MatFys. Week 5, Dec , 2005

Transcription

1 MatFys Week 5, Dec. 1-18, 005 Lectures During this week s lectures we shall finish our treatment of classical mechanics and begin the quantum theory part of the course with an introduction to Hilbert spaces. Monday Dec. 1. We prove a version of Noether s theorem on the connection between symmetries of a Lagrangian dynamical system and the existence of conserved quantities (so-called first integrals of motion) as given in section 0 of VIA. Some supplementary notes will be handed out. Wednesday Dec. 14. We introduce the notion of Hilbert space as a generalisation of finite dimensional inner product spaces. These will serve as a replacement in quantum mechanics of the state space of classical mechanics. We expect to cover approximately the first half of Ch. 3 of the notes. These are written in Danish. The same material (approximately) is covered by Ch.4 in the book N. Young: An introduction to Hilbert space (Cambridge 1988) or by Ch. in S.K.Berberian: Introduction to Hilbert space (Amer. Math. Soc. 1999). Tutorials Monday Dec. 1. Solve the following exercises: Exercise 5A Consider the planar pendulum with configuration space S 1 and Lagrange function given by L(v) = m R v mgh(v), v T S 1, 1

2 where is the standard norm on the tangent spaces of S 1 (induced from R ) and h is the height function, i.e. and R is the length of the pendulum. h(v) = Ry, for v T (x,y), a) Show that the coordinate representation of L w.r.t. the coordinate system z given by z(cos θ, sin θ) = θ + π, θ ] 3π, π [. is L z 1 (θ, θ) = 1 mr θ + mgr cos θ. b) Show that the eq. of motion for the pendulum is θ + g sin θ = 0. R c) Find the conserved energy expressed in terms of θ. Exercise 6A Show that the geodesic equation ( ) d g ij q j 1 g jk q j q k = 0 dt q i can be rewritten as j q i + j,k where the Christoffel symbols Γ i jk Γ jk = 1 l j,k Γ i jk q j q k = 0, are given by g il ( glk q j + g jl q g ) jk k q l where (g ij ) denotes the inverse of the matrix (g ij ). Exercise 7A Consider on S with standard metric (induced from R 3 ) the coordinate system y defined by for (θ, ϕ) ] π, π [ ] π, π[. y(cos θ cos ϕ, cos θ sin ϕ, sin θ) = (θ, ϕ),

3 a) Show that = sin θ cos ϕ sin θ sin ϕ y 1 x1 x + cos θ x 3 = cos θ sin ϕ + cos θ cos ϕ y 1 x1 x, where x = (x 1, x, x 3 ) is the standard coordinate system on R 3. b) Show that the components (matrix elements) of the metric on S w.r.t. y are given by g 11 = 1, g = cos θ, g 1 = g 1 = 0. c) Show that the geodesic equation for S w.r.t. y has the form θ + (cos θ sin θ) ϕ = 0 ϕ ( tan θ) θ ϕ = 0. Find the (non-trivial) solutions for which θ is constant. d) Determine the Christoffel symbols w.r.t. the coordinate system y. Exercise 8A Consider the spherical pendulum with configuration space S and Lagrange function given by L(v) = m R v mgh(v), v T S, where is the standard norm on the tangent spaces of S (induced from R 3 ) and h is the height function, i.e. h(v) = Rz, for v T (x,y,z) S, where R is the length of the pendulum. a) Show that the coordinate representation of L w.r.t. the coordinate system w given by w(cos θ cos ϕ, cos θ sin θ, sin θ) = (θ + π, ϕ), (θ, ϕ) ] 3π, π [ ]0, π[. is L w 1 (θ, ϕ, θ, ϕ) = 1 mr ( θ + ϕ cos θ) + mgr cos θ. 3

4 b) Show that the eqs. of motion for the pendulum are θ + ϕ cos θ sin θ + g R sin θ = 0 ϕ ( tan θ) ϕ θ = 0. Wednesday Dec. 14. Solve the following exercises: Exercise 9A Let (M, g) be a Riemannian manifold and consider the length functional L defined by L(γ) = b a γ(t) γ(t) dt, defined on differentiable curves γ : [a, b] M, and let us assume γ(t) 0 for all t [a, b]. As usual, p denotes the norm on T p M defined by the metric g. a) Verify that L is invariant under reparametrisations of the curve, that is L(γ) = L(γ ϕ), if ϕ : [c, d] [a, b] is a bijective differentiable function with ϕ > 0. Hint. Verify and use that the tangent vector to γ ϕ at t equals ϕ(t) γ(ϕ(t)). b) Show that the Euler-Lagrange equations for L w.r.t. a coordinate system x with usual notation take the form d j g ij q j g jk j,k q j q k q dt i = 0. j,k g jk q j q k j,k g jk q j q k c) Verify that if the parametrisation is chosen such that γ(t) is constant then the Euler-Lagrange equation in b) is identical to the geodesic equation. Exercise 30A Consider on the surface of revolution M = {(x, y, z) x + y = f(z), a < z < b} 4

5 with standard metric (induced from R 3 ) the cylindrical coordinate system w defined by w(f(z) cos ϕ, f(z) sin ϕ, z) = (ϕ, z) for (ϕ, z) ] π, π[ ]a, b[. a) Show that = f(z) sin ϕ + f(z) cos ϕ w 1 x1 x = f (z) cos ϕ w x + f (z) sin ϕ 1 x + x. 3 b) Show that the components of the metric w.r.t. w are g 11 = f(z) g = 1 + f (z), g 1 = g 1 = 0. c) Show that the geodesic equations have the form d dt (f(z) ϕ) = 0 d dt ((1 + f (z) )ż) = f(z)f (z) ϕ + f (z)f (z)ż. d) Derive Claireaut s theorem (see VIA p. 86) f(z(t)) sin α(t) = constant, where α(t) is the angle between the tangent γ(t) to a solution γ and the meridian vector w γ(t) Hint. It may be usefull to consider instead the angle β = π α between the curve and the horizontal vector and use conservation of energy w 1 (i.e. of speed) and the first equation of motion. Exercise 31A Consider the rod in the Example on p.84 of VIA with configuration space R S 1 and Lagrange function L = T = m 1 (ẋ + ẏ ) + m (ż + u ) where (x, y) and (z, u) are the locations of the two particles at the ends of the rod in R. 5

6 a) Let the coordinate system w on R S 1 be defined by w(x, y, cos θ, sin θ) = (x, y, θ), (x, y) R, θ ] π, π[. Show that the coordinate representation of L w.r.t. w is L = m 1 + m (ẋ + ẏ ) + m R θ + m R θ(ẏ cos θ ẋ sin θ). b) Verify that the motion described by this Lagrangian can be interpreted as the geodesic motion for a suitable metric on R S 1 and determine the metric components from the expression found in a). c) Determine the equations of motion for the system. Exercise 3A Consider a rigid body (in R 3 ) consisting of N particles. a) Show that the configuration space of the body is R 3 S S 1 (or equivalently R 3 SO(3)). b) Assume that the body is not acted on by external forces such that its Lagrange function is L = T = N i=1 m i ẋ i, where x 1,..., x N are the positions of the particles and m 1,..., m N their masses. c) Show that invariance under translations implies by Noether s theorem that the total momentum N P = m i ẋ i is conserved. i=1 Hint. Define h u (p, q) = (p + ue, q) for p R 3 and q SO(3), where e R 3 is fixed, and use that x i = p + f i (q) for suitable functions f i (which you need not determine). Bergfinnur Durhuus 6