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 + x2sinθx2 = -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) = -sinθλ22 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 xλ 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)!