Calculus of Variations: An Introduction to Lagrange's Contributions

PHY 711 Classical Mechanics and Mathematical Methods 10-10:50 AM MWF in Olin103 Lecture notes for Lecture 2 Chapter 3.17 of F&W Introduction to the calculus of variations 1. Mathematical construction 2. Practical use 3. Examples 8/28/2024 PHY 711 Fall 2024 -- Lecture 2 1

2 8/28/2024 PHY 711 Fall 2024 -- Lecture 2

4 PM Olin 101 Refreshments at 3:30 PM Olin Lobby 8/28/2024 PHY 711 Fall 2024 -- Lecture 2 3

Your questions From Thomas At the end of slide 22 I am confused why you set dy/dx=K' and why that is a useful trick? From Conall Calculus of Variation- when trying to minimize y(x), can we always utilize the Euler- Lagrange equation...or do these only apply under specific conditions? For a later lecture -- a. Holonomic Constraints vs non-holonomic constraints. ...holonomic constraints are constraints of motion that can be expressed with an equation that relates coordinates and time? Non-Holonomic constraints also affect motion but are not as easily definable with math? b. The book states " ...k equations that relate the coordinates and possibly time.." When the author refers to coordinates, they are referring to those that pertain to the position of the particles, correct? I guess I am a little confused about the systems in question here. 8/28/2024 PHY 711 Fall 2024 -- Lecture 2 4

More questions From Julia My question is would the calculus of variations method still work for the bead on a string example if there is friction? And what are some real examples of when the calculus of variations method is used? 8/28/2024 PHY 711 Fall 2024 -- Lecture 2 5

The calculus of variation as a mathematical construction. Today and Friday, we will focus on the idea and process, before applying it to physical systems next week. 8/28/2024 PHY 711 Fall 2024 -- Lecture 2 6

According wikipedia Joseph-Louis Lagrange (born Giuseppe Luigi Lagrangia or Giuseppe Ludovico De la Grange Tournier; 25 January 1736 10 April 1813), also reported as Giuseppe Luigi Lagrange or Lagrangia,was an Italian mathematician and astronomer, later naturalized French. He made significant contributions to the fields of analysis, number theory, and both classical and celestial mechanics. 8/28/2024 PHY 711 Fall 2024 -- Lecture 2 7

According to Wikipedia Leonard Euler (April 7, 1707-September 18, 1783) Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in many other branches of mathematics such as analytic number theory, complex analysis, and infinitesimal calculus. He introduced much of modern mathematical terminology and notation, including the notion of a mathematical function. He is also known for his work in mechanics, fluid dynamics, optics, astronomy and music theory. 8/28/2024 PHY 711 Fall 2024 -- Lecture 2 8

In Chapter 3, the notion of Lagrangian dynamics is developed; reformulating Newton s laws in terms of minimization of related functions. In preparation, we need to develop a mathematical tool known as the calculus of variation . Minimization of a simple function local minimum dV = 0 dx global minimum 8/28/2024 PHY 711 Fall 2024 -- Lecture 2 9

Minimization of a simple function , ) ( function a Given x V value(s) the find of V x x for which ( minimized is ) maximized) (or . dV = Necessary condition : 0 dx local minimum dV = 0 dx global minimum 8/28/2024 PHY 711 Fall 2024 -- Lecture 2 10

Functional minimization of an integral relationship Consider a family of functions ( ), with fixed end points y x dy dx = = ( ) y x and ( ) and an integral form ( ), y x , . y y x y L x i i f f dy dx Find the function ( ) which extremizes y x ( ), y x , . L x = Necessary condition: 0 L Example : 1 , 1 ( ) dx ( ) dy ) 2 2 = + L ( 0 , 0 8/28/2024 PHY 711 Fall 2024 -- Lecture 2 11

Difference between minimization of a function V(x) and the minimization in the calculus of variation. Minimization of a function V(x) Know V(x) Find x0 such that V(x0) is a minimum. Calculus of variation For i x x want to find a function ( ) x y x f that minimizes an integral that depends on ( ). The analysis involves deriving and solving a differential equation for the function ( ). y x y x 8/28/2024 PHY 711 Fall 2024 -- Lecture 2 12

