By Garrett P.
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88) The Poisson bracket (85) may then be re-expressed as: {«,t;}q,p = Xtw- 1 X 1 ,. (89) The phase space, which we denoted by T is itself a differential manifold and the coordinates (q, p) refer to a specific coordinate chart. ,n [Pi, i = n + l , n + 2 , . . ,2n. ,n i = n + l , n + 2 , . ,2n, ( 91 ) where again the e* are the canonical basis vectors of E 2 ". The cotangent space T*F has the dual basis, {drj1}^, such that W;^-) = sij. (92) 21 In terms of the q and p , dql, drf = i — 1,2,... ,n dpi-n, i — n+ l,ra + 2 , .
The forces acting on it are: • —mgk = force due to gravity (k is the unit vector in the vertical direction), • t = tension on the string, • a = centrifugal force. The equation of motion is mi = —mgk + t + a, (4) We set up the coordinate system in a way such that the origin of coordinates coincides with the undisturbed position of the mass centre of the pendulum, the X 3 -axis is aligned with the vertical direction and the motion of the pendulum takes place in the X2X3-plane. Then, in coordinate form, (4) becomes, mx\ — 0, mx2 = —||t|| sin# + ||a|| sin#, mx3 — —mg + ||t|| cos# — ||a|| cos#, (5) where 0 is the angular displacement, • a X2 a t~X3 sin 0 = —, cos o = — - — .
2 using Newton's laws of motion, and this time tackle the problem using first the Lagrangian and then the Hamiltonian formalisms. Clearly, L = T - V = \mt292 - mgl{l - cos 9), (32) and for small 9. L=\ml292-\mgl92, (33) from which 9 JL = mee\ ^ = _mfl#. (34) Hence, the Lagrangian equation of motion, d (dh\ dt \dqi J dL dqi with qi = 9, becomes m£29 + mgl6 = 0, which is the same as (10). Next, using the Hamiltonian approach, 8L = ^ = mi2e, 39 (35) V so that, 1 2 2 H = T + V = ^mC»2n2 9' + ^mgtB2 p2$ - + \mgt92.