Question

Mechanics. Need help with c) and d)

1. A particle of mass m moves in three dimensions, and has position r(t)-(x(t), y(t), z(t)) at time t. The particle has potential energy V(x, y, 2) so that its Lagrangian is given by where i d/dt, dy/dt, dz/dt (a) Writing q(q2.93)-(r, y, z) and denoting by p (p,P2, ps) their associated canonical momenta, show that the Hamiltonian is given by (show it from first principles rather than using the energy) H(q,p)H(g1, 92,9q3, P1, P2,Ps) 5 marks] (b) Write down the six Hamilton's equations for the motion of the particle. [3 marks] (c) Suppose the particle is subject to a constant force FFi F2j +F3k, with Fi, F2, F constants i. Choosing V(0,0, 0)0, show that the potential energy of the particle is 3 marks] ii. Solve Hamilton's equations for this particular case if the particle is initially located at (ro, Vo, 2o) with velocity (vo, Vvo, Vz0). [5 marks] ii. Consider the case of projectile motion for the particle by setting F1-F20 and Fs-mg i 2 marks] (d) Consider the force F(ri+yj+2k) (k constant, k >0) acting on the particle. i. Choosing V(0,0, 