Rotational motion of a rigid body spinning around a rotational axis ˆn;

Physics 106a, Caltech 15 November, 2018 Lecture 14: Rotations The motion of solid bodies So far, we have been studying the motion of point particles, which are essentially just translational. Bodies with spatial extent (either rigid or elastic) can move in other ways: Translational motion: straight line (as in the motion of the center of mass of a solid body or collection of point particles; we often refer to this as trivial motion); falling motion (as in a cannonball in a constant gravitational field, with a parabolic trajectory in space); Keplerian orbital motion, in a 1/r potential, for bound orbits; Keplerian scattering (unbound orbits ); Rotational motion of a rigid body spinning around a rotational axis ˆn; Rigid bodies are composed of a collection of point masses where the distance between each point is fixed. Vibrational motion of an elastic body, which we will describe as a collection of normal modes, with each mode being a simple harmonic oscillator. Rotational Motion Here we will focus on describing rotational motion, covering these topics: Physics Transformation Ideas Rotational symmetry Rotated system vectors, tensors rotation matrices (group structure) Dynamics in rotating frame Rotating system fictitious forces Dynamics of rotating bodies Rotational Symmetry Rotating axes tied to dynamics Euler angles moment of inertia angular momentum We use the rotational symmetry of space to write laws of motion in vector notation. m a = F. (1) Whatever direction we apply the force, we get the same magnitude acceleration and in the direction of the force. This brings us to the question: 1

What is a vector? P r O Our template for a vector is the straight arrow OP joining point O to point P. This is the displacement vector r. Often we will call the point O the origin. The vector has a magnitude (which we can measure with a calibrated ruler): for a displacement we call the magnitude the length r, and write r 2 = r 2. The vector also has a direction: if we rotate the vector we get a different vector (pointing in a different direction). Thus the use of vectors is intimately connected with the physics of rotations. A vector is a quantity that behaves in the same way as r under rotations: the length is unchanged, and if the direction starts off along some r, the rotated direction is along the rotated r. A scalar on the other hand only has a magnitude and is unchanged by rotations. Other quantities (tensors) are changed by rotations in a more complicated way (see later). We add two vectors r 3 = r 1 + r 2 by laying them tail to head and forming the third leg of the triangle. r 1 +r 2 r 2 r 1 Multiplication by a scalar gives a vector in the same (positive scalar) or opposite (negative scalar) direction with length multiplied by the magnitude of the scalar. We may define the dot product of two vectors by a construction such as r 1 r 2 = 1 2 [ ( r1 + r 2 ) 2 r 1 2 r 2 2 ]. (2) Alternatively we can calculate the product of the lengths times the cosine of the angle between the the vectors: r 1 r 2 = r 1 r 2 cos θ 12. The length of a vector can be expressed as r 2 = r r r 2. By subtraction and multiplication by the scalar dt 1 we can form the velocity and acceleration vectors Newton tells us the law of motion [ ] r(t + δt) r(t) v = lim δt 0 δt = d r dt, d v a = dt. (3) m a = F (4) where m is a scalar and F (which we must know how to evaluate). For example for two particles separated by the vector r we might have F = GMm r 3 r or F = K r (5) 2

for gravity or a spring. These are clearly vectors too (scalars r). The interpretation of this law is given in the first section. The use of vectors, simplifies the formulation of the law, relying on the isotropy of space. Without the symmetry, vectors wouldn t be any help. Vectors thus provide a coordinate-axis independent way of specifying Newton s law of motion, and other physics. We think of r, v, F, E, B... as having a physical significance independent of any particular coordinate representation. Later on we will discuss pseudovectors, and then tensors. A scalar is a 0th-rank tensor, and a vector is a 1st-rank tensor. Components Often it is convenient to use components of vectors with respect to a particular choice of Cartesian axes. Define î, ĵ, ˆk as three unit length (î î = 1, etc.) orthogonal (î ĵ = 0, etc.) reference vectors an orthonormal basis. Sometimes I ll use the notation ê i with ê 1 = î, ê 2 = ĵ, ê 3 = ˆk. Then for a general vector a (not meant to imply acceleration) we can write a = a 1 î + a 2 ĵ + a 3ˆk. (6) The a i, i = 1, 2, 3 are called the components of the vector with respect to the basis. We might also use the notation a x for a 1, a y = a 2 etc. Since the axes are orthonormal the components are also given by projection a 1 = a î, etc. (7) We often think of the three components (a 1, a 2, a 3 ) and a as equivalent. Using the orthonormality of the basis vectors, the dot product of two vectors can be written in terms of the components 3 a b = a i b i. (8) i=1 The isotropy of space means that no particular choice of basis is singled out. We are led to ask for a given vector a how the components with respect to different bases are related. Rotation of coordinate axes rotated k unrotated k' j a x' axis i' i x axis j' Consider a set of basis vectors î, ĵ, ˆk and a rotated set î, ĵ, ˆk. The components a i and a i with respect to these axes are given by a = a 1î + a 2ĵ + a 3ˆk = a 1 î + a 2 ĵ + a 3ˆk. (9) 3

