Mechanics Phsics 151 Lecture 8 Rigid Bod Motion (Chapter 4) What We Did Last Time! Discussed scattering problem! Foundation for all experimental phsics! Defined and calculated cross sections! Differential cross section and impact parameter! Rutherford scattering! Translated into laborator sstem! Angular translation + Jacobian! Shape of σ(θ) changes σ ( Θ ) = Nhits s sin = I σ ds Θ dθ
Goals For Toda! Start discussing rigid-bod motion! Multi-particle sstem with fixed shape! Concentrate on representing the rotation! Which generalized coordinates should we use?! Define Euler angles! Define infinitesimal rotation! Will use this for angular velocities, etc! Toda s lecture is largel mathematical! Assume knowledge of linear algebra Rigid Bod! Multi-particle sstem with fixed distances! Constraints: r = r r = const for all i, j ij i j! How should we define generalized coordinates?! How man independent coordinates are there?! If ou start from 3N and subtract the number of constraints 2 N( N 1) 7N N 3N = 0 for N 7 2 2 Not all the constraints are independent! Right answer: 3 translation and 3 rotation = 6 Toda s theme
2-D Rotation! 2-dimensional rotation is specified b a 2 2 matrix x cosθ sinθ x = sinθ cosθ j i x j i i i j x θ x = j i j j i! Tr the same thing with 3-d rotation z z x x 3D Rotation! Vector r is represented in x--z and x - -z as r = xi+ j + zk = x i + j + z k! Using angles θ ij between two axes x = r i = xi i + j i + zk i = cosθ x + cosθ + cosθ z 11 12 13 = cosθ21x+ cosθ22 + cosθ23z z = cosθ x+ cosθ + cosθ z x θ x 11 or 31 32 33 x cosθ11 cosθ12 cosθ13 x = cosθ cosθ cosθ 21 22 23 z cosθ31 cosθ32 cosθ 33 z z θ 13 z θ 12
3D Rotation! Simplif formulae b renaming ( x, z, ) ( x, x, x) 1 2 3 x z x1 x2 x3! Rotation is now expressed b x = cosθ x = a x = a x! We got 9 parameters a ij to describe a 3-d rotation! Onl 3 are independent (,, ) (,, ) i ij j ij j ij j j j Einstein convention: Implicit summation over repeated index Constraints of Rotation! Rotation cannot change the length of an vector! Exactl the constraints we need for rigid bod motion 2 r = xx = xx i i i i! Using the transformation matrix x = ax x = x a x a x i ij j therefore aa i i ij j ik k 1 ( j = k) = δ 0 ( j k) ij ik jk! Matrix A = [a ij ] is orthogonal AA! = 1 Transpose of A 6 conditions reduces free parameters from 9 to 3
Orthogonal Matrix! Goldstein Section 4.3 covers algebra of matrices! You must have learned this alread! Orthogonal matrix A satisfies AA! = 1! Consider the determinants 2 AA! = A! A = A =1 A =± 1! A = +1 " proper matrix! A = 1 " improper matrix A a a a 11 12 13 = a21 a22 a 23 a31 a32 a 33 aa = δ ij ik jk Transposed matrix Space Inversion! Space inversion is represented b 1 0 0 r = r = Sr 0 1 0 r S = 1 0 0 1! S is orthogonal Doesn t change distances! But it cannot be a rotation! Coordinate axes invert to become left-handed! Orthogonal matrices with A = 1 does this! Rigid bod rotation is represented b proper orthogonal matrices
Rotation Matrix! A operating on r can be interpreted as! Rotating r around an axis b an angle! Positive angle = clockwise rotation! Rotating the coordinate axes around the same axis b the same angle in the opposite direction! Positive angle = counter clockwise rotation! Both interpretations are useful! We are more interested in the latter for now! How do we write A with 3 parameters?! There are man was = r Ar Euler Angles! Transform x--z to x - -z in 3 steps ( x, z, ) x ( ξ, η, ζ ) ( ξ, η, ζ ) ( x,, z ) z ζ φ ξ η Rotate CCW b φ around z axis Rotate CCW b θ around ξ axis Rotate CCW b ψ around ζ axis ζ z θ x ξ η z x Dx CDx Ax = BCDx z x ψ x
