COMMONWEALTH OF AUSTRALIA Copyright Regulations 1969

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1 COMMONWEALTH OF AUSTRALA Copyright Regulations 1969 Warning - Do not remove this notice This material has been reproduced and communicated to you by or on behalf of the University of New South Wales pursuant to Part VB of the Copyright Act 1968 (The Act). The material in this Communication may be subject to copyright under the Act. Any further reproduction or communication of this material by you may be the subject of copyright protection under the Act. have read the above statement and agree to abide by its restrictions.

2 THE UNVERSTY OF NEW SOUTH WALES SCHOOL OF PHYSCS FNAL EXAMNATON NOVEMBER 2014 PHYS3510 Advanced Mechanics, Fields and Chaos Time Allowed - 2 hours Total number of questions- 4 Answer ALL questions All questions ARE of equal value Candidates may bring their own approved calculators. Answers must be written in ink. Except where they are expressly required, pencils may only be used for drawing, sketching or graphical work

3 FORMULA SHEET Eul er-lagrange equations Polar coordinates Spherical polar coordinates x = rsin8cosif> y = rsin8sinif> z = rcos8 Canonical Transformations 1) F;(q,Q,t) af; p=- aq; P. =- af; aqi K=H + af; at af p.= - 2 aqi Q = af2 ap K=H+ af2 at af q.=-- 3 api P. = _ af3 aq; K =H + af3 at af q.= api Q. = af4 ap. K = H + af4 at Poisson Bracket Hamilton-Jacobi Theory Canonical transformation to H(q,p,t) Q. P. > H(q,p) =constant P; (constant of motion) New Hamiltonian New equations of motion. ak Q.=-=0 ' ap. ak P=--=0 aq; K = H(P;) = a 1. ak Q. =-=V. ap,. ak P=-- =0, aqi

4 With solutions Generati ng Function Q. ={3 P, = yl > Hamilton's Principle S(q,P,r) (us) as Hq-L +-=0 Hamilton-Jacobi equation aql' ar New constant momenta P 1 = y 1 (a.,...,a,) (one choice) Y; = a, Q. = V 1 l + f3, P = Y; lamilton's Characteristic W (q, P) 1-1 ( q,,- aw) -a 1 = 0 ijql P, = Y 1 (ap...,a,) y, =a, Hamilton-Jacobi solution S = S(q 1,y 1,t) W = W(q11 y,) First half of transformation as p. =- aq, Second half of transformation as Q, =- = {3, ar, aw p=- aq, aw Q, =a= vl(y j)t + {3, Y, Action-angle Variables aw w. = - ' aj. Euler-Lagrange equation for fields Mathematical identities sin 2 Q + cos 2 Q = l l + cos2 = 2 cos 2 if> d -tanx = sec 2 x dx d Sin X=--;=== dx ~ d -1 l -tan x a-- dx + x 2 tan 2 Q+l= sec 2 Q 1-cos2 = 2sin 2 if> d -cotx = cosec 2 x dx d COS X = - --====-- dx..j1- x 2 d cot x =--- dx 1 + x 2

5 QUESTON 1. (25 marks) A sphere of radius a and mass rn rests on top of a fixed sphere of radius b. The first sphere is sl ightly displaced so that it rol ls without slipping down the second sphere. (a) f x is the distance between the centres of the two spheres show that the Lagrangian for the small sphere is given by (b) What are the constraint equations and the initial conditions for? (c) Use the method of Lagrange multipliers to show that the equations of motion for the system in spherical polar coordinates are given by r : :t (m.x)- ( mx4/- mgcos ) = J.., : d( 2 " 2" ") - mx cf>+ ~ ma (8+ ) -mgxsin =b\ dt : d ( 2.. ) - ~rna (8 + ) =-a\ dt (d) Solve these equations to determine where the smaller sphere leaves the larger sphere?

6 QUESTON 2. (25 marks) (a) s the following transformation cano nical Q = p + iaq p = p- iaq? 2ia f not can it be made canonical? (b) Find the F 2 (q,p) generating function that generates this canonical transformation. (c) Use this transformation to transform the Hamiltonian for the harmonic oscillator 1 ( 2 2 2) H =2 p +mw q, into a new Hamiltonian K(Q,P). (d) Find and solve the equations of motion for Q and P. (e) Write down the solution for the original variables q and p.

7 QUESTON 3. (25 marks) (a) Find the frequencies of a two-dimensional simple harmo nic oscillator with mass m and unequal force constants k. 1 and k 2 in the x and y directions respectively, using the method of action-angle variables. The Hamiltonian is given by 1 ( 2 2) 1 ( 2 2) H = 2m p x + p Y + 2 klx + kzy. (b) f the Lagrangian density for displacements of an elastic rod is given by find the Lagrangian equations of motion. What is the physical meaning of this result?

8 QUESTON 4. (25 marks) (a) Determine the fixed points and calculate their stability properties for the equations x = x - xy. y = xy - y (b) T he stability of an iterative mapping xn+ = j(x 1,) can be determined by calculating the Lyapunov exponent defined by N -1 J.. = lim - "" lnlf '(x;)l N--+00 N ~ i=o Find the Lyapunov exponent as a function of,u for the two fixed points of the quadratic map, xn +l =,uxn(l-xn). Discuss the stability of the fixed points as,u,2? (c) lfthe derivative of two applications of the quadratic map at the 2-cycle is f 1 ~ = 4 + 2,u-,u2, explain the behaviour of the Lyapunov exponent as,u, 3 from above, and also as,u increases from 3. (d) What are tangent bifurcations and pitchfork bifurcations and how do they arise? (e) For the quadratic map we can describe the type of bifurcations that lead to cycles of a particular length working sequentially from small to larger cycles. Complete all the missing entries in the table. cycle length periodic 'prime' number of created by created by points points n-cycles n 2n tangent pitchfork

9 (f) Define the unstable manifold of a fixed point. Prove that unstable manifolds from different fixed points do not intersect.

Physics 5153 Classical Mechanics. Canonical Transformations-1

1 Introduction Physics 5153 Classical Mechanics Canonical Transformations The choice of generalized coordinates used to describe a physical system is completely arbitrary, but the Lagrangian is invariant