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On the Numerical Solution of the Euler–Lagrange Equations
SIAM Journal on Numerical Analysis, 1995This paper is concerned with the numerical solution of the Euler-Lagrange equations of Mechanics, i.e. second-order ordinary differential equations where the variables satisfy some holonomic constraints. Typically such a set of equations in autonomous form can be written as \(M(x)x'' + F'(x)^ T z = G(x,x')\), together with \(F(x) = 0\), where \(F ...
Rabier, Patrick J. +1 more
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Fractional Euler–Lagrange equations revisited
Nonlinear Dynamics, 2012zbMATH Open Web Interface contents unavailable due to conflicting licenses.
Herzallah, Mohamed A. E. +1 more
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On the validity of the Euler–Lagrange equation in a nonlinear case
Nonlinear Analysis: Theory, Methods & Applications, 2010zbMATH Open Web Interface contents unavailable due to conflicting licenses.
BONFANTI, GIOVANNI, MAZZOLA, MARCO
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1983
Jakob Bernoulli’s solution of 1696 to his brother Johann’s problem of the brachistochrone (§1.2) marked the introduction of variational considerations. However, it was not until the work of Euler (c. 1742) and Lagrange (1755) that the systematic theory now known as the calculus of variations emerged.
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Jakob Bernoulli’s solution of 1696 to his brother Johann’s problem of the brachistochrone (§1.2) marked the introduction of variational considerations. However, it was not until the work of Euler (c. 1742) and Lagrange (1755) that the systematic theory now known as the calculus of variations emerged.
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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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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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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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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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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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