Using the orthonormality we can calculate e.g. a 1 = a 1 î î + a 2 î ĵ + a 3 î ˆk (10) or in matrix notation with U the rotation matrix a 1 a 1 a 2 = U a 2 (11) a 3 a 3 î î î ĵ î ˆk U = ĵ î ĵ ĵ ĵ ˆk. (12) ˆk î ˆk ĵ ˆk ˆk This defines the rotation matrix in terms of the direction cosines between the basis vectors. The matrix notation means 3 a i = U ij a j U ij a j (13) j=1 where the last expression uses the Einstein repeated subscript convection that we are to sum over any (pairwise) repeated subscript. I will use this convention from now on. Note that we have kept the vector fixed, and rotated the basis vectors. This is called a passive rotation. For passive rotations, we are making use of the rotation matrix to transform the vector components in the rotated coordinate basis to the vector components in the unrotated basis; going in the other direction makes use of the inverse of the rotation matrix U 1. For active rotations, below, the rotation matrix will allow us to go from the unrotated vector to the rotated vector, in a fixed coordinate basis. Rotation matrix The rotation matrix U is an orthogonal matrix. Define the transpose Ũ by Ũij = U ji. Then you can check UŨ = ŨU = I (14) with I the unit matrix (ones down the diagonal, zeroes elsewhere) I ij = δ ij = { 1 if i = j 0 if i j (15) (the Kronecker delta). Equivalently the rows (and columns) are orthonormal U ik U jk = δ ij etc. (remember k-summed) (16) The matrix U ij contains 9 numbers and Eqs. (16) are six constraints (the similar statements for columns turn out not to be independent). Thus there are 3 numbers that specify U. One choice of parameters is the direction of the axis of rotation ˆn and the angle of the rotation φ. Another choice is the Euler angles which we will consider in Lecture 17. Note that if we allow the elements of U to be complex (as in quantum mechanics), an orthogonal matrix is generalized to a unitary matrix U, such that Ũ = U 1 so that Ũ U = 1. We define the Hermitian congugate matrix U Ũ so that U U = 1. Unitary matrices (or operators) are used everywhere in QM, especially for rotations, translations, and time evolution of quantum states. 4

Example: Consider a rotation by φ about the z direction. The z component of a vector is unchanged, and the transformation of the x, y components is easily calculated by trigonometry. Suppose the vector projected onto the x, y plane makes an angle θ to the (rotated) î axis. Then it will make an angle θ + φ to the (unrotated) î axis. a 1 = a cos(θ + φ) = a(cos θ cos φ sin θ sin φ) = a 1 cos φ a 2 sin φ a 2 = a sin(θ + φ) = a(cos θ sin φ + sin θ cos φ) = a 1 sin φ + a 2 cos φ This gives the rotation matrix U(φ, ˆk) = cos φ sin φ 0 sin φ cos φ 0 0 0 1 j. (17) The general expression for the rotation matrix for an angle φ about an axis ˆn is (see Assignment 6) U ij = (1 cos φ)ˆn iˆn j + cos φ δ ij sin φ ɛ ijkˆn k. (18) From this it follows that Tr U = 1 + 2 cos φ. Also ˆn is the eigenvector of U with eigenvalue 1. (See the slides for a different argument leading to these results.) Rotation axis and angle Rotation angle: Rotation through same the angle φ about a different axis ˆn is Uˆn = ŪRˆnŪ 1 with ˆn = Ū ˆn j' a θ φ i i' The trace is invariant under this transformation Tr Uˆn = Tr Uˆn The rotation angle can be found from Tr U = 1 + 2 cos φ Rotation axis: The rotation axis is left unchanged by U: U ˆn = ˆn The rotation axis is the eigenvector of U corresponding to eigenvalue 1 (for a proper rotation) Improper rotations include parity inversions, which turn right-handed coordinate systems into left-handed ones. { +1 proper rotation det U = 1 improper rotation 5

