M3/4A16 GEOMETRICAL MECHANICS, Part 1 (2009) Page 1 of 5 UNIVERSITY OF LONDON Course: M3/4A16 Setter: Holm Checker: Gibbons Editor: Chen External: Date: January 27, 2008 BSc and MSci EXAMINATIONS (MATHEMATICS) May-June 2009 M3/4A16 GEOMETRICAL MECHANICS, Part 1 Setter s signature Checker s signature Editor s signature................................................
c 2009 University of London M3/4A16 Page 1 of 5 UNIVERSITY OF LONDON BSc and MSci EXAMINATIONS (MATHEMATICS) May-June 2009 This paper is also taken for the relevant examination for the Associateship. M3/4A16 GEOMETRICAL MECHANICS, Part 1 Date: Time: Credit will be given for all questions attempted but extra credit will be given for complete or nearly complete answers. Calculators may not be used.
M3/4A16 GEOMETRICAL MECHANICS, Part 1 (2009) Page 2 of 5 1. The 3D eikonal equation for an optical ray path r(s) R 3 is given by ( d n(r) dr ) = ṙ 2 n ds ds r where ṙ = dr/ds with ṙ = 1. (a) Prove that the 3D eikonal equation preserves ṙ = 1. (b) (c) Explain how the solutions for vectors ṙ and r arrange themselves geometrically, relative to the prescribed gradient n/ r. Derive the 3D eikonal equation from Fermat s principle in the form B 1 0 = δs = δ A 2 n2 (r(τ)) dr dτ dr dτ dτ, with new arclength parameter dτ = nds (optical pathlength). (d) 1. Take the fibre derivative of the Lagrangian to define the canonical momentum; 2. Legendre transform this version of Fermat s principle to determine its Hamiltonian; 3. Write Hamilton s canonical equations for it and 4. Use them to recover the 3D eikonal equation. (e) For L = r p, compute for n = n(r) with r = r. dl { } dτ = L, H
M3/4A16 GEOMETRICAL MECHANICS, Part 1 (2009) Page 3 of 5 2. A steady Euler fluid flow in a rotating frame satisfies u (v dx) + dπ = 0, with π := p + 1 2 u 2 u v, where u is Lie derivative with respect to the divergenceless vector field u = u, with u = 0, and v = u + R, with Coriolis parameter curlr = 2Ω. (a) (b) (c) Write out the Lie derivative relation u (v dx) + dπ = 0 above in two vector forms. In the first form, use the dynamic definition of the Lie-derivative. In the second form, use Cartan s formula in Cartesian coordinates. Explain geometrically what the Cartan version of the steady flow relation means in terms of the vectors u, curl v and (p + 1 2 u 2 ). Show that the steady flow relation u (v dx) + dπ = 0 above implies that the exact two-form defined by dq dp := d(v dx) = curlv ds is invariant under the flow of the divergenceless vector field u. (d) Show that Cartan s formula for the Lie derivative in the steady Euler flow condition implies the Hamiltonian formula and identify the function H. u (curlv ds) = u (dq dp ) = dh (e) (f) Use the result of (2c) to write u Q = u Q and u P = u P in terms of the partial derivatives of H. Write u Q = u Q and u P = u P in terms of a canonical Poisson bracket.
M3/4A16 GEOMETRICAL MECHANICS, Part 1 (2009) Page 4 of 5 3. (a) Compute the Poisson bracket table among the quantities x 1 = 1 2 ( q 2 + p 2 ), x 2 = 1 2 ( p 2 q 2 ), x 3 = p q, with canonical variables (q, p) T R 2. (b) Derive the Poisson bracket for smooth functions on R 3 by changing variables (q, p) T R 2 to (x 1, x 2, x 3 ) R 3 by using the chain rule. Show that the corresponding Hamiltonian vector field for a function H : R 3 R may be expressed as a divergenceless vector field X H = {, H} = S 2 H with S 2 = x 2 1 x 2 2 x 2 3 0. (1) Explain why S 2 0 and describe its level set geometrically. (c) Show that the flow of the divergenceless vector field X H preserves volume in 3D. (d) Consider the linear Hamiltonian (whose level sets form planes in R 3 ) H = ax 1 + bx 2 + cx 3 (2) with constant values of (a, b, c). Compute the equation of motion on the space of variables x R 3 obtained by setting d/dt = X H using the Hamiltonian vector field (1) and the linear Hamiltonian (2). (e) Specialise to find the solution of this equation of motion for (a, b, c) = (1, 0, 0).
M3/4A16 GEOMETRICAL MECHANICS, Part 1 (2009) Page 5 of 5 4. Symplectic 2 2 time-dependent matrices M i (t) with i = 1, 2, 3 satisfy the relation M i JM T i = J with J = ( ) 0 1. 1 0 (a) By using this relation, show that the quantities m i = ṀiM 1 i sp(2) satisfy (Jm i ) T = Jm i for i = 1, 2, 3. (b) The matrices X i = Jm i satisfy X T i = X i for i = 1, 2, 3. Show that J[m i, m j ] = [X i, X j ] J := 2sym(X i JX j ) = X i JX j X j JX i where [m i, m j ] := m i m j m j m i and sym denotes symmetric part. (c) If X = JṀM 1 for derivative Ṁ = M(s, σ)/ s σ=0 and Y = JM M 1 for variational derivative δm = M = M(s, σ)/ σ σ=0, show that equality of cross derivatives in s and σ when evaluated at σ = 0 implies the relation δx := X = Ẏ + [X, Y] J where [X, Y] J := 2sym(XJY). Hint: Define m = ṀM 1 and n = M M 1, subtract cross derivatives m ṅ and then use the result from the previous part. (d) (e) Use the previous relation to compute the Euler-Poincaré equation for evolution resulting from Hamilton s principle ( ) l 0 = δs = δ l(x(s)) ds = tr X δx ds Specialise the Euler-Poincaré equation to the case that l(x) = 1 2 tr(x2 ), where tr denotes trace of a matrix.