Mechanics Rigid Body Motion, Lecture Notes - Physics, Study notes of Mechanics

Download Mechanics Rigid Body Motion, Lecture Notes - Physics and more Study notes Mechanics in PDF only on Docsity! Mechanics Physics 151 Lecture 8 Rigid Body Motion (Chapter 4) What We Did Last Time ! Discussed scattering problem ! Foundation for all experimental physics ! Defined and calculated cross sections ! Differential cross section and impact parameter ! Rutherford scattering ! Translated into laboratory system ! Angular translation + Jacobian ! Shape of σ(Θ) changes ( ) sin s ds d σ Θ = Θ Θ hitsN I σ= ⋅ 2-D Rotation ! 2-dimensional rotation is specified by a 2×2 matrix ! Try the same thing with 3-d rotation cos sin sin cos x x y y x y θ θ θ θ ′     =    ′ −     ′ ′⋅ ⋅   =   ′ ′⋅ ⋅   i i i j j i j j θ x x′ y y′ i ′j ′ij x x′ y y′z z′ 3D Rotation ! Vector r is represented in x-y-z and x’-y’-z’ as ! Using angles θij between two axes x y z x y z′ ′ ′ ′ ′ ′= + + = + +r i j k i j k 11 12 13cos cos cos x x y z x y zθ θ θ ′ ′ ′ ′ ′= ⋅ = ⋅ + ⋅ + ⋅ = + + r i i i j i k i 21 22 23cos cos cosy x y zθ θ θ′ = + + 31 32 33cos cos cosz x y zθ θ θ′ = + + x x′ y y′z z′ 11θ 12θ 13θ 11 12 13 21 22 23 31 32 33 cos cos cos cos cos cos cos cos cos x x y y z z θ θ θ θ θ θ θ θ θ ′           ′ =      ′           or 3D Rotation ! Simplify formulae by renaming ! Rotation is now expressed by ! We got 9 parameters aij to describe a 3-d rotation ! Only 3 are independent 1 2 3( , , ) ( , , )x y z x x x→ 1 2 3( , , ) ( , , )x y z x x x′ ′ ′ ′ ′ ′→ cosi ij j ij j ij j j j x x a x a xθ′ = = =∑ ∑ Einstein convention: Implicit summation over repeated index Space Inversion ! Space inversion is represented by ! 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 body rotation is represented by proper orthogonal matrices 1 0 0 0 1 0 0 0 1 −   ′ = − = ≡ −   −  r r Sr r 1= −S Rotation Matrix ! A operating on r can be interpreted as ! Rotating r around an axis by an angle ! Positive angle = clockwise rotation ! Rotating the coordinate axes around the same axis by 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 many ways ′ =r Ar Euler Angles ! Transform x-y-z to x’-y’-z’ in 3 steps ( , , )x y z ( , , )ξ η ζ ( , , )ξ η ζ′ ′ ′ ( , , )x y z′ ′ ′ Rotate CCW by φ around z axis Rotate CCW by θ around ξ axis Rotate CCW by ψ around ζ’ axis z x y ξ η ζ z x y ξ ′ η′ζ ′ φ θ z x y z′ x′ y′ ψ x Dx CDx =Ax BCDx Euler’s Theorem ! In other words ! Arbitrary 3-d rotation equals to one rotation around an axis ! Any 3-d rotation leaves one vector unchanged ! For any rotation matrix A ! There exists a vector r that satisfies ! A has an eigenvalue of 1 The general displacement of a rigid body with one point fixed is a rotation about some axis Ar = r Eigenvector with eigenvalue 1 Euler’s Theorem ! If a matrix A satisfies ! Since ! For odd-dimensioned matrices Ar = r ( ) 0−A 1 r = 0 or 0 or 0− = = =A 1 r A -1 1− =A A! ( )− − − − − − A 1 A = 1 A A 1 A = 1 A A 1 = 1 A ! ! ! ! − = −M M 0− − − =A 1 = A 1 Q.E.D. Rotation Vector? ! Euler’s theorem provides another way of describing 3-d rotation ! Direction of axis (2 parameters) and angle of rotation (1) ! It sounds a bit like angular momentum ! Critical difference: commutativity ! Angular momentum is a vector ! Two angular momenta can be added in any order ! Rotation is not a vector ! Two rotations add up differently depending on which rotation is made first Infinitesimal Rotation ! A vector r is rotated by (1 + ε) as ! Euler’s theorem says this equals to a rotation by an infinitesimal angle dΦ around an axis n ( )′ = +r 1 ε r 3 2 1 3 1 2 2 1 3 0 0 0 d d x d d x d d x d d Ω − Ω       ′≡ − = = − Ω Ω =       Ω − Ω    r r r ε r ×r Ω n r dr dΦ d d= Φr r ×n d d= ΦΩ n Axial Vector ! dΩ behaves pretty much like a vector ! dΩ rotates the same way as r with coordinate rotations ! Space inversion S reveals difference ! Ordinary vector flips ! dΩ doesn’t ! Such a “vector” is called an axial vector ! Examples: angular momentum, magnetic field ′ = = −r Sr r ( )d d d d d ′ ′ ′= × ′= − = − × = × r r Ω r r Ω r Ω d d′ =Ω Ω Parity ! Parity operator P represents space inversion ( , , ) ( , , )x y z x y z→ − − −P EigenvalueParityQuantity +1PV* = V*Axial vector −1PV = −VVector −1PS* = −S*Pseudoscalar +1PS = SScalar *V × V = V * *S⋅V V = * *S V = V * *S V = V* * S⋅V V = *V × V = V etc.