Rotation of vector rotated k a a' unrotated i j Instead of rotating the axes/basis vectors we may discuss rotations of the vector, and then evaluate components with respect to one choice of basis. This is called an active rotation. Suppose the vector a is rotated into a new vector a by the same physical rotation as we used for the basis vectors (e.g. the same rotation axis and angle of rotation). We now have the components ( a) i of a and ( a ) i of a defined by a = ( a) 1 î + ( a) 2 ĵ + ( a) 3ˆk (19) a = ( a ) 1 î + ( a ) 2 ĵ + ( a ) 3ˆk (20) Using the fact that the components of the rotated vector with respect to rotated axes are the same as the components of the unrotated vector with respect to unrotated axes you can show ( a) i = U ij ( a ) j (21) with U the same rotation matrix. Notice however that the expression relates the rotated to the unrotated, wheres for the passive rotation the expression relates the unrotated to the rotated. As is clear intuitively, rotating the coordinate axes has the same effects on the components as the negative rotation acting on the vector: U 1 (φ, ˆn) = U( φ, ˆn). Applied to both sides of F = m a a passive rotation says that we get the same component expression e.g. F x = ma x with respect to any choice of basis vectors, whereas an active rotation says that a rotated force gives a rotated acceleration. Alternative definition of a vector Some texts prefer to define a vector as a quantity defined by three numbers (a 1, a 2, a 3 ) that transform according to the rules Eq. (13). I prefer the geometric definition. It seems inelegant to define a quantity used to eliminate the need for components in terms of components! For an exposition of the geometric definition of 4-vectors in special relativity see the first few pages of the notes to Ph136a (we will return to this next term). Pseudovectors We see that we can define a vector as a geometric object that behaves under a 3-dimensional rotation (with continuous parameters ˆn and φ) in a well-defined way. We may ask about how such objects behave under Parity, which is a discrete 3-dimensional coordinate transformation: r r, or x x, y y, z z. Such a transformation (which turns a right-handed coordinate 6

system to a left-handed one), obviously flips the sign of a position vector r. It also flips the sign of a velocity or acceleration vector: v v, a a. Other vectors can behave differently. For example, a vector formed from the cross-product of two vectors, such the angular momentum L = r p, does not flip sign; it is a pseudovector. It is still a vector, in that it transforms under rotations just like other vectors; but it has a handedness, also known as a chirality or helicity. Rotations conserve handedness, while parity flips it. Other examples include the magnetic field B = A, where A is the vector potential of electrodynamics. Three of the four fundamental interactions of nature (gravity, electromagnetism, and the strong nuclear interaction) conserve parity; the weak nuclear interaction violates it maximally (Nobel Prize, 1957). We will encounter other pseudovectors, below. Group properties Rotations form a group. A group is defined as a collection of elements (here different rotations) with the properties: multiplication: the product U 2 U 1 is defined as first apply U 1 and then apply U 2. Clearly with U 3 some other rotation; U 2 U 1 = U 3 (22) associative rule: it can be shown (U 1 U 2 )U 3 = U 1 (U 2 U 3 ) ; (23) identity: the identity exists do nothing, or rotate about any axis by an angle of zero: U(0, ˆn) = I. inverse: the inverse rotation is a rotation about the same axis through the negative angle: U 1 (φ, ˆn) = U( φ, ˆn). Note that multiplication of rotations is not commutative in general U 1 U 2 U 2 U 1 (24) the order of rotations about different axes does matter. (Try π/2 rotations about two axes at π/2.) Multiplication is also rather complicated: it is not easy to see what U 3 is if U 1, U 2 are specified by axis/angles ˆn 1, θ 1 and ˆn 2, θ 2. Rotations form a continuous or Lie group since the elements are specified by continuous parameters (direction of axis and angle). Since (at least some of) the elements are non-commuting, it is a non-abelian group. In particular, the set of all possible proper rotations (det U = +1) comprises the special orthogonal group in 3 dimensions, SO(3). Including improper rotations gives the orthogonal group O(3). The U here denote physical rotation operations. The 3 3 matrices U ij give a representation of the group the effect of the operations on an orthonormal triad of 3-vectors. We could also look at the effect of rotations on other quantities, such as a tensor (see later), or the 2p or 3d wave functions in hydrogen (see Ph125), or spinors (see Ph205). These would give different representations. 7

