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7.2. CALCULUS OF VARIATIONS c 2006 Gilbert Strang constant: the Euler-Lagrange equation (2) is d dx @F @u0 = d dx u0 p 1+(u0)2 = 0 or u0 p 1+(u0)2 = c: (4) That integration is always possible when F depends only on u0 (@F=@u = 0). It leaves the equation @F=@u0 = c. Squaring both sides, u is seen to be linear: (u0) 2= c (1+(u0)2) and u0 = c p 1 c2 and u = c p 1 c2 x+d: (5)

Euler-Lagrange Equation The calculus of variations is thus all about solving optimization problems. The point is to nd the function that optimizes the functional. To under-stand this method, we rst look at the basic one-dimensional functional, also known as the objective functional: J[y] = Z b a L(x;y;y0)dx (3.1)

Calculus of Variations and Euler-Lagrange Equation Introduction: Calculus of variations is a field of mathematical analysis that uses variations, which are small changes in functions and functional, to find maxima and minima of functional: mappings from a set of functions to the real numbers.

After Euler saw the 1755 work of the 19-year-old Lagrange, Euler dropped his own partly geometric approach in favor of ...

Calculus of Variations and Euler-Lagrange Equation Introduction: Calculus of variations is a field of mathematical analysis that uses variations, which are small changes in functions and functional, to find maxima and minima of functional: mappings from …

Euler’s equation for integrals involving several functions. The simplest integral in the Calculus of Variations. involves a single function y = f(x). In applications these integrals arise in cases involving only a single functional dependence. An example of a single functional dependence is a curve in the plane defined by y = f(x).

The functionals dealt with in the calculus of variations are of the form The goal is to find a y(x) that minimizes Г, or maximizes it. Used in deriving the Euler-Lagrange equation b³ , ( ), ( ) a * ªº¬¼f x F x y x y x dx

The stationary path of the functional S is determined by solving this first-order differential equation. 26. Page 26. 4 Euler–Lagrange equation. Equations (23) ...

Finding the extrema of functionals is similar to finding the maxima and minima of functions. The maxima and minima of a function may be located by finding the points where its derivative vanishes (i.e., is equal to zero). The extrema of functionals may be obtained by finding functions where the functional derivative is equal to zero. This leads to solving the associated Euler–Lagrange equation.

Jul 21, 2014 · Multiply the equation by , and integrate over the domain D. After integrating by parts, we find the variational problem as follow. with T = T 1 on B 1. TT= 1 −⋅∇kn T q 2 = −⋅∇kn T h T T()= 3 − δT 0 [() (,) 1 2 2 T DT δτkT fxTdTd #$’(∇+ %%)* ∫∫ 23 2 23 1 ()]0 BB2 ++∫∫q Td h T T dσσ−=! Diffusion Equation Applications

Definition 2 Let Ck[a, b] denote the set of continuous functions defined on the interval a≤x≤b which have their first k-derivatives also continuous on a≤x≤b ...

This implies that f(x) ≡ 0 on [a, b]. Euler-Lagrange Equation (Necessary Condition for Extremum). Theorem: If y(x) is an extremizing function for ...

Calculus of Variations Understanding of a Functional Euler-Lagrange Equation – Fundamental to the Calculus of Variations Proving the Shortest Distance Between Two Points – In Euclidean Space The Brachistochrone Problem – In an Inverse Square Field Some Other Applications Conclusion of Queen Dido’s Story

Functions that maximize or minimize functional may be found using the Euler–Lagrange equation of the calculus of variations. In differential calculus we ...