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PHY411-AST233 Lecture Notes

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PHY411-AST233 Lecture Notes PHY411-AST233 Lecture notes Alice Quillen March 23, 2020 Contents 1 Infinitesimal Transformations and Lie Groups 2 1.1 Lie Groups . 2 1.2 Rotation Example . 5 1.3 Example: Locomotion of Deformable Jello Cubes . 7 1.4 The Jello cube! . 9 1.5 Mechanical Connection . 14 1.6 A fibre bundle and a Connection . 16 2 Rigid Bodies 17 2.1 Configuration space . 18 2.2 Generators of Rotation Matrices . 19 2.3 The kinetic energy . 21 2.4 Example for the Moment of Inertia tensor . 22 2.5 Stability of a Freely Rotating Body . 22 2.6 Rigid Body motion from a Lagrangian . 24 2.7 A Hamiltonian, a moment map . 26 3 Rotational Dynamics 29 3.1 Euler angles . 29 3.1.1 Rotations . 30 3.2 Lagrangian and Hamiltonian in terms of Euler angles . 31 3.2.1 Axisymmetric bodies . 32 3.3 Torque free rotation of the axi-symmetric body . 34 3.4 Euler angles of the torque free axi-symmetric body . 35 3.5 Frequencies seen by an external viewer - asteroid rotation . 37 3.6 Long axis and short axis modes . 38 3.7 Comments on Damping of Wobbling . 40 3.8 Andoyer-Deprit variables . 40 3.8.1 Principal axis rotation . 45 1 3.9 Potential energy of a rigid body from an external force field . 45 3.10 MacCullagh's Formula . 46 3.10.1 For an axisymmetric body . 49 3.11 Precession of an axisymmetric body in a circular orbit . 49 3.12 Quaternions . 51 3.13 A Lie-Poisson numerical method for Solid Body rotation with Quaturnions 57 4 Tidal Evolution 58 4.1 Gravitational Potential Expansion in Legendre Polynomials . 58 4.2 Expansion of the gravitational potential of a point mass in spherical harmonics 59 4.3 Tidal deformation of a spherical body . 60 4.4 Love numbers for fluid bodies . 61 4.4.1 Love numbers for an incompressible homogeneous elastic body . 63 4.5 Rotational Deformation . 63 4.6 Tidal spin down . 65 4.6.1 Conventions for delay . 67 4.6.2 Associated orbital drift rate during spin down . 68 4.7 Other timescales . 68 5 Spin Resonances 68 5.1 Types of Spin Resonance . 69 5.2 Spin-Orbit resonance . 70 5.3 The dissipative spin-orbit resonance problem . 71 6 Problems 72 1 Infinitesimal Transformations and Lie Groups 1.1 Lie Groups A Lie Group is a group that is also a finite dimensional differentiable manifold. The group operations are smooth functions. A continuous group is a group where continuity is imposed on the elements of the group in the sense that a small change in one of the factors of a product produces a correspondingly small change in their product. Lie algebra operates on the tangent space of the Lie group at the identity element and this algebra captures the local structure of the Lie Group. Elements of the Lie algebra are elements of the group that are \infinitesimally close" to the identity, and the Lie bracket of the Lie algebra is related to the commutator of two such infinitesimal elements. An r parameter group on an n dimensional space (x) gives transformations x0 = f(x; a) 2 where a is the r parameters. We desire that for a a vector of zeros, we have the identity x = f(x; 0; 0; 0::::) Infinitesimal generators can be computed as @f Li = @a i ai=0 where evaluate at aj = 0 for j 6= i. The above f is a matrix or an operator and L has the dimension of f. Taking into account that f gives a vector we can also write this as @fj @ Li = (1) @a @x i ai=0 j (using summation notation for j) where @ gives a basis for the tangent space, and these @xj can be used to operate on any function. Above we have written Li as a differential operator. It operates on a function of x. With g(x) a function, then Lig(x) is a number so Lig(x) is a function of x. Using two different differential operators Li;Lj we can operate sequentially on a function Li(Ljg(x)). Let us consider two vector operators and write them more simply as @ A = ai @xi @ B = bj @xj where ai; bi are functions of x and we have used summation notation. Our generators Li;Lj are