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 Review Ch. 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 Transformations Sect. 1.3 Arbitrary point P in 3d space, labeled with Cartesian coordinates (x 1,x 2,x 3 ). Rotate axes to (x 1,x 2,x 3 ). Figure has 2d Illustration  Easy to show that (2d): x 1 = x 1 cosθ + x 2 sin θ x 2 = -x 1 sin θ + x 2 cos θ = x 1 cos(θ + π/2) + x 2 cosθ

Direction Cosines Notation: Angle between x i axis & x j axis  (x i,x j ) Define the Direction Cosine of the x i axis with respect to the x j axis: λ ij  cos(x i,x j ) For 2d case (figure):  x 1 = x 1 cosθ + x 2 sinθ x 2 = -x 1 sinθ + x 2 cosθ = x 1 cos(θ +π/2) + x 2 cosθ  λ 11  cos(x 1,x 1 ) = cosθ λ 12  cos(x 1,x 2 ) = cos(θ - π/2) = sinθ λ 21  cos(x 2,x 1 ) = cos(θ + π/2) = -sinθ λ 22  cos(x 2,x 2 ) = cosθ

So: Rewrite 2d coordinate rotation relations in terms of direction cosines as: x 1 = λ 11 x 1 + λ 12 x 2 x 2 = λ 21 x 1 + λ 22 x 2 Or: x i = ∑ j λ ij x j (i,j = 1,2) Generalize to general rotation of axes in 3d: Angle between the x i axis & the x j axis  (x i,x j ). Direction Cosine of x i axis with respect to x j axis: λ ij  cos(x i,x j ) Gives: x 1 = λ 11 x 1 + λ 12 x 2 + λ 13 x 3 ; x 2 = λ 21 x 1 + λ 22 x 2 + λ 23 x 3 x 3 = λ 31 x 1 + λ 32 x 2 + λ 33 x 3 Or:x i = ∑ j λ ij x j (i,j = 1,2,3)

Arrange direction cosines into a square matrix: λ 11 λ 12 λ 13 λ = λ 21 λ 22 λ 23 λ 31 λ 32 λ 33 Coordinate axes as column vectors: x 1 x 1 x = x 2 x = x 2 x 3 x 2 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 & x 1, x 2, x 3  α,β,γ Direction cosines of line  cosα, cosβ, cosγ Trig manipulation (See Prob. 1-2) gives: cos 2 α + cos 2 β + cos 2 γ = 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 (x 1,x 2,x 3 ) to (x 1,x 2,x 3 ). Describe by giving the direction cosines of all angles between original axes (x 1,x 2,x 3 ) & final axes (x 1,x 2,x 3 ). 9 direction cosines: λ ij  cos(x i,x j ) Not all 9 are independent! Can show: 6 relations exist between various λ ij : 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 λ ij λ kj = δ ik (c) δ ik  Kronecker delta: δ ik  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 λ ij λ ik = δ jk (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)!