Introduction to Calculus of Variations

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A short introduction to calculus of variations.

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Introduction to Calculus of Variations

  1. 1. Σ YSTEMSIntroduction to Calculus of Variations Dimitrios Papadopoulos Delta Pi Systems Thessaloniki, Greece
  2. 2. Overview ◮ What is calculus of variations? ◮ The Case of One Variable ◮ The Case of Several Variables ◮ The Case of n Unknown Functions ◮ Lagrange Multipliers Delta Pi Systems
  3. 3. What is calculus of variations? ◮ Calculus of variations deals with problems where functionals appear. ◮ A functional is a kind of function, where the independent variable is itself a function (or a curve). ◮ Historical examples: shortest path, the problem of brachistochrone, the isoperimetric problem. ◮ In calculus of variations lie the origins of many modern scientific fields, such as the finite element method, the level set method, and optimal control of partial differential equations. Delta Pi Systems
  4. 4. Calculus of Variations - The Case of One Variable ◮ The integral b I= f (y, y, x)dx ˙ (1) a has an extremum if the Euler-Lagrange differential equation is satisfied ∂f d ∂f − ( )=0 (2) ∂y ˙ dx ∂ y ◮ Find the shortest plane curve joining two points A and B, i.e. find the curve y = y(x) for which the functional b b dx2 + dy 2 = 1 + y ′2 dx (3) a a achieves its minimum. Delta Pi Systems
  5. 5. Calculus of Variations - The Case of Several Variables ◮ The functional I[z] = F (x, y, z, zy , zx )dxdy (4) R has an extremum if the partial differential equation is satisfied ∂ ∂ Fz − Fz − Fz = 0 (5) ∂x x ∂y y ◮ Find the surface of least area spanned by a given contour I[z] = 2 2 1 + zx + zy dxdy (6) R r(1 + q 2 ) − 2spq + t(1 + p2 ) = 0 (7) where p = zx , q = zy , r = zxx , s = zxy , t = zyy Delta Pi Systems
  6. 6. Calculus of Variations - The Case of n Unknown Functions ◮ The functional b I[y1 , . . . , yn ] = F (x, y1 , . . . , yn , y1 , . . . , yn )dx ˙ ˙ (8) a leads to a system of n second-order differential equations d Fyi − Fy′ = 0 (i = 1, . . . , n) (9) dx i ◮ The functional I[z1 , . . . , zn ] = F (x, y, z1 , . . . , zn , z1,x , . . . , zn,x , z1,y , . . . , zn,y )dxdy R (10) leads to a system of n partial differential equations. Delta Pi Systems
  7. 7. Lagrange MultipliersGiven the functional b J[y] = F (x, y, y ′ )dx, alet the admissible curves satisfy the conditions b y(a) = A, y(b) = B, K[y] = G(x, y, y ′ )dx = l awhere K[y] is another functional, and let J[y] have an extremum for y = y(x).Then, if y = y(x) is not an extremal of K[y], there exists a constant λ suchthat y = y(x) is an extremal of the functional b (F + λG)dx, ai.e., y = y(x) satisfies the differential equation d d Fy − Fy′ + λ(Gy − Gy′ ) = 0. dx dx Delta Pi Systems
  8. 8. Bibliography 1. I.M. Gelfand and S.V. Fomin, Calculus of Variations. 2. R. Courant and D. Hilbert, Methods of Mathematical Physics. 3. F. Riesz and B. Sz-Nagy, Functional Analysis. 4. R. Bellman, Dynamic Programming. Delta Pi Systems
  9. 9. Contact usDelta Pi SystemsOptimization and Control of Processes and SystemsThessaloniki, Greecehttp://www.delta-pi-systems.eu Delta Pi Systems

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