Example: (1,1) ( ) ( ) 2 2 = + L dx dy ( ) 0,0 2 1 dy dx = + 1 dx Sample functions: 0 1 1 4 = = + = ( ) 1 1.4789 y x x L dx 1 x 0 1 = = 1 1 + = = ( ) 2 1.4142 y x x L dx 2 0 1 = = 1 4 + = 2 2 ( ) 1.4789 y x x L x dx 3 0 8/28/2024 PHY 711 Fall 2024 -- Lecture 2 13

Calculus of variation example for a pure integral function dy function the Find ( which ) extremizes ( ), , y x L y x x dx x dy dy f x where ( ), , ( ), , . L y x x f y x x dx dx dx i = Necessary condition : 0 L + At any let , ( ) ( ) x ( ) x y x y x y x ( ) ( ) ( ) dy x dy dy x + dx dx dx Formally : x f f dy f = + . L y dx ( ) / y dy dx dx dy x , , x x y i dx 8/28/2024 PHY 711 Fall 2024 -- Lecture 2 14

Comment on partial derivatives -- function ( ( , ) l im da a da , ) b f a + ( , ) f a b f f a da b f a 0 b f a f b = + df da d b b a a b 8/28/2024 PHY 711 Fall 2024 -- Lecture 2 15

Comment about notation concerning functional dependence and partial derivatives Suppose , , represent independent variables that determine a function : We write ( , , ). A partial derivative with respect to implies that we hold , fixed and infinitessimally change y z f x 0 , lim y z x x x y z f x y z f x x + , , ) x y z ( ( , , f x ) f x y z = , y z 8/28/2024 PHY 711 Fall 2024 -- Lecture 2 16

After some derivations, we find , x f f dy f = + L y dx ( ) / y dy dx dx dy x , x x y i dx x f d f f = dx y = 0 for all x x x ( ) i f / y dx dy dx dy x , , x x y i dx f d f = 0 for all x x x ( ) i f / y dx dy dx dy , , x x y dx Note that this is a total derivative 8/28/2024 PHY 711 Fall 2024 -- Lecture 2 17

Some derivations -- Consider the term x f dy dx f dx : ( ) / dy dx x , x y i * = dy dx d dx If ( ) is a well-defined function, then y x y x x f dy dx f d dx f f dx = y dx ( ) ( ) / / dy dx dy dx x x , , x y x y i i x d dx f d dx f f y dx = y ( ) ( ) / / dy dx dy dx x , , x y x y i 8/28/2024 PHY 711 Fall 2024 -- Lecture 2 18

Note that the infinitessimal variation of the function ( ) notation is meant to imply a general y y x 8/28/2024 PHY 711 Fall 2024 -- Lecture 2 19

Clarification -- what is the meaning of the following statement: dy d y dx dx = Up to now, the operator is not well defined and meant to represent dy da a general infinitessimal difference. Suppose t hat ,where y a appears in the functional form somehow. For most functional forms ( , ) y x a d dxda 2 2 ( , ) y x a dadx d dxda d that one can think of, = . One can show this to be 2 2 ( , ) a ( , ) y x a dadx y x d ( ) ( ) x = = + 1 a a the case even for ( , ) where = 1 ln . y x a x x a (Note that here we are being imprecise wrt partial and total derivatives.) 8/28/2024 PHY 711 Fall 2024 -- Lecture 2 20

Some derivations (continued)-- x d dx f d dx f f y dx y ( ) ( ) / / dy dx dy dx x , , x y x y i x f x f d dx f f y dx = y ( ) ( ) / / dy dx dy dx x , , x y x y i x i x d dx f f y dx = 0 ( ) / dy dx x , x y i Euler-Lagrange equation: f y d dx f = 0 for all x x x ( ) i f / dy dx dy dx , x , x y 8/28/2024 PHY 711 Fall 2024 -- Lecture 2 21