Euler Angles cosφ sinφ 0 D = sinφ cosφ 0 0 0 1 1 0 0 C = 0 cosθ sinθ 0 sinθ cosθ! Definition of Euler angles is somewhat arbitrar! Ma rotate around different axes in different order! Man conventions exist Watch out! cosψ sinψ 0 B = sinψ cosψ 0 0 0 1 cosψ cosφ cosθ sinφsinψ cosψ sinφ + cosθ cosφsinψ sinψ sinθ A = sinψ cosφ cosθ sinφcosψ sinψ sinφ cosθ cosφcosψ cosψ sinθ + sinθsinφ sinθ cosφ cosθ Rigid Bod Motion! Motion of a rigid bod can be described b:! Define x - -z axes (bod axes) attached to the rigid bod! Same direction as x--z (space axes) at t = 0! Origin fixed at one point of the rigid bod (e.g. CoM)! Use R(t) to describe the motion of the origin! Use A(t) to describe the rotation of the x - -z axes! Use Euler angles φ(t), θ(t), ψ(t)! A(0) = 1 " φ(0) = θ(0) = ψ(0) = 0! 6 independent coordinates (x,, z, φ, θ, ψ)
Euler s Theorem The general displacement of a rigid bod with one point fixed is a rotation about some axis! In other words! Arbitrar 3-d rotation equals to one rotation around an axis! An 3-d rotation leaves one vector unchanged! For an rotation matrix A! There exists a vector r that satisfies! A has an eigenvalue of 1 Ar = r Eigenvector with eigenvalue 1 Euler s Theorem! If a matrix A satisfies Ar = r ( A 1) r = 0 A 1= 0 or r = 0 or A-1 = 0 1! Since A = A! ( A 1) A=1! A! A 1 A! = 1 A! A 1 = 1 A! For odd-dimensioned matrices A 1 = A 1 =0 Q.E.D. M = M
Rotation Vector?! Euler s theorem provides another wa of describing 3-d rotation! Direction of axis (2 parameters) and angle of rotation (1)! It sounds a bit like angular momentum! Critical difference: commutativit! Angular momentum is a vector! Two angular momenta can be added in an order! Rotation is not a vector! Two rotations add up differentl depending on which rotation is made first Infinitesimal Rotation! Small (infinitesimal) rotations are commutative! The can be represented b vectors! We also need them to describe how a rigid bod changes orientation with time! Infinitesimal rotation must be close to non-rotation x = x + ε x or x = ( 1+ ε) x ε " 1 i i ij j! Two successive infinitesimal rotations make ( 1+ ε1)( 1+ ε2) = 1+ ε1+ ε2 + εε 1 2 = 1+ ε + ε 1 2! Obviousl commutative ij 2 nd order of ε vanishes
Infinitesimal Rotation! Inverse of an infinitesimal rotation is ( 1+ ε) 1 = 1 ε 1! Using A = A! 1+ ε! = 1 ε ε! = ε ε is antismmetric! We can write ε as 0 dω3 dω2 ε = d 3 0 d Ω Ω1 dω2 dω1 0 ( 1+ ε)( 1 ε) = 1+ ε ε= 1 dω = ( dω1, dω2, dω3) behaves almost like a vector We ll see Infinitesimal Rotation! A vector r is rotated b (1 + ε) as r = ( 1+ ε) r 0 dω3 dω2 x1 dr r r = εr = d 3 0 d 1 x Ω Ω 2 = r dω dω2 dω1 0 x 3! Euler s theorem sas this equals to a n rotation b an infinitesimal angle dφ dφ around an axis n dr = r ndφ dω = ndφ dr r
Axial Vector! dω behaves prett much like a vector! dω rotates the same wa as r with coordinate rotations! Space inversion S reveals difference! Ordinar vector flips r = Sr = r dω = dω! dω doesn t ( dr) = r dω = dr = r dω = r dω! Such a vector is called an axial vector! Examples: angular momentum, magnetic field Parit! Parit operator P represents space inversion P ( x, z, ) ( x,, z) Quantit Scalar Pseudoscalar Vector Axial vector Parit PS = S PS * = S * PV = V PV * = V * Eigenvalue +1 1 1 +1 * V V=V V V = S * * * V V =V * * V V = S * * S V=V * * S V =V etc.
Summar! Discussed 3-dimensional rotation! Preparation for rigid bod motion! Movement in 3-d + Rotation in 3-d = 6 coordinates! Looked for was to describe 3-d rotation! Euler angles one of the man possibilities! Euler s theorem! Defined infinitesimal rotation dω! Commutative (unlike finite rotation)! Behaves as an axial vector (like angular momentum)! Read to go back to phsics