Typical Problem: Consider a definite integral that depends on an unknown function \(y(x)\), as well as its derivative \(y'(x)=\frac \right]. The mathematical techniques developed to solve this type of problem are collectively known as the calculus of variations. As a motivating example, let us consider the problem of finding the shortest. One example is finding the curve giving the shortest distance between two points - a straight line, of course, in Cartesian geometry (but can you prove it?) but less obvious if the two points lie on a curved surface (the problem of finding geodesics.) We will now generalise this to functionals. The general setup: functionals and boundary conditions isoperimetric problems, geodesic problems. Many problems involve finding a function that maximizes or minimizes an integral expression. The basic setup: Bernoulli and the Brachistochrone. MATH0043 Handout: Fundamental lemma of the calculus of variations.8.4 Generalization of the geodesic concept. Minimization of functionals, Euler Lagrange equations, sufficient conditions for a minimum, geodesic, isoperimetric and time of transit problems, variational. 8.2 A system of differential equations for geodesic curves. Minimization problems that can be analyzed by the calculus of variations serve to char- acterize the equilibrium congurations of almost all continuous physical systems, ranging through elasticity, solid and uid mechanics, electro-magnetism, gravitation, quantum me- chanics, string theory, and many, many others. 1.3 The Euler-Lagrange differential equation. The bred space for the problem will be (M × N,M), which has dimension m n and the mapping now serves as a ber. For solving the inverse problem of calculus of variation we would like to use the formalism of calculus of variations on bred manifolds. The Euler-Lagrange Equation, or Euler’s Equation 1.1 The fundamental problem and lemma of calculus of variations. geodesic curve x(t) on (M, M), x(t) is a geodesic curve on (N, N). It arose out of the necessity of looking at physical problems in which an optimal solution is sought e.g. These curves are called geodesics, and the study. MATH0043 §2: Calculus of Variations MATH0043 §2: Calculus of Variations The calculus of variations is a subject as old as the Calculus of Newton and Leibniz. A few of the problems of the calculus of variations are very old, and were considered and partly solved.
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