This document provides an overview of calculus of variations, which generalizes the method of finding extrema of functions to functionals. It discusses how functionals take on extreme values when their path or curve satisfies certain necessary conditions, analogous to single-variable calculus. These necessary conditions are derived by applying the calculus of variations methodology to functionals dependent on a path and finding the Euler-Lagrange equation. Several examples from physics are described where extremizing a functional corresponds to minimizing time, length, or other physical quantities.
Hello, I am Subhajit Pramanick. I and my friend, Sougata Dandapathak, both presented this ppt in our college seminar. It is basically based on the origin of calculus of variation. It consists of several topics like the history of it, the origin of it, who developed it, application of it, advantages and disadvantages etc. The main aim of this presentation is to increase our mathematical as well as physical conception on advanced classical mechanics. We hope you will all enjoy by reading this presentation. Thank you.
Partial Differential Equation plays an important role in our daily life.In mathematics, a partial differential equation (PDE) is a differential equation that contains unknown multivariable functions and their partial derivatives. PDEs are used to formulate problems involving functions of several variables, and are either solved by hand, or used to create a computer model. A special case is ordinary differential equations (ODEs), which deal with functions of a single variable and their derivatives.
PDEs can be used to describe a wide variety of phenomena such as sound, heat, diffusion, electrostatics, electrodynamics, fluid dynamics, elasticity, or quantum mechanics. These seemingly distinct physical phenomena can be formalised similarly in terms of PDEs. Just as ordinary differential equations often model one-dimensional dynamical systems, partial differential equations often model multidimensional systems. PDEs find their generalisation in stochastic partial differential equations.
Hello, I am Subhajit Pramanick. I and my friend, Sougata Dandapathak, both presented this ppt in our college seminar. It is basically based on the origin of calculus of variation. It consists of several topics like the history of it, the origin of it, who developed it, application of it, advantages and disadvantages etc. The main aim of this presentation is to increase our mathematical as well as physical conception on advanced classical mechanics. We hope you will all enjoy by reading this presentation. Thank you.
Partial Differential Equation plays an important role in our daily life.In mathematics, a partial differential equation (PDE) is a differential equation that contains unknown multivariable functions and their partial derivatives. PDEs are used to formulate problems involving functions of several variables, and are either solved by hand, or used to create a computer model. A special case is ordinary differential equations (ODEs), which deal with functions of a single variable and their derivatives.
PDEs can be used to describe a wide variety of phenomena such as sound, heat, diffusion, electrostatics, electrodynamics, fluid dynamics, elasticity, or quantum mechanics. These seemingly distinct physical phenomena can be formalised similarly in terms of PDEs. Just as ordinary differential equations often model one-dimensional dynamical systems, partial differential equations often model multidimensional systems. PDEs find their generalisation in stochastic partial differential equations.
Lesson 15: Exponential Growth and Decay (Section 021 slides)Matthew Leingang
Many problems in nature are expressible in terms of a certain differential equation that has a solution in terms of exponential functions. We look at the equation in general and some fun applications, including radioactivity, cooling, and interest.
Describes the mathematics of the Calculus of Variations.
For comments please contact me at solo.hermelin@gmail.com.
For more presentations on different subjects visit my website on http://www.solohermelin.com
Differential Equations Lecture: Non-Homogeneous Linear Differential Equationsbullardcr
A lecture I presented in Differential Equations, Spring 2006. This was supplemented with a hands-on solution to a random problem with variables designated by students in the class.
This presentation gives example of "Calculus of Variations" problems that can be solved analytical. "Calculus of Variations" presentation is prerequisite to this one.
For comments please contact me at solo.hermelin@gmail.com.
For more presentations on different subjects visit my website at http://www.solohermelin.com.
Lesson 15: Exponential Growth and Decay (Section 021 slides)Matthew Leingang
Many problems in nature are expressible in terms of a certain differential equation that has a solution in terms of exponential functions. We look at the equation in general and some fun applications, including radioactivity, cooling, and interest.
Describes the mathematics of the Calculus of Variations.
For comments please contact me at solo.hermelin@gmail.com.
For more presentations on different subjects visit my website on http://www.solohermelin.com
Differential Equations Lecture: Non-Homogeneous Linear Differential Equationsbullardcr
A lecture I presented in Differential Equations, Spring 2006. This was supplemented with a hands-on solution to a random problem with variables designated by students in the class.
This presentation gives example of "Calculus of Variations" problems that can be solved analytical. "Calculus of Variations" presentation is prerequisite to this one.
For comments please contact me at solo.hermelin@gmail.com.
For more presentations on different subjects visit my website at http://www.solohermelin.com.
