Question

1. A particle of mass m is attracted to a fixed point O by an inverse square force (F(r) = -k/r²). Solve this problem using Hamilton’s equations. (Hint: Use 2-dimensional cylindrical coordinates.)

Answer 1

Tom plane the As shown in figure , we have used polar coordinates (1,0). The velocity of particle is given by, - J = i en troeo so, T = 1 mlů ²+ 91282) 2 Now, we can write the Lagrangion L as, L= T-U ml eiz +912 2) - (-k) and for = 2L - mi Po = a1 - mr²o force ark Now, it is evident that, L is not a se time, and W is not of the velouties, H is equal to the of the particle , l.pl H= Ttu function of a function total Charge

the deri depinition So, either from this or from the of the Hamiltonian, we find, - , PLE 022 2 m 2012 and, the Hamilton's equations are, ů = 2H - Pr apr m o = ah OPomy? r = -24 29 - Por m2 -OH The Hust two equations define the definitions of the momenta. The last equation indicates that so is constant of motion as we expected because o is cyclic coordinate. Integrating the last equation, and then using the second equall Wo find that) Po = r² 8c constant - I
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