A short introduction to calculus of variations.

1. Σ YSTEMS
Introduction to Calculus of Variations
Dimitrios Papadopoulos
Delta Pi Systems
Thessaloniki, Greece

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
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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.
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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.
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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
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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.
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7. Lagrange Multipliers
Given the functional
b
J[y] = F (x, y, y ′ )dx,
a
let the admissible curves satisfy the conditions
b
y(a) = A, y(b) = B, K[y] = G(x, y, y ′ )dx = l
a
where 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 such
that y = y(x) is an extremal of the functional
b
(F + λG)dx,
a
i.e., y = y(x) satisfies the differential equation
d d
Fy − Fy′ + λ(Gy − Gy′ ) = 0.
dx dx
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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.
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9. Contact us
Delta Pi Systems
Optimization and Control of Processes and Systems
Thessaloniki, Greece
http://www.delta-pi-systems.eu
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