First principle, power rule, derivative of constant term, product rule, quotient rule, chain rule, derivatives of trigonometric functions and their inverses, derivatives of exponential functions and natural logarithmic functions, implicit differentiation, parametric differentiation, L'Hopital's rule
Image sciences, image processing, image restoration, photo manipulation. Image and videos representation. Digital versus analog imagery. Quantization and sampling. Sources and models of noises in digital CCD imagery: photon, thermal and readout noises. Sources and models of blurs. Convolutions and point spread functions. Overview of other standard models, problems and tasks: salt-and-pepper and impulse noises, half toning, inpainting, super-resolution, compressed sensing, high dynamic range imagery, demosaicing. Short introduction to other types of imagery: SAR, Sonar, ultrasound, CT and MRI. Linear and ill-posed restoration problems.
This is meant for university students taking either information technology or engineering courses, this course of differentiation, Integration and limits helps you to develop your problem solving skills and other benefits that come along with it.
Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics
2. Motivation
• Dirichlet Principle – One stationary
ground state for energy
• Solutions to many physical problems
require maximizing or minimizing some
parameter I.
• Distance
• Time
• Surface Area
• Parameter I dependent on selected
path u and domain of interest D:
I = ò F x u u dx
D
• Terminology:
( , , x )
• Functional – The parameter I to
be maximized or minimized
• Extremal – The solution path u
that maximizes or minimizes I
3. Analogy to Calculus
• Single variable calculus:
• Functions take extreme values on
bounded domain.
• Necessary condition for extremum
at x0, if f is differentiable:
f ¢( x0 ) = 0
• Calculus of variations:
• True extremal of functional for
unique solution u(x)
• Test function v(x), which vanishes
at endpoints, used to find extremal:
( , , )
b
w( x) =u( x) +e v( x) I ée ù = F x w w dx
ë û ò
x
a
• Necessary condition for extremal:
dI 0
de =
4. Solving for the Extremal
• Differentiate I[e]:
b b
dI de ( e ) = d de ò F ( x , w , w )
dx = ò ¶ F ¶ w ¶ F ¶ w ¶ w ¶ e + ¶ w ¶
e
dx æ ö
ç ¸
ç ¸
è ø
x x
a a x
• Set I[0] = 0 for the extremal, substituting terms for e = 0 :
ew(e ) v( x)
¶ = 0 ( ) ¶ w v x e
¶ = ¶ ( ) ( ) x x
w e v x e
¶ æç ö¸ = x ¶ 0 x
( ) è ø
dI F v F v dx de u u
= æ ¶ + ¶ ö
ò ¶ ¶ 0
æç ö¸ ç ¸ è ø ç ¸
• Integrate second integral by parts:
w v x e
¶ ¶
æç ö¸ = è ø
wæçè0öø¸ u( x) =
wx æçè0öø¸ ux ( x) =
0
b
x
a x
è ø
b b
F vdx F v dx u u
ò ¶ ¶ ¶ + ò¶
x
= a a x
b b b b
F v dx F v d F vdx d F vdx u u dx u dx u
ò¶ ¶ = ¶ - ò ¶ = - ò ¶ x
¶ ¶ ¶
é ù æ ö æ ö
ê ú ç ¸ ç ¸
ê ú ç ¸ ç ¸
ë û è ø è ø
a x x a a x a x
ò ¶ ¶ - ò F
= 0
0
u
x
b b
a a
F vdx d æ ¶ ö
u dx
ç ¸ vdx è ¶ ø
F d F
u dx u
¶ - ¶
¶ ¶ ò =
x
b
a
vdx é æ öù
ê çç ¸¸ú êë è øúû
5. The Euler-Lagrange
• Since v(x) is an arbitraryE funqctionu, thae ontlyi woay fnor the integral to be zero is
for the other factor of the integrand to be zero. (Vanishing Theorem)
0
vdx é æ öù
F d F
u dx u
¶ - ¶
¶ ¶ ò =
ê ¸¸ú êë çç è x
øúû
b
a
• This result is known as the Euler-Lagrange Equation
¶ F = d é ¶ F
ù ¶ u dx ë¶ ê u
ú x
û
• E-L equation allows generalization of solution
extremals to all variational problems.
6. Functions of Two Variables
• Analogy to multivariable calculus:
• Functions still take extreme
values on bounded domain.
