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Published in:
2018 | OriginalPaper | Chapter
1. The Euler-Lagrange Equation
Author : Hansjörg Kielhöfer
Published in: Calculus of Variations
Publisher: Springer International Publishing
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Abstract
In order to give the functionals a domain of definition, we need to introduce suitable function spaces. First of all we require that the Lagrange function or Lagrangian, Here \([a, b] = \{x|a\le x\le b\}\) is a compact interval in the real line.
$$ J(y) = \int ^b_a F(x,y, y') dx $$
$$ F : [a, b] \times {\mathbb {R}}\times {\mathbb {R}}\rightarrow {\mathbb {R}}, {\qquad }\text {is continuous.} $$