x)\cdot\text{grad}(\partial_ 1L)+b(x)\cdot\text{grad}(L)=0\), is introduced and is shown to be locally solvable when \(a(x)\) and \(b(x)\) are analytic vector functions. A general solution is constructed when the equation is linearized with respect to \(x_ 1\).
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2018
In order to give the functionals $$ J(y) = \int ^b_a F(x,y, y') dx $$ a domain of definition, we need to introduce suitable function spaces. First of all we require that the Lagrange function or Lagrangian, $$ F : [a, b] \times {\mathbb {R}}\times {\mathbb {R}}\rightarrow {\mathbb {R}}, {\qquad }\text {is continuous.} $$ Here \([a, b] = \{
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In order to give the functionals $$ J(y) = \int ^b_a F(x,y, y') dx $$ a domain of definition, we need to introduce suitable function spaces. First of all we require that the Lagrange function or Lagrangian, $$ F : [a, b] \times {\mathbb {R}}\times {\mathbb {R}}\rightarrow {\mathbb {R}}, {\qquad }\text {is continuous.} $$ Here \([a, b] = \{
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Reduction of Euler-Lagrange Equations in Gauge Theories
International Journal of Modern Physics A, 2003We present a reduction procedure to the so-called canonical form for the Euler-Lagrange equations of a general gauge theory. The reduction procedure reveals constraints in the Lagrangian formulation of singular theories and, in that respect, is similar to the Dirac procedure in the Hamiltonian formulation.
Geyer, B., Gitman, D., Tyutin, I.
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On the Euler-Lagrange equation of a functional by Pólya and Szegö
Calculus of Variations and Partial Differential Equations, 2016In this paper, the classes of sets \[ \begin{aligned} &{\mathcal K}^3=\big\{K\subset{\mathbb R}^3,\;K\text{ convex and compact, }{\mathcal H}^2(K)>0\big\}\\ &{\mathcal K}^3_0=\big\{K\in{\mathcal K}^3,\;K\text{ has a nonempty interior}\big\}\end{aligned} \] are considered, together with the shape cost functional \[ {\mathcal F}(K)={(\operatorname{Cap}K)^
FUSCO, NICOLA, Zhong, Xiao
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The Euler-Lagrange Equations and Characteristics
1985In the preceding chapter, we considered the problem of minimizing the functional $$J\left( y \right) = \int\limits_0^t {L\left[ {x\left( t \right), y\left( t \right)} \right] dt}$$ (1) subject to relations of the form $$frac{{d{{x}_{i}}}}{{dt}} = {{g}_{i}}\left( {x, y} \right), {{x}_{i}}\left( 0 \right) = {{c}_{i}}, i = l, 2, ..., n,$$
Richard Bellman, George Adomian
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Euler-Lagrange Equations for Hypergestures
2017This chapter deals with a model from mathematical physics of string theory that describes the transition from symbolic reality to physical reality of musical gestures. We demonstrate, using multidimensional Fourier theory and Green functions, that the physical gesture can be viewed as a function of a potential and the symbolic gesture. The role of this
Guerino Mazzola +6 more
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On the Metrics and Euler-Lagrange Equations of Computational Anatomy
Annual Review of Biomedical Engineering, 2002▪ Abstract This paper reviews literature, current concepts and approaches in computational anatomy (CA). The model of CA is a Grenander deformable template, an orbit generated from a template under groups of diffeomorphisms. The metric space of all anatomical images is constructed from the geodesic connecting one anatomical structure to another in the
Michael I, Miller +2 more
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2015
Abstract. In this paper we examine the Euler- Lagrange equation and by expressing the fundamental thermo of calculus of variations, we calculate the Euler- Lagrange equation for the simplest problem of calculus of variations and by offering an example we will discuss the specific modes of Euler - Lagrange equation.
FATHI POOR, Zahra +2 more
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Abstract. In this paper we examine the Euler- Lagrange equation and by expressing the fundamental thermo of calculus of variations, we calculate the Euler- Lagrange equation for the simplest problem of calculus of variations and by offering an example we will discuss the specific modes of Euler - Lagrange equation.
FATHI POOR, Zahra +2 more
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A generalized Euler–Lagrange equation
Mathematical Proceedings of the Cambridge Philosophical Society, 1969AbstractA generalized Euler–Lagrange equation is presented. It provides a unified approach to boundary value problems in potential theory, diffusion, magnetostatics, and integral equations.
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On generalizations of the Euler–Lagrange equation
Nonlinear Analysis: Theory, Methods & Applications, 2001Ledzewicz, Urszula, Schaettler, Heinz
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