12-267/Derivation of Euler-Lagrange

DBN: Classes: 2012-13: 12-267 / Navigation Additions to this web site no longer count towards good deed points. # Week of... Notes and Links 1 Sep 10 About This Class.  Monday: Introduction and the Brachistochrone.  Tuesday: More on the Brachistochrone, administrative issues. Tuesday Notes.  Friday: Some basic techniques: first order linear equations. 2 Sep 17  Monday: Separated equations, escape velocities. HW1.  Tuesday: Escape velocities, changing source and target coordinates, homogeneous equations.  Friday: Reverse engineering separated and exact equations. 3 Sep 24  Monday: Solving exact equations, integration factors. HW2.  Tuesday: Statement of the Fundamental Theorem. Class Photo.  Friday: Proof of the Fundamental Theorem. 4 Oct 1  Monday: Last notes on the fundamental theorem. HW3.  Tuesday Hour 1: The chain law, examples of variational problems.  Tuesday Hour 2: Deriving Euler-Lagrange.  Friday: Reductions of Euler-Lagrange. 5 Oct 8 Monday is thanksgiving.  Tuesday: Lagrange multiplyers and the isoperimetric inequality. HW4.  Friday: More Lagrange multipliers, numerical methods. 6 Oct 15  Monday: Euler and improved Euler.  Tuesday: Evaluating the local error, Runge-Kutta, and a comparison of methods.  Friday: Numerical integration, high order constant coefficient homogeneous linear ODEs. 7 Oct 22  Monday: Multiple roots, reduction of order, undetermined coefficients.  Tuesday: From systems to matrix exponentiation. HW5. Term Test on Friday. 8 Oct 29  Monday: The basic properties of matrix exponentiation.  Tuesday: Matrix exponentiation: examples.  Friday: Phase Portraits. HW6. Nov 4 was the last day to drop this class 9 Nov 5  Monday: Non-homogeneous systems.  Tuesday: The Catalan numbers, power series, and ODEs.  Friday: Global existence for linear ODEs, the Wronskian. 10 Nov 12 Monday-Tuesday is UofT November break. HW7.  Friday: Series solutions for [math]\displaystyle{ y'=f(x,y) }[/math]. 11 Nov 19  Monday: [math]\displaystyle{ \pi }[/math] is irrational, more on the radius of convergence.  Tuesday (class): Airy's equation, Fuchs' theorem.  Tuesday (tutorial): Regular singular points. HW8.  Friday: Discussion of regular singular points.. 12 Nov 26  Monday: Frobenius series by computer. Qualitative Analysis Handout (PDF).  Tuesday: The basic oscillation theorem. Handout on the Frobenius Method. HW9.  Friday: Non-oscillation, Sturm comparison. 13 Dec 3  Monday: More Sturm comparisons, changing the independent variable.  Tuesday: Amplitudes of oscillations. Last class was on Tuesday! F1 Dec 10 F2 Dec 17 The Final Exam (time, place, style, office hours times) Register of Good Deeds Add your name / see who's in!

Disclamer: This is a student prepared note based on the lecure of Tuesday October 2nd.

For a function [math]\displaystyle{ y(x) }[/math] defined on [math]\displaystyle{ [a, b] }[/math] to be an extremum of [math]\displaystyle{ J(y) = \int_a^b F(x, y, y') dx }[/math], it must be that for any function [math]\displaystyle{ h(x) }[/math] defined on [math]\displaystyle{ [a, b] }[/math] that preserves the endpoints of [math]\displaystyle{ y }[/math] (that is, [math]\displaystyle{ h(a) = 0 }[/math] and [math]\displaystyle{ h(b) = 0 }[/math]), we have [math]\displaystyle{ \frac{d}{d \epsilon } J(y + \epsilon h) }[/math][math]\displaystyle{ |_{\epsilon = 0} = 0 }[/math].

[math]\displaystyle{ \frac{d}{d \epsilon } J(y + \epsilon h) }[/math][math]\displaystyle{ |_{\epsilon = 0} }[/math]

[math]\displaystyle{ = \frac{d}{d \epsilon } \int_a^b F(x, y + \epsilon h, y' + \epsilon h') dx |_{\epsilon = 0} }[/math]

Let [math]\displaystyle{ F_n }[/math] signify F differentiated with respect to its nth variable.

[math]\displaystyle{ = \int_a^b (F_1 \cdot 0 + F_2 \cdot h + F_3 \cdot h') dx |_{\epsilon = 0} }[/math]

[math]\displaystyle{ = \int_a^b (F_2(x, y, y') \cdot h + F_3(x, y, y') \cdot h') dx }[/math]

[math]\displaystyle{ = \int_a^b (F_2 \cdot h - [\frac{d}{dx} F_3] \cdot h) dx + F_3 \cdot h |_a^b }[/math] (integrating by parts)

Due to the constraints of [math]\displaystyle{ h(a) = 0 }[/math] and [math]\displaystyle{ h(b) = 0 }[/math], [math]\displaystyle{ F_3 \cdot h |_a^b = 0 }[/math].

[math]\displaystyle{ = \int_a^b (F_2 - \frac{d}{dx} F_3) \cdot h dx }[/math]

As this must be equal to 0 for all h satisfying the endpoint constraints, we must have that [math]\displaystyle{ F_2 = \frac{d}{dx} F_3 }[/math], or in other terms, [math]\displaystyle{ F_y - \frac{d}{dx} F_y' = 0 }[/math].

Special cases (without derivations):

In the case that F does not depend on y', we have [math]\displaystyle{ F_y = 0 }[/math]

In the case that F does not depend on y, we have [math]\displaystyle{ F_{y'} = c }[/math]

In the case that F does not depend on x, we have [math]\displaystyle{ F - y'F_{y'} = c }[/math]