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 $y'=f(x,y)$. 11 Nov 19  Monday: $\pi$ 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 $y(x)$ defined on $[a,b]$ to be an extremum of $J(y)=\int _{a}^{b}F(x,y,y')dx$, it must be that for any function $h(x)$ defined on $[a,b]$ that preserves the endpoints of $y$ (that is, $h(a)=0$ and $h(b)=0$), we have ${\frac {d}{d\epsilon }}J(y+\epsilon h)$$|_{\epsilon =0}=0$.

${\frac {d}{d\epsilon }}J(y+\epsilon h)$$|_{\epsilon =0}$

$={\frac {d}{d\epsilon }}\int _{a}^{b}F(x,y+\epsilon h,y'+\epsilon h')dx|_{\epsilon =0}$

Let $F_{n}$ signify F differentiated with respect to its nth variable.

$=\int _{a}^{b}(F_{1}\cdot 0+F_{2}\cdot h+F_{3}\cdot h')dx|_{\epsilon =0}$

$=\int _{a}^{b}(F_{2}(x,y,y')\cdot h+F_{3}(x,y,y')\cdot h')dx$

$=\int _{a}^{b}(F_{2}\cdot h-[{\frac {d}{dx}}F_{3}]\cdot h)dx+F_{3}\cdot h|_{a}^{b}$ (integrating by parts)

Due to the constraints of $h(a)=0$ and $h(b)=0$, $F_{3}\cdot h|_{a}^{b}=0$.

$=\int _{a}^{b}(F_{2}-{\frac {d}{dx}}F_{3})\cdot hdx$

As this must be equal to 0 for all h satisfying the endpoint constraints, we must have that $F_{2}={\frac {d}{dx}}F_{3}$, or in other terms, $F_{y}-{\frac {d}{dx}}F_{y}'=0$.

Special cases (without derivations):

In the case that F does not depend on y', we have $F_{y}=0$

In the case that F does not depend on y, we have $F_{y'}=c$

In the case that F does not depend on x, we have $F-y'F_{y'}=c$