Background• The Physics Knowledge Expected for this Course:

Newton’s Laws of Motion the “Theme of the Course”

– Energy & momentum conservation– Elementary E&M

• The Math Knowledge Expected for this Course:– Differential & integral calculus

– Differential equations

– Vector calculus

– See the Math Review in Chapter 1!!

Math ReviewCh. 1: Matrices, Vectors, & Vector Calculus

• Definition of a Scalar: Consider an array of

particles in 2 dimensions,

as in Figure a. The particle

masses are labeled by their

x & y coordinates as

M(x,y)

• If we rotate the

coordinate axes, as in

Figure b, we find

M(x,y) M(x,y) That is, the masses are

obviously unchanged

by a rotation of

coordinate axes. So, the masses are Scalars!

• Scalar Any quantity which is invariant under a coordinate transformation.

Coordinate TransformationsSect. 1.3

• Arbitrary point P in 3d space, labeled with Cartesian coordinates (x1,x2,x3). Rotate axes to (x1,x2,x3). Figure has 2d Illustration

• Easy to show that (2d): x1 = x1cosθ + x2sin θ x2 = -x1sin θ + x2cos θ = x1cos(θ + π/2) + x2cosθ

Direction Cosines• Notation: Angle between xi axis & xj axis (xi,xj)

• Define the Direction Cosine of the xi axis with respect to the xj axis:

λij cos(xi,xj)

• For 2d case (figure): x1 = x1cosθ + x2sinθ

x2 = -x1sinθ + x2cosθ = x1cos(θ +π/2) + x2cosθ

λ11 cos(x1,x1) = cosθ

λ12 cos(x1,x2) = cos(θ - π/2) = sinθ

λ21 cos(x2,x1) = cos(θ + π/2) = -sinθ

λ22 cos(x2,x2) = cosθ

• So: Rewrite 2d coordinate rotation relations in terms of direction cosines as:

x1 = λ11 x1 + λ12 x2

x2 = λ21 x1 + λ22 x2

Or: xi = ∑j λij xj (i,j = 1,2)• Generalize to general rotation of axes in 3d: • Angle between the xi axis & the xj axis (xi,xj).

Direction Cosine of xi axis with respect to xj axis:

λij cos(xi,xj) Gives:

x1 = λ11x1 + λ12x2 + λ13x3 ; x2 = λ21x1+ λ22x2 + λ23x3

x3 = λ31x1 + λ32x2 + λ33x3

• Or: xi = ∑j λijxj (i,j = 1,2,3)

• Arrange direction cosines into a square matrix:

λ11 λ12 λ13

λ = λ21 λ22 λ23

λ31 λ32 λ33

• Coordinate axes as column vectors:

x1 x1

x = x2 x = x2

x3 x2

• Coordinate transformation relation: x = λ x

λ Transformation matrix or rotation matrix

Example 1.1

Work this example

in detail!

Rotation Matrices Sect. 1.4

• Consider a line

segment, as in Fig. Angles between line

& x1, x2, x3 α,β,γ

• Direction cosines of line cosα, cosβ, cosγ• Trig manipulation (See Prob. 1-2) gives:

cos2α + cos2β + cos2γ = 1 (a)

• Also, consider 2

line segments

direction cosines: cosα, cosβ, cosγ, &

cosα, cosβ, cosγ• Angle θ between

the lines:

• Trig manipulation (Prob. 1-2) gives:

cosθ = cosα cosα +cosβcosβ +cosγcosγ (b)

Arbitrary Rotations• Consider an arbitrary rotation from axes

(x1,x2,x3) to (x1,x2,x3).

• Describe by giving the direction cosines of all angles between original axes (x1,x2,x3) & final axes (x1,x2,x3). 9 direction

cosines: λij cos(xi,xj)

• Not all 9 are independent! Can show:

6 relations exist between various λij:

Giving only 3 independent ones.• Find 6 relations using Eqs. (a) & (b) for each primed axis in

unprimed system. • See text for details & proofs!

• Combined results show:

∑j λij λkj = δik (c)

δik Kronecker delta: δik 1, (i = k); = 0 (i k).

(c) Orthogonality condition.

Transformations (rotations) which satisfy (c)

are called

ORTHOGONAL TRANSFORMATIONS.

• If consider unprimed axes in primed system, can also show: ∑i λij λik = δjk (d)

(c) &(d) are equivalent!

• Up to now, we’ve considered

P as a fixed point & rotated the

axes (Fig. a shows for 2d)

• Could also choose the axes

fixed & allow P to rotate

(Fig. b shows for 2d) • Can show (see text) that: Get the

same transformation whether rotation acts on the frame of reference (Fig. a) or the on point (Fig. b)!