User:EGM6341.s11.TEAM1.WILKS/Mtg41

EGM6321 - Principles of Engineering Analysis 1, Fall 2010

Mtg 41: Tue, 30Nov10

Page 41-1

Note: Generating functions

Eq.(5)p.40-3 : General function for $\left\{P_{n}\right\}\Rightarrow \ (1-2\mu \ \rho \ +\rho ^{2})^{-{\frac {1}{2}}}\$

Eq.(6)p.40-4 : General function for "r choose k" ${r \choose k}\Rightarrow \ (1+x)^{r}\$

Note: Inverse square law, pg.40-1

Newton’s law is a claim—that could have been wrong—about the actual relation between the force F on a particle with mass m and its acceleration a. One tests it by calculating the acceleration with a presumed force and comparing it to the measured value. The test will fail if either Newton’s law or the presumed force is wrong. Could F =ma be tested more generally, without recourse to positing forces and looking at actual solutions? It seems not. Kane, String Theory, Physics Today, November 2010



Where $F=kr\$ and $d(t)=r\$

File:Ssc2008-19a.jpg

Where ${\underline {f}}={\frac {-GMm{\underline {{\hat {r}}\ }}}{r^{2}}}\$

Page 41-2

2 recurrence relationships:

$\displaystyle {\begin{aligned}\mu \ P_{n}'-P_{n-1}'=nP_{n}\end{aligned}}$

(RR1)

Note: not useful to generate $P_{n}\$ from previously known $P_{k},k=0,1,...,n-1\$ but useful to obtain Legendre differential equation together with recurrence relationship 2.

$\displaystyle {\begin{aligned}(n+1)P_{n+1}-(2n+1)\mu \ P_{n}+nP_{n-1}'=0\end{aligned}}$

(RR2)

Note: RR2 useful to generate $\left\{P_{n}\right\}\$ knowing $P_{0}(x)=1,P_{1}(x)=x\$

HW7.9: Generate $\left\{P_{2},...,P_{6}\right\}\$ using RR2 starting from $P_{0},P_{1}\$ cf Eq.(4) - Eq.(6) p.36-2

Derive RR1: $A(\mu \ ,\rho \ ):=1-2\mu \ \rho \ +\rho ^{2}\$

Where $-x=-2\mu \ \rho \ +\rho ^{2}\$

Then from Eq.(6) and Eq.(7) p.40-3: ${\frac {1}{\sqrt {A(\mu \ ,\rho \ )}}}=\alpha _{0}+\alpha _{1}(2\mu \ \rho \ -\rho ^{2})+\alpha _{2}(2\mu \ \rho \ -\rho ^{2})^{2}+...\$

Where $(2\mu \ \rho \ -\rho ^{2})^{2}=4\mu ^{2}\rho ^{2}-4\mu \ \rho ^{3}+\rho ^{4}\$