Loading [MathJax]/jax/output/HTML-CSS/jax.js
Skip to main content
Library homepage
 

Text Color

Text Size

 

Margin Size

 

Font Type

Enable Dyslexic Font
Mathematics LibreTexts

1.2: Ordinary Differential Equations

( \newcommand{\kernel}{\mathrm{null}\,}\)

Set

$$E(v)=\int_a^bf(x,v(x),v'(x))\ dx\]

and for given ua, ubR

$$V=\{v\in C^2[a,b]:\ v(a)=u_a,\ v(b)=u_b\},\]

where y and f is sufficiently regular. One of the basic problems in the calculus of variation is

(P) minvVE(v).

Euler equation

Let uV be a solution of (P), then

$$\frac{d}{dx}f_{u'}(x,u(x),u'(x))=f_u(x,u(x),u'(x))\]

in (a,b).

Exercise 1.2.1: Proof

For fixed ϕC2[a,b] with ϕ(a)=ϕ(b)=0 and real ϵ, |ϵ|<ϵ0, set g(ϵ)=E(u+ϵϕ). Since g(0)g(ϵ) it follows g(0)=0. Integration by parts in the formula for g(0) and the following basic lemma in the calculus of variations imply Euler's equation.

Admissible Variations

Figure 1.2.1.1: Admissible Variations

Basic lemma in the calculus of variations. Let hC(a,b) and

bah(x)ϕ(x) dx=0

for all ϕC10(a,b). Then h(x)0 on (a,b).

Proof. Assume h(x0)>0 for an x0(a,b), then there is a δ>0 such that (x0δ,x0+δ)(a,b) and h(x)h(x0)/2 on (x0δ,x0+δ).
Set

$$
\phi(x)
=\left\{(δ2|xx0|2)2x(x0δ,x0+δ)0x(a,b)[x0δ,x0+δ] \right. .
\]

Thus ϕC10(a,b) and

$$\int_a^b h(x)\phi(x)\ dx\ge \frac{h(x_0)}{2}\int_{x_0-\delta}^{x_0+\delta}\phi(x)\ dx>0,\]

which is a contradiction to the assumption of the lemma.

Contributors and Attributions


This page titled 1.2: Ordinary Differential Equations is shared under a not declared license and was authored, remixed, and/or curated by Erich Miersemann.

Support Center

How can we help?