Dr. Wang Xingbo Fall ， 2005 Mathematical & Mechanical Method in Mechanical Engineering

Functional and Calculus of Variation Mathematical & Mechanical Method in Mechanical Engineering Introduction to Calculus of Variations Find the shortest curve connecting P = ( a, y ( a )) and Q = ( b, y ( b )) in XY plane

The arclength is Mathematical & Mechanical Method in Mechanical Engineering Introduction to Calculus of Variations The problem is to minimize the above integral

A function like J is actually called a functional. y(x) is call a permissible function Mathematical & Mechanical Method in Mechanical Engineering Introduction to Calculus of Variations A functional can have more general form

We will only focus on functional with integral Mathematical & Mechanical Method in Mechanical Engineering Introduction to Calculus of Variations A increment of y(x) is called variation of y(x), denoted as δy(x)

consider the increment of J[y(x)] caused by δy(x) ΔJ [y(x)]= J [y(x)+δy(x)]- J [y(x)] ΔJ [y(x)]= L[y(x), δy(x)]+β[y(x),δy(x)] max|δy(x)| Mathematical & Mechanical Method in Mechanical Engineering Introduction to Calculus of Variations If β[y(x),δy(x)] is a infinitesimal of δy(x), then L is called variation of J[y(x)] with the first order, or simply variation of J[y(x)],denoted by δJ[y(x)]

1.Rules for permissible functions y(x) and variable x. Mathematical & Mechanical Method in Mechanical Engineering Introduction to Calculus of Variations δx=dx

Rules for functional J. Rules for functional J. Mathematical & Mechanical Method in Mechanical Engineering Introduction to Calculus of Variations δ 2 J =δ(δJ), …, δ k J =δ(δ k-1 J) δ(J 1 + J 2 )= δJ 1 +δJ 2 δ(J 1 J 2 )= J 1 δJ 2 + J 2 δJ 1 δ(J 1 / J 2 )=( J 2 δJ 1 - J 1 δJ 2 )/ J 2 2

Rules for functional J and F Mathematical & Mechanical Method in Mechanical Engineering Introduction to Calculus of Variations

If J[y(x)] reaches its maximum (or minimum) at y 0 (x), then δJ[y 0 (x)]=0. If J[y(x)] reaches its maximum (or minimum) at y 0 (x), then δJ[y 0 (x)]=0. Mathematical & Mechanical Method in Mechanical Engineering Introduction to Calculus of Variations Let J be a functional defined on C 2 [a,b] with J[y(x)] given by Let J be a functional defined on C 2 [a,b] with J[y(x)] given by How do we determine the curve y(x) which produces such a minimum (maximum) value for J? How do we determine the curve y(x) which produces such a minimum (maximum) value for J?

The Euler-Lagrange Equation Mathematical & Mechanical Method in Mechanical Engineering Introduction to Calculus of Variations

Let M(x) be a continuous function on the interval [a,b], Suppose that for any continuous function h(x) with h(a) = h(b) = 0 we have Mathematical & Mechanical Method in Mechanical Engineering Fundamental principle of variations Then M(x) is identically zero on [a, b]

choose h(x) = -M(x)(x - a)(x - b) Then M(x)h(x) ≥ 0 on [a, b] Mathematical & Mechanical Method in Mechanical Engineering Fundamental principle of variations 0 = M(x)h(x) = [M(x)] 2 [-(x - a)(x - b)]  M(x)=0 If the definite integral of a non-negative function is zero then the function itself must be zero

Example :Prove that the shortest curve connecting planar point P and Q is the straight line connected P and Q Mathematical & mechanical Method in Mechanical Engineering Introduction to Calculus of Variations

Mathematical & mechanical Method in Mechanical Engineering Introduction to Calculus of Variations y(x)=ax+b

Mathematical & mechanical Method in Mechanical Engineering Beltrami Identity. If then the Euler-Lagrange equation is equivalent to :

The Brachistochrone Problem Mathematical & mechanical Method in Mechanical Engineering Introduction to Calculus of Variations Find a path that wastes the least time for a bead travel from P to Q

Let a curve y(x) that connects P and Q represent the wire Mathematical & mechanical Method in Mechanical Engineering The Brachistochrone Problem

By Newton's second law we obtain Mathematical & mechanical Method in Mechanical Engineering The Brachistochrone Problem

Euler-Lagrange Equation Mathematical & mechanical Method in Mechanical Engineering The Brachistochrone Problem

Mathematical & mechanical Method in Mechanical Engineering The Brachistochrone Problem The solution of the above equation is a cycloid curve

Mathematical & mechanical Method in Mechanical Engineering Integration of the Euler-Lagrange Equation Case 1. F(x, y, y’) = F (x) Case 2. F (x, y, y’) = F (y) :F y (y)=0 Case 3. F (x, y, y’) = F (y’) :

Mathematical & mechanical Method in Mechanical Engineering Integration of the Euler-Lagrange Equation Case 4. F (x, y, y’) = F (x, y) F y (x, y) = 0 y = f (x) Case 5. F (x, y, y’) = F (x, y’)

Mathematical & mechanical Method in Mechanical Engineering Integration of the Euler-Lagrange Equation Case 6 F (x, y, y ’ ) = F (y, y ’ )

Mathematical & mechanical Method in Mechanical Engineering The Euler-Lagrange Equation of Variational Notation

Mathematical & mechanical Method in Mechanical Engineering The Lagrange Multiplier Method for the Calculus of Variations Conditions

The minimize problem of following functional is equal to the conditional ones. Mathematical & mechanical Method in Mechanical Engineering The Lagrange Multiplier Method for the Calculus of Variations where λ is chosen that y(a)=A, y(b)=B

Example Mathematical & Mechanical Method in Mechanical Engineering The Lagrange Multiplier Method for the Calculus of Variations The Lagrange Multiplier Method for the Calculus of Variations E-L equation is under Leads to

Mathematical & Mechanical Method in Mechanical Engineering Variation of Multi-unknown functions Variation of Multi-unknown functions

The Euler-Lagrange equation for a functional with two functions y 1 (x),y 2 (x) are The Euler-Lagrange equation for a functional with two functions y 1 (x),y 2 (x) are Mathematical & Mechanical Method in Mechanical Engineering Variation of Multi-unknown functions Variation of Multi-unknown functions

Mathematical & Mechanical Method in Mechanical Engineering Higher Derivatives Higher Derivatives

What is the shape of a beam which is bent and which is clamped so that y (0) = y (1) = y’ (0) = 0 and y’ (1) = 1. Mathematical & Mechanical Method in Mechanical Engineering Example

Class is Over! See you! Mathematical & Mechanical Method in Mechanical Engineering