Clarfication Why does this term go to zero? x d dx f d dx f f y dx y ( ) ( ) / / dy dx dy dx x , , x y x y i x f x f d dx f f y dx = y ( ) ( ) / / dy dx dy dx x , , x y x y i x i x d dx f f y dx = 0 ( ) / dy dx x , x y i Answer -- By construction ( ) y x = = ( ) 0 y x i f 8/28/2024 PHY 711 Fall 2024 -- Lecture 2 22

Recap -- L x f f dy f = + y dx ( ) / y dy dx dx dy x , , x x y i dx x f d f f = dx y = 0 for all x x x ( ) i f / y dx dy dx dy x , , x x y i dx f d f = 0 for all x x x ( ) i f / y dx dy dx dy , , x x y dx Here we conclude that the integrand has to vanish at every argument in order for the integral to be zero a. Necessary? b. Overkill? 8/28/2024 PHY 711 Fall 2024 -- Lecture 2 23

= = Example : End points - - ) 0 ( y ; 0 ) 1 ( y 1 2 2 1 dy dy dy = + = + 1 ( ), , 1 L dx f y x x dx dx dx 0 f d f = 0 ( ) / y dx dy dx dy , , x x y dx / d dy dx = 0 ( ) dx 2 + 1 / dy dx Solution: + = / dy dx dy dx K = ' K K ( ) 2 2 1 K 1 / dy dx = ( ) y x x = + ( ) y x ' K x C 8/28/2024 PHY 711 Fall 2024 -- Lecture 2 24

Example : Lamp shade shape y(x) x 2 2 dy dy dy f x = + = + 2 1 ( ), , 1 A x dx f y x x x dx dx dx i f d f y = 0 ( ) / y dx dy dx dy , , x x y dx xi yi / d xdy dx = 0 ( ) dx 2 + 1 / dy dx x xf yf 8/28/2024 PHY 711 Fall 2024 -- Lecture 2 25

/ d dx xdy dx = 0 ( ) 2 + 1 / dy dx / xdy dx = K 1 ( ) 2 + 1 / dy dx 1 dy dx = 2 x 1 K 1 2 x x K = + ( ) y x ln 1 K K 2 1 2 K 1 1 8/28/2024 PHY 711 Fall 2024 -- Lecture 2 26

General form of solution -- 2 x x K = + ( ) y x ln 1 K K 2 1 2 K 1 1 = = + S uppose 1 and 2 3 K K 1 2 + 2 + 3 = ( ) y x ln 2 1 x x 8/28/2024 PHY 711 Fall 2024 -- Lecture 2 27

+ 2 + 3 = ( ) y x ln 2 1 x x 2 2 dy dx = + = 2 1 15.02014144 A x dx 1 (according to Maple) 8/28/2024 PHY 711 Fall 2024 -- Lecture 2 28

Another example: (Courtesy of F. B. Hildebrand, Methods of Applied Mathematics) ( ) x ( ) 0 ( ) 1 = = Consider curves all with and 0 1 y y y minimize that integral the : 2 1 dy = 2 constant for 0 I ay dx a dx 0 Euler - Lagrange equation : 2 d y 2 + = 0 ay dx ( ) sin a x = ( ) ( ) y x sin a 8/28/2024 PHY 711 Fall 2024 -- Lecture 2 29

dy Review for : ( ), , , f y x x dx x dy f x necessary a condition extremize to : f y(x), ,x dx dx i f d f dy = 0 Euler-Lagrange equation ( ) / y dx dy dx dy , , x x y dx dy Note that for ( ), , , f y x x dx df f dy f d f = + + ( ) / dx y dx dy dx dx dx x d f dy f d dy f dy = + + ( ) ( ) / / dx dy dx dx dy dx dx dx x d f f Alternate Euler-Lagrange equation = f ( ) / dx dy dx dx x 8/28/2024 PHY 711 Fall 2024 -- Lecture 2 30