Infinitesimal rotation Infinitesimal rotations are a vector!" n!a # a Rotations by an infinitesimal angle δφ about some axis ˆn are easier to deal with: they can in fact be described in terms of an infinitesimal rotation vector δ φ = δφ ˆn. Let s look at this in active form. Consider the change δ a in a vector a due to an infinitesimal rotation δφ about an axis specified by the unit vector ˆn (see figure). The vector a is at an angle θ to ˆn. The change δ a is perpendicular to a and ˆn and is of magnitude a sin θδφ, and so δ a = δφˆn a = δ φ a with δ φ = δφˆn. (25) δ φ is a pseudovector it behaves like ˆn a vector under rotations, but under inversions r r it does not change sign (make some sketches). We also know this because the operation requires a right hand rule which changes under inversion. For successive infinitesimal rotations a a a we just add the δ φ vectors: a = a + δ φ 2 a = ( a + δ φ 1 a ) + δ φ 2 ( a + δ φ 1 a ) (26) = a + (δ φ 1 + δ φ 2 ) a + O(δ φ 2 ). (27) This also implies that infinitesimal rotations commute the order does not matter. These results are not true for finite rotations φ 1, ˆn 1 and φ 2, ˆn 2. Angular velocity Rotation rate is defined as a small rotation divided by a small time increment and can be characterized in terms of the angular velocity ω = dφ/dt, which is also a (pseudo-)vector (vector δφ divided by scalar δt). Rotation matrix for infinitesimal rotation: a in our notation convention, so that The rotated vector is a and the unrotated one or in component notation with respect to a fixed basis a = a + δ φ a (28) ( a) i = (δ ij ɛ ijk δφ k )( a ) j (29) with ɛ ijk the Levi-Civita symbol 0 if any repeated indices ɛ ijk = 1 if ijk is an even permutation of 123 1 if ijk is an odd permutation of 123 (30) 8

The corresponding rotation matrix is δu = I + δ φ M (31) with M a vector of matrices with components 0 0 0 0 0 1 0 1 0 M 1 = 0 0 1, M 2 = 0 0 0, M 3 = 1 0 0 (32) 0 1 0 1 0 0 0 0 0 or (M k ) ij = ɛ ijk. (33) (You can check the general U given in terms of direction cosines Eq. (12) reduces to Eq. (31) for small rotations.) We can build up a finite rotation from a succession of infinitesimal rotations about a fixed axis: for a rotation φ about ˆn build up as N rotations through δφ = φ/n for N ( U = lim I + φ M) N N N ˆn = e φˆn M, (34) where the exponential of a matrix is just a compact way of writing the power series e A = 1 + A + 1 2! A2 + (35) You can check that finite rotations about the same axis do commute. Since we can form a general rotation from the matrices M giving an infinitesimal rotation the M are called generators of the Lie group (actually representations of the generators). You can show that M i M j M j M i = c k ijm k (36) with the structure constant c k ij is this case given by ck ij = ɛ ijk. This type of expression for the commutation rule of the generators is called the Lie algebra and is a fundamental property of the group. The relation c k ij = ɛ ijk essentially defines the group SO(3). In QM, we have the commutation relation [L i, L j ] = i ɛ ijk L k ; the matrices i M represent the angular momentum operators L, and they generate finite rotations, just as in classical mechanics. Noether s theorem for rotation revisited Define the rotation transformed paths in coordinate notation R i (δφ, t) = δu ij r j (37) with δu given by Eq. (31). For a system with rotational symmetry Noether s theorem gives the conserved quantities R i I k = p i δφ k = p i (M k ) ij r j (remember, i, j summed) (38) δφ=0 = ɛ ijk p i r j = ( r p) k (39) so that I is the angular momentum vector L = r p. We ve given the argument for a single particle: summing over many particles is a trivial extension. 9