in this form. @ @ ABg = ai bj g @xi @xj 2 @ g @bj @g = aibj + ai @xixj @xi @xj 2 @ @bj @ AB = aibj + ai @xi@xj @xi @xj If we take the commutator 2 2 @ @bj @ @ @ai @ [A; B] = aibj + ai − aibj − bj @xi@xj @xi @xj @xi@xj @xj @xi @bj @aj @ = ai − bi @xi @xi @xj 3 We notice that the second derivatives cancel. That means that using the commutator our infinitesimal generators can operate on each other and give us another generator. The infinitesimal operators form a Lie algebra with commutators (called Lie brackets) that satisfy k [Li;Lj] = cijLk k with cij called structure constants. The commutator is a binary operator that satisfies the Jacobi identity, is antisymmetric and is bilinear w.r.t to multiplication with elements of a field (typically real or complex numbers). On a matrix Lie group, we can define an exponential map for any operator in the Lie algebra, giving us a map between elements of the Lie algebra and the Lie Group. For u in the Lie algebra 2 3 eu = 1 + u + u2 + u3:::: 2 3! and this gives an element in the Lie group. In this sense, elements of the Lie algebra are infinitesimal generators for the group. u2 v2 euev = (1 + u + ::::)(1 + v + ::::) 2 2 1 1 e(u+v) = 1 + u + v + (u + v)2:::: = 1 + u + v + (u2 + v2 + uv + vu):::: 2 2 We notice that euev 6= eu+v Locally 1 1 1 euev = exp u + v + [u; v] + [[u; v]; v] − [[u; v]; u]::: 2 12 12 known as the Baker-Campbell-Hausdorff formula. Consider two near identity operators A; B and alternating the order that the transfor- mations are done (see Figure 1 and 7). To second order A2 B2 A2 B2 e−Ae−BeAeB ≈ I − A + I − B + I + A + I + B + 2 2 2 2 A2 A2 B2 B2 = I − A − B + A + B + + + + + AB − AB − BA + AB − A2 − B2 2 2 2 2 = I + [A; B] ∼ e[A;B] The commutator helps us compute what happens when infinitesimal transformation don't commute. 4 Figure 1: Alternating infinitesimal transformations need not commute. 1.2 Rotation Example Rotation matrices in 3 dimensions form a Lie group. We define the special orthogonal group SO(3) as 3 T T A 2 R ; such that detA = 1;A A = AA = I Matrices A operate on x a 3D Cartesian coordinate. Rotation by angle φ about the z axis 0 cos φ − sin φ 0 1 R(φ) = @ sin φ cos φ 0 A (2) 0 0 1 Near the identity 0 1 0 1 − sin φ − cos φ 0 0 −1 0 dR(φ) = cos φ − sin φ 0 = 1 0 0 (3) dφ @ A @ A φ!0 0 0 0 0 0 0 φ=0 is the generator for R(φ). Expanding R(φ) for small φ (to first order in φ) 0 1 −φ 0 1 ^ R(φ) ∼ @ φ 1 0 A = I + φLz (4) 0 0 1 with 0 0 −1 0 1 ^ Lz = @ 1 0 0 A 0 0 0 5 a matrix describing the infinitesimal rotation about the z axis. We can similarly find 0 0 0 1 1 ^ Ly = @ 0 0 0 A −1 0 0 0 0 0 0 1 ^ Lx = @ 0 0 −1 A 0 1 0 for rotations about x and y axes. For a small rotation φ about z 0 x = (I + φL^ z)x 0 x0 1 2 0 0 −1 0 13 0 x 1 0 x − φy 1 @ y0 A = 4I + φ @ 1 0 0 A5 @ y A = @ y + φx A z0 0 0 0 z z Suppose we have a function F (x; y; z) and we transform it for a small φ. @F @F F (x0; y0; z0) = F (x − yφ, y + xφ, z) = F (x; y; z) − y φ + x φ @x @y The change in F @F @F ∆F = −y + x φ @x @y We associate the operator Lz = −y@x + x@y (5) with our infinitesimal transformation. And this is a vector corresponding to the z compo- nent of angular momentum. So if the coordinate system is rotated by a small amount φ about the z axis we can compute the change to any function of the coordinates with ∆F = LzF φ where Lz is the differential operator as defined in equation 5 Now let us find the operator using equation 1. Our function 0 x0 1 0 x 1 f(x; y; z) = @ y0 A = R(φ) @ y A z0 z 0 x cos φ − y sin φ 1 = @ x sin φ + y cos φ A z 6 0 −x sin φ − y cos φ 1 @f = x cos φ − y sin φ @φ @ A 0 0 1 −y @f = x @φ @ A φ=0 0 @fx @fy Lz = @x + @y @φ φ=0 @φ φ=0 = −y@x + x@y consistent with what we have above in equation 5. This example illustrates two way of describing the infinitesimal operators. One way with matrices. The other way with differential operators. In both cases we can think of the infinitesimal operators as lying in the tangent space of the manifold.
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