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Unformatted text preview: ' $Vv?o"n.=OonE—VCQVEI v Q? v ? Lv”? g; 9 V©C©E o ao C 2_.1 Use the WKB eikonal method to ﬁnd an approximate solution for 9(t), correct to
ﬁrst order The lowest order solution 18 not acceptable Your solution should \(é dLj
satisfy the conditions 6(0)0 and (d9/dt)(0)= 1. Assume that (dco/dt)(0)= —0 I” ’15“? at) gt WWW 0 aft/5 a) >>jw/ r5523): WKB solution to ﬁrst order including initial conditions 61%)  t; [LiarsWM] {to/H)V2 2i Write down the Hamiltonian function for this system, H(p,9,t). Using your solution for 9(t), calculate explicitly, to lowest nonvanishing order only, the
energy of the mass, h(t), The latter function is the “h” function as deﬁned by GPS. .
_ . 9,2 . Z Z
Mic}: {1/er 501972 a: 9 /Z + (/0 6/; lg: iwcoszfj +1 001 sthlfj/eu : 5059/?—
2 .2 H(p,9,t): fl/z + oﬂt? 92
h(t)=(::>w(~é) Now, make a sketch of centours of constant H for the two cases t = 0 and t >> 1:.
Calculate the value of h(t) for these two cases; then mark the position of the particle
on your H—contours, ie, identify which H surface it begins on and where it ends up.
Your two H—contours should be approximately to scale, showing the distortion as well as H—Values and where “h” is. Two sketches _2._3 From your WKB solution, calculate to lowest nonvanishing order the action, A(t) = f p619, for this system at any time, where the integral is over one full period.
[The integral is doable even if omega is not known explicitly] Show that, to the
order of your calculation, the action is a constant of the motion. What is the
interpretation of fpdf? and its constancy in phase space? Complete the sentence
below. Complete the sentence: “As time evolves, the H surfaces slowly distort but the particle stays on those H surfaces such that W I; if F65? “4‘36? .”
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 Fall '11
 Hassam
 mechanics

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