Question

A particle is moving in a potential V ) given by the figure below. (b)Λ. Sketch approximately the phase portrait 0 <q < 10. Label regions where (a) the particle has enough energy to make it over both local maxima. (b) the particle is confined to the left of both maxima. (c) the particle is confined to the right of both maxima. (d) the particle is confined bewteen the maxima. (e) the particle has enough enough energy to make it over the left maximum, but is confined to the left of the right maximum. Identify any fixed points and separatrices, and whether fixed points are stable or unstable.

Answer 1

1. 
   - 2 10,-4) 2 kor @ the total onergy is greater the both maxima. So the particle will not be bound. But it's Velocity alll not be same as the kinetic enesay will be minimum when potential is maximum and vice versa. So the phase portrait should look lke. 6 8 10.
   
   6 . this means the ub who wil ni particle has a energy less than the smaller que uil chon minima marimai and it starts from q<2 . then it cle will at man go to 9=2, then it's total energy will be equal to potential. So then it all tun arand. O Different portraits for diff. values of vill be Ei tuone month @ this case the energy aull len of equal to the maximum potential and the particle starts from 977 this will be same as for the previous case Ira a> different portraits for different energy. au h ig bori u avioles no f the se dalier
   
   o in will about and the be the the case the particle bound and oscillates minima at q =45 amp. Marimum space it can get (2,0) qiw ten atro TVC to all have shown that hold for three different energies, I Be care full, that is can extend In this only to q=2 point. VA UN Separatri tronatlab reg the fixed points 9=2 and 7 are q= 4.5 is bortro are g = 2 4.57 unstable stable.