Dr. Wang Xingbo Fall ， 2005

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

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

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

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

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

6. Mathematical & Mechanical Method in Mechanical Engineering Introduction to Calculus of Variations 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)| 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)]

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

8. Mathematical & Mechanical Method in Mechanical Engineering Introduction to Calculus of Variations Rules for functional J. δ2 J =δ(δJ), …, δkJ =δ(δk-1J) δ(J1+ J 2)= δJ 1+δJ 2 δ(J 1 J 2)= J 1δJ 2+ J 2δJ 1 δ(J 1/ J 2)=( J 2δJ 1- J 1δJ 2)/ J 22

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

10. Mathematical & Mechanical Method in Mechanical Engineering Introduction to Calculus of Variations If J[y(x)] reaches its maximum (or minimum) at y0(x), then δJ[y0(x)]=0. Let J be a functional defined on C2[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?

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

12. Mathematical & Mechanical Method in Mechanical Engineering Fundamental principle 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 Then M(x) is identically zero on [a, b]

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

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

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

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

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

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

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

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

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

22. 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’) :

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

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

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

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

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

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

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

30. Mathematical & Mechanical Method in Mechanical Engineering Variation of Multi-unknown functions The Euler-Lagrange equation for a functional with two functions y1(x),y2(x) are

31. Mathematical & Mechanical Method in Mechanical Engineering Higher Derivatives

32. Mathematical & Mechanical Method in Mechanical Engineering Example 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.

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