Thanks. I hit a wall when reading Nakahara's Geometry, Topology and Physics. In chapter 10 he takes the variation of the action of a scalar field and the Einstein-Hilbert action, equates them both to zero and ends up with the Field Equations.dextercioby said:An easy book on the subject is Weinstock's "Calculus of Variations with Applications to Physics and Engineering".
Daniel.
Calculus of Variations is a branch of mathematics that deals with finding the optimal value of a functional (a mathematical expression involving a function), usually in the form of a function or a curve, by varying the input function. It involves finding a function that minimizes or maximizes a certain integral or functional.
Calculus of Variations was first developed by mathematicians such as Johann Bernoulli, Leonhard Euler, and Joseph-Louis Lagrange in the 17th and 18th centuries. It was further developed by mathematicians such as Adrien-Marie Legendre and Carl Friedrich Gauss in the 19th century. Today, it is an essential tool in many areas of mathematics and physics.
Calculus of Variations has many applications in various fields such as physics, engineering, economics, and optimization. It is used to solve problems involving optimal control, mechanics, economics, and shape optimization, among others.
Some key concepts in Calculus of Variations include the Euler-Lagrange equation, which is used to find the critical points of a functional, and the fundamental lemma of the calculus of variations, which states that the variation of a functional must equal zero at the extremal points.
Some common techniques used in Calculus of Variations include the method of variation of parameters, the method of Lagrange multipliers, and the calculus of variations on Banach spaces. These techniques are used to solve problems involving different types of functionals and constraints.