Simple Nature - Light and Matter

18 bestt = t

19 c1 = bestc1

20 c2 = bestc2

21 c3 = (b-c1*a-c2*a**2)/(a**3)

22 print(c1, c2, c3, bestt)

23 if (bestj == 1) and (bestk == 1) :

24 d = d*.5

Derivation of the steady state for damped, driven oscillations

Using the trig identities for the sine of a sum and cosine of a sum, we can change equation [2]

on page 181 into the form

[

(−mω^2 +k) cosδ−bωsinδ−Fm/A

]

sinωt

+

[

(−mω^2 +k) sinδ+bωcosδ

]

cosωt= 0.

Both the quantities in square brackets must equal zero, which gives us two equations we can

use to determine the unknownsAandδ. The results are

δ= tan−^1

bω

mω^2 −k

= tan−^1

ωωo

Q(ω^2 o−ω^2 )

and

A=

Fm

√

(mω^2 −k)^2 +b^2 ω^2

=

Fm

m

√

(ω^2 −ω^2 o)^2 +ω^2 oω^2 Q−^2

.

Proofs relating to angular momentum

Uniqueness of the cross product

The vector cross product as we have defined it has the following properties:

(1) It does not violate rotational invariance.

(2) It has the propertyA×(B+C) =A×B+A×C.

(3) It has the propertyA×(kB) =k(A×B), wherekis a scalar.

Theorem:The definition we have given is the only possible method of multiplying two vectors

to make a third vector which has these properties, with the exception of trivial redefinitions

which just involve multiplying all the results by the same constant or swapping the names

of the axes. (Specifically, using a left-hand rule rather than a right-hand rule corresponds to

multiplying all the results by−1.)

Proof:We prove only the uniqueness of the definition, without explicitly proving that it has

properties (1) through (3).

Using properties (2) and (3), we can break down any vector multiplication (Axˆx+Ayˆy+

Azˆz)×(Bxˆx+Byˆy+Bzˆz) into terms involving cross products of unit vectors.

A “self-term” likeˆx×xˆmust either be zero or lie along thexaxis, since any other direction

would violate property (1). If was not zero, then (−ˆx)×(−xˆ) would have to lie in the opposite

1024 Chapter Appendix 2: Miscellany