• Necessary condition for extremum
at x0, if f is differentiable:
( 0, 0 ) ( 0, 0 ) 0 x y f x y = f x y =
• Calculus of variations method similar:
( , , , , ) x y
I = òò F x y u u u dxdy w( x, y) = u ( x, y) +e v ( x, y)
D
( ) ( , , , , ) x y
æ ¶ ¶ ¶ ¶ ¶ ¶ ö = = çç + + ¸¸ è ¶ ¶ ¶ ¶ ¶ ¶ ø
dI d òò w e
F x y w w w dxdy òò
F w F w F dxdy
d e d e x y
w e w e w
e
D D x y
F vdxdy F v dxdy F v dxdy
u u u
¶ + ¶ + ¶ =
òò ¶ òò ¶ òò ¶
0 x y
D D x D y
¶ é ¶ ù é ¶ ù = ê ú + ê ú ¶ ë¶ û êë¶ úû
òò 0
x y
é¶ F æ - d ¶ F ö d æ ¶ F öù ê ç ¸- çç ¸¸ú vdxdy
= êë ¶ u dx è ¶ u ø dy è ¶ u
øúû
D x y
F d F d F
u dx u dy u
7. Further Extension
• With this method, the E-L equation can be extended to N variables:
¶ F N
é = å
d ¶ F
ù ê ú
¶ u i = 1 dq u
i êë¶ q
úû
i
• In physics, the q are sometimes referred to as generalized position
coordinates, while the uq are referred to as generalized momentum.
• This parallels their roles as position and momentum variables when solving
problems in Lagrangian mechanics formulism.
8. Limitations
• Method gives extremals, but doesn’t indicate maximum or minimum
• Distinguishing mathematically between max/min is more difficult
• Usually have to use geometry of physical setup
• Solution curve u must have continuous second-order derivatives
• Requirement from integration by parts
• We are finding stationary states, which vary only in space, not in time
• Very few cases in which systems varying in time can be solved
• Even problems involving time (e.g. brachistochrones) don’t change in time
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20. Calculus of Variations
Examples in Physics
Minimizing, Maximizing, and Finding Stationary Points
(often dependant upon physical properties and
geometry of problem)
21. Geodesics
A locally length-minimizing curve on a surface
Find the equation y = y(x) of a curve joining points (x1, y1) and (x2, y2) in order
to minimize the arc length
ds = dx2 + dy2 dy and
dy dx y ( x) dx
so
= = ¢
dx
( )
= + ¢
= ò = ò + ¢
ds y x dx
L ds y x dx
( )
2
2
Geodesics minimize path length
1
1
C C
22. Fermat’s Principle
Refractive index of light in an inhomogeneous
medium
, where v = velocity in the medium and n = refractive index
Time of travel =
v c = n
T dt ds 1
nds
ò ò ò
ò
= = =
v c
C C C
( ) ( ) 2
= + ¢
T n x , y 1
y x dx
C
Fermat’s principle states that the path must minimize the time of travel.
23. Brachistochrone Problem
Finding the shape of a wire joining two given points such that
a bead will slide (frictionlessly) down due to gravity will result
in finding the path that takes the shortest amount of time.
The shape of the wire will minimize
time based on the most efficient
use of kinetic and potential energy.
dt s y x dx
( )
2
( ) ( )
2
1 1
v ds
dtd
v v
1 1
, C C
T dt y x dx
v x y
=
= = + ¢
= ò = ò + ¢
24. Principle of Least Action
Energy of a Vibrating String
• Calculus of
variations can
locate saddle points
• The action is
stationary
Action = Kinetic Energy – Potential Energy
é æ ¶ ö 2 æ ¶ ö 2
ù = ê ç ¸ - ú êë è ¶ ø è ç ¶ ø ¸ úû
A u u T u dxdt
t x
at ε = 0
r
d A( u +
e
v)
d
e
Explicit differentiation of A(u+εv) with
respect to ε
A u u v T u v dxdt
= éêr æç ¶ ö¸æç ¶ ö¸- æç ¶ ö¸æç ¶ ö¸ùú = ë è ¶ øè ¶ ø è ¶ øè ¶ øû òò
[ ] 0
t t x x
Integration by parts
= ér ¶ - ¶ ù = ê ¶ ¶ ú ë û òò
[ ] [ ] 2 2
A u u T u v dxdt
2 2 0
t x
2 2 D
v is arbitrary inside the boundary D
[ ]
òò
D
D
2 2
2 2 u T u 0
t x
r ¶ - ¶ =
¶ ¶
This is the wave equation!
25. Soap Film
When finding the shape of a soap bubble that spans a wire
ring, the shape must minimize surface area, which varies
proportional to the potential energy.
Z = f(x,y) where (x,y) lies over a plane region D
The surface area/volume ratio is minimized
in order to minimize potential energy from
cohesive forces.
{( ) ( ) ( )}
x , y bdy D ;
z h x
A u 2 u 2
dxdy
1 x y
D
Î =
= òò + +