Slideshare uses cookies to improve functionality and performance, and to provide you with relevant advertising. If you continue browsing the site, you agree to the use of cookies on this website. See our User Agreement and Privacy Policy.

Slideshare uses cookies to improve functionality and performance, and to provide you with relevant advertising. If you continue browsing the site, you agree to the use of cookies on this website. See our Privacy Policy and User Agreement for details.

Successfully reported this slideshow.

Like this presentation? Why not share!

- Calculus of variations by sauravpatkotwar 1384 views
- Calculus of variations & solution m... by José Pallo 3534 views
- Introduction to CAGD for Inverse Pr... by Delta Pi Systems 539 views
- Introduction to inverse problems by Delta Pi Systems 920 views
- Calculus of variations by Solo Hermelin 1633 views
- Optmization techniques by Deepshika Reddy 1992 views

1,793 views

Published on

A short introduction to calculus of variations.

No Downloads

Total views

1,793

On SlideShare

0

From Embeds

0

Number of Embeds

414

Shares

0

Downloads

31

Comments

0

Likes

2

No embeds

No notes for slide

- 1. Σ YSTEMSIntroduction 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 Delta Pi Systems
- 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. 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. 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. 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. 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. 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. Contact usDelta Pi SystemsOptimization and Control of Processes and SystemsThessaloniki, Greecehttp://www.delta-pi-systems.eu Delta Pi Systems

No public clipboards found for this slide

Be the first to comment