1 PHYS 705: Classical Mechanics Rigid Body Motion Introduction + Math Review
2 How to describe a rigid body? Rigid Body - a system of point particles fixed in space i r ij j subject to a holonomic constraint: r ij c ij for all i, j pairs Question: How many independent coordinates are needed to specify the system? Let count them: - N particles at most 3N degrees of freedom 1 2 N N 1 1 1 2 N N N - there are of the constraint equations: - For N large, but many equations are not independent! From first inspection, it is not clear how many independent coordinates are there. r ij c ij
3 How to describe a rigid body? A better way to think about this i 1 d 1 d 3 d 2 2 ANY point i within the rigid body can be located by the distances from three fixed reference points! d, d, d 1 2 3 3 So, we only need to specify the coordinates of the three reference points; then ALL points in the at most 9 dofs rigid body are fixed by the constraint equations, r ij c ij
4 How to describe a rigid body? Again, NOT ALL 9 coordinates for the 3 ref pts are independent! Let count... - 3 coordinates needed to specify the 1 st point - 2 more coordinates to fix the 2 nd point (Since point #2 can be anywhere on the surface of a sphere (fixed distance) centered on the 1 st point.) 1 r 12 2-1 more coordinate to fix the 3 rd point r 13 r 23 (Since point #3 must lie on a circle with 3 fixed distances to the first two points.) Total 6 degrees of freedom
5 How to describe a rigid body? Thus, a rigid body needs only six independent generalized coordinates to specify its configurations. Independent on how many particles even in the limit of a continuous body. The next question is: How should one specify these 6 coordinates?
6 Fixed and Body Axes for a rigid body We will use two sets of coordinates: - 1 set of external fixed coordinates (unprimed) - 1 set of internal body coordinates (primed) z ' z x, yz, x ', y', z' (As the name implied, the body axes are attached to the rigid body.) y ' 3 coords to specify the origin o of the body axes wrt the fixed x o x ' o y system 3 coords to specify the orientation of the body axes wrt to the translated fixed system (dotted axes).
7 Orientation of the Body Axes There are many ways to describe the orientation of the body axes Most often used: Euler s angles (later) but similar to (roll, pitch, yaw) - A sequence of 3 rotations in a standard order to get from the fixed axes (shifted) to the body axes. - This gives a choice for the 3 orientation coordinates
8 Orientation of the Body Axes There are many ways to describe the orientation of the body axes Alternatively, one can use 2 coordinates to specify the orientation of the zˆ ' axis of rotation (direction points along) and the 3 rd coordinate to specify the angle of rotation. zˆ '
9 Euler s and Chasles's Theorems Useful general principle for the analysis of rigid bodies Euler s Theorem: A general displacement of a rigid body with 1 pt fixed in space is equivalent to a rotation about some axis. Chasles s Theorem: A general displacement = a translation + a rotation
10 Math Review: Coordinate Transformation - For an actual physical object in nature, the intrinsic properties of the object should not depend on how we chose to describe it! - Specifically, one s choice of the coordinate system shouldn t matter the description of motion should be the same in any coords we choose. - If there is a transformation linking two coordinate systems, physical properties should be unaffected by the transformation operation. e.g., mass m makes no difference to a particular choice of coords Simple quantities like m are called scalars.
11 Invariance and Coordinate Transformation Next, more complex quantities : vectors (a set of n numbers ) z z ' A y ' Different from a scalar, the actual description of A a vector depends on the choice of the coord system, i.e., the actual components. x x ' y BUT, there exists a well defined transformation for vectors such that Vector equation such as F = ma (physical laws) remain unchanged in different reference frames! Scalars, Vectors, and more complicated objects such as: Matrices and Tensors used in defining physical quantities are restrained by how they remain invariant under a coordinate transformation.
12 Notation z z ' y ' From now on, we will interchangeably label our axes with indices, i.e., y x, yz, x, x, x 1 2 3 x x ' x ', y', z' x', x', x' 1 2 3
13 Rotation To describe the orientation of a rigid body, the transformation of interest is: Rotation x 2 ' x 2 P x 2 x 1 ' Go from specifying a point P in unprimed system to its x ' 1 specification in primed system x 1 ' 1 x 1 x, Relating from : x1 x2 green line = 2 x sin red line = x1 cos x' r ed + green = x cos x 1 1 2 sin
14 Rotation x x, x Similarly, relating from : ' 2 1 2 x ' 2 x 2 x 1 P x 2 x ' 1 x 1 ' green line = red line = x 2 x 1 sin cos x ' 2 x 1 x 1 x' red green = x cos x sin x sin x cos 2 2 1 1 2
15 Rotation These two transformation equations can be put in a compact matrix form: x ' 1 cos sin x1 x1 x ' R sin cos x x 2 2 2 R is called a rotation matrix and it takes P from the unprimed frame to the prime frame. Putting all entries in terms of cosine, we have: x ' 1 cos cos 2 x1 x ' 2 cos 2 cos x2
16 Rotation This is a particular elegant way to express the rotation matrix where are called the directional cosines and it is the cosine of the angle between the corresponding axes! x ' 1 cos cos 2 x1 a11 a12 x1 x ' 2 cos 2 cos x2 a21 a22 x2 a ij i.e., a cosine of angle between x' and x axes ij i j a11 x ' 1 x 1 a12 x 2 2 x ' 1 a21 x ' 2 2 x 1 a22 x ' 2 x 2
17 Rotation - Throughout our discussion, the point P is fixed and the prime frame rotates x ' 2 x 2 P counter-clockwise This is the passive point of view. (convention: + counterclockwise) x 1 ' x 1 - One could equally consider the x 2 transformation (rotation) as taking the point P (or vector) and rotating it by in the clockwise direction in the same frame. P P ' This is the active point of view. (convention: + clockwise) x 1 Either way, the math is the same!
18 Rotation - In 3 dimensions, it is particular simple to get the matrix for a rotation about one of the coordinate axes: Just put a 1 in the diagonal corresponding to that axis and squeeze the 2D rotation matrix into the rest of the entries. i.e., to rotate about the x 3 axis: cos sin 0 sin cos 0 0 0 1 x 2 i.e., to rotate about the axis: cos 0 sin 0 1 0 sin 0 cos
19 Rotation Examples: x 3 x 2 x ' 3 0 1 0 90 use 1 0 0 x ' 2 0 0 1 x ' 1 x 1 x 1 x 3 x 2 x ' 2 90 1 0 0 use 0 0 1 x ' 3 0 1 0 x ' 1 (recall: + is counter-clockwise in passive view)
20 Rotation To do compound rotations, the matrices multiply together but need to pay attention to the order of multiplication. 1 st Rotation: x 3 x 2 x ' 3 0 1 0 90 x ' 1 0 0 x ' x 2 0 0 1 x ' 1 x 1 2 nd Rotation: x ' 3 x ' 2 x ' 1 x '' 2 90 1 0 0 x ' 0 0 1 x '' x ' 1 0 1 0 x '' 3
21 Rotation Combining the two rotations: 2 nd 1 st 1 0 0 0 1 0 0 1 0 x '' 0 0 1 1 0 0 x 0 0 1 x 0 1 0 0 0 1 1 0 0 IMPORTANT : Rotations do not commute reverse order gives different results. x 1 first x ' 3 then x 3 90 x ' 2 90 x '' 1 x '' 2 x 2 x ' 3 x '' 3 x 1 x ' 1 x ' 0 1 01 0 0 0 0 1 1 0 0 0 0 1 x 1 0 0 x 0 0 10 1 0 0 1 0 NOT the same as before!
22 Euler s Angles - However, a general rotation about an arbitrary axis is not quite so simple. In general all entries will be nonzero! - But, one can build up a general rotation as separate successive rotations. - There are many conventions... We will choose one called the,, Euler s Angles consisting of a particular sequence of 3 rotations (D, C, B) along three principle axes: ' ' D C B
Euler s Angles Start with the fixed axes D C B xyz,, The first rotation D is about by an angle : ( space axes) x, y, z,, z ' ' cos sin 0 D sin cos 0 0 0 1 23 The second rotation C is about the new (new x-axis) by an angle :,, ', ', ' The third rotation B is about the new (new z-axis) by an angle : x y z ' ', ', ' ', ', ' ( body axes) demo: http://demonstrations.wolfram.com/eulerangles/ C B 1 0 0 0 cos sin 0 sin cos cos sin 0 sin cos 0 0 0 1
Euler s Angles D C B ' ' 24 - Then the general rotation from the fixed axes to the body x', y', z ' axes is given by the product: A(,, ) B( ) C( ) D( ),, xyz,, (The sequence of rotated angles are called the Euler s angles.) (note the order of operations!) A cos cos cossinsin cos sin cos cossin sinsin sin cos cossincos sin sin cos coscos sin cos sinsin sin cos cos x' body Ax( fixed)
25 Back to a bit of Math Review: Vectors & Matrices Write a general matrix multiplication out: - The i th component of is: a ij 3 ' i ax ij j j1 axes in the primed and unprimed frames. For a general transformations, the nine matrix elements, x where are some functions of the directional cosines between the a, i 1,2,3and j 1,2,3 ij x ' can be unrelated parameters But, for rotations, they will not be all independent since the transformation of vectors must satisfy the property that the length of vectors must remain invariant
26 A bit of Math Review: Vectors & Matrices Recall the picture for vectors under rotations - In the unprimed frame, let x x1, x2, x3 - In the primed frame, we have x ' x ', x', x' 1 2 3 x ' 2 x 2 Vector with the same magnitude x ' 1 x 1 This gives the following condition: 2 2 2 2 x' x' xi i i i x
27 A bit of Math Review: Vectors & Matrices Putting in the transformation equation for explicitly, we have, 2 i For the RHS to be equal to 2 x ' ax ij j ax ij j ax ik k i i j i j k We need to have the following condition: x ' aa xx i j k i ij ik j k x 2 i x 2 i aa ij ik jk 0 1 for j for j k k orthogonality condition where jk is the kronecker delta.
28 A bit of Math Review: Vectors & Matrices Comments: 1. To simplify the summation signs, we will use the Einstein convention (sum over repeated indices), i.e., eg x xx 2.., i i i i aa and we have the orthogonality condition: i sums over ij ik jk 2. Although we have 3 x 3 = 9 of the orthogonality conditions for jk, 1,2,3 a ij But, since are just numbers and (commute), we only have one equation for each unique jk pair and this gives six independent conditions (pairs of jk). aa aa i j ik ik i j 9 a ij and 6 conditions 3 free parameters (3 Euler s angles)
29 A bit of Math Review: Vectors & Matrices In Einstein s notation, we can rewrite the transformation as: x ' ax with aa ij ik jk i ij j A vector is a rank-1 tensor meaning that we need only one to transform it. a ij We will define a rank-2 tensor as an object that transform according to: I ' a a I ij ik jl kl with similar orthogonality conditions on a both so that tensor equations ik and a I ij involving will be invariant. jl (more on tensors later)
30 A bit of Math Review: Vectors & Matrices Now, we will review some general notations and properties of matrices and transformations: 1. Matrix Multiplication: C c a b ij ik kj a T ij AB (sum over k) 2. Multiplication does not commute: AB a BA 3. Multiplication is associative: ABC A BC T 4. A is the transpose T T T AB B A 5. is the inverse, where ji T ik kj ki jk jk ki ab ab ba summing index diff 1 1 1 A AA I or ij jk 1 0 0 1 1 st index 2 nd index ab ba ba ik kj kj ik jk ki ab ik kj cjl aik bc kj jl a a ik
31 A bit of Math Review: Vectors & Matrices Continuing 1 T 6. For an orthogonal matrix, we have : A A a 1 ij a ji Tr A a 7. Trace: ii (sum over i) 8. Determinant: det A a for n1 k 1 A a A det 1 det minor for n1 k 1k 1k minor A 1k a a am1 a 11 1n mn
32 A bit of Math Review: Vectors & Matrices example: a b det A ad bc ad bc c d a a a a a a a a a a a a a a a 11 12 13 21 22 22 23 21 23 21 22 23 11 12 13 a32 a33 a31 a33 a31 a23 31 32 33 a a a 9. Other properties of det: det A 1 1 (can be expanded in any row or column) det T det A For an orthogonal matrix, A det A det AB det Adet B T 2 1 T AA I AA det I det A det A det A So, det A 1 if A is an orthogonal matrices!
33 A bit of Math Review: Vectors & Matrices 10. Levi Civita Tensors: ijk 1, ijk is of forward permutation 1, ijk is of backward permutation 0, otherwise ijk In terms of (123, 231, 312) (321, 213, 132) det A ijkaa 1i 2 ja3k C AB c a b i ijk j k c a b a b a b a b e.g., 1 123 2 3 132 3 2 2 3 3 2 A useful identity: ijk lmk il jm im jl (sum over k) (1 st & 2 nd match) (mixed)
34 A bit of Math Review: Vectors & Matrices Example using ijk ABCD ijkab j k ilmcd l m ijkilmabcd j k l m jkilmiabcd j k l m ab cd a bcd l m l m m l l m abcd jl km jm kl j k l m AC BD AD BC (dot product sums over i) (they are numbers order does not matter) (permuting does not change the sum) (use pervious identity) (collapse delta function ) (more exercise in homework)
35 A bit of Math Review: Eigenvalues & Eigenvectors n 11. Given a matrix A, x is an eigenvector with eigenvalue if Ax A - Thinking in terms of active transformations, a vector is an eigenvector of a transformation A if A merely stretch it by a factor. 12. Eigenvalues can be founded by solving the characteristic equation of the matrix A: det AI 0 (since A is n dimensional, in general, there will be n roots) det A (including multiplicity (degeneracy)) 13. 1 2 n
36 A bit of Math Review: Eigenvalues & Eigenvectors 14. All of the following statements are equivalent: 1 - A is invertible meaning A exists and such that AA det A 0 columns - of A are linearly independent rows T - A is invertible - A is nonsingular - For a given non-zero vector b, a unique vector x s.t. Ax - All eigenvalues 0 1 I b
37 A bit of Math Review: Similarity Transformation 15. Similarity Transformations - Given an n-dim vector space V, choose a particular basis In the active sense, a linear transformation A (such as a rotation) takes a vector x in V into another vector x in V - Now, if one choose a different basis The same transformation will in general be represented by a different matrix, let say B. - We would say that A and B represent the same transformation if A is similar to B, i.e., a nonsingular matrix C such that 1 A C BC (this is called a Similarity Transform)
38 A bit of Math Review: Similarity Transformation Looking at this closer C basis #1 basis #2 (vectors transform under A) x Ax (vectors transform under B) Cx B C Ax -If A and B are the same, then we want CAx B Cx A 1 C BC (If this holds, then A and B will have the same active effects on vectors.)
39 A bit of Math Review: Similarity Transformation Important properties of a Similarity Transformation: also, 1 A C BC B det det det since det 1 1 1 A C BC i cki c Tr Tr c b c b b kj Tr jk B b kk jk k ij jk 1 C 1/ det C Similarity Transformations are important: Sometimes, there is a best or most natural basis in which to represent a linear transformation. It is advantageous in many cases if the linear transformation can be expressed as a diagonal matrix. ij i Tr index
40 A bit of Math Review: Similarity Transformation For a given symmetric matrix B, there exist a matrix A which is similar to B such that 0 1 1 0 n A C BC A is diagonal and its entries are the eigenvalues - All eigenvalues (multiplicity possible) are real - The columns of C are the eigenvectors of B For non-symmetric matrices, s might be complex. If one is restricted to real-value matrices, then the best that one can do is that A is block diagonal (Jordan form) with 2 x 2 blocks corresponding to pairs of complex conjugate s.
41 Back to Rigid Body (Finite Rotations) Finally, recall Euler/Chasles Theorem: Any general displacement of a rigid body with 1 point fixed is equivalent to a rotation about a given axis. So, instead of 3 Euler rotations,,, one could do ONE rotation if one know where the axis is located. fixed at this point
42 Back to Rigid Body (Finite Rotations) There is a formula that relates these 4 angles and we can obtain this equation by considering the following Similarity Transformation: cos sin 0 full Similarity Transformation sin cos 0 Euler matrix 0 0 1 zˆ ' rotate about zˆ ' by fixed at this point
43 Back to Rigid Body (Finite Rotations) In particular, we will have: 1 BCD S R( ) S -BCD are the three Euler rotations -S is a nonsingular matrix for the Similar Transformation -R() is the (simple) rotation matrix along the given axis zˆ ' BCD cos cos cossinsin cos sin cos cossin sinsin sin cos cossincos sin sin cos coscos sin cos sinsin sin cos cos R cos sin 0 sin cos 0 0 0 1
44 Back to Rigid Body (Finite Rotations) Since the trace is invariant under a Similarity Transformation, we have 1 BCD S RS R Tr Tr Tr RHS: Tr R cos sin 0 Tr sin cos 0 2cos 1 0 0 1 LHS: BCDcos cos cos sin si Tr n sin sin cos cos cos cos cos cos cos
45 Back to Rigid Body (Finite Rotations) So, the LHS simplify to, BCD Tr cos 1cos cos Now, RHS+1 = LHS +1 gives, 1cos 1cos 2 cos 1 cos 1 cos cos 1 2cos 1 1 2 2 cos 2 2 cos 2 2 2 2 2 cos cos cos cos 2 2 2 2