Background

2. 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!!

3. 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)

4. 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.

5. 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

6. Direction Cosines • Notation:Angle between xi axis & xjaxis  (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

7. So: Rewrite 2d coordinate rotation relations in terms of direction cosines as: x1 = λ11 x1 +λ12 x2 x2 = λ21 x1 +λ22 x2 Or: xi = ∑jλij xj (i,j = 1,2) • Generalize to general rotation of axes in 3d: • Angle between the xi axis & the xjaxis  (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 = ∑jλijxj (i,j = 1,2,3)

8. Arrange direction cosines into a square matrix: λ11 λ12 λ13 λ = λ21 λ22 λ23 λ31 λ32 λ33 • Coordinate axes as column vectors: x1x1 x =x2 x =x2 x3x2 • Coordinate transformation relation: x = λ x λ Transformation matrix or rotation matrix

9. Example 1.1 Work this example in detail!

10. 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)

11. Also, consider 2 line segments  direction cosines: cosα, cosβ, cosγ, & cosα, cosβ, cosγ • Angle θbetween the lines: • Trig manipulation (Prob. 1-2) gives: cosθ = cosαcosα +cosβcosβ +cosγcosγ (b)

12. 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!

13. Combined results show: ∑jλij 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: ∑iλij λik = δjk (d) (c) &(d) are equivalent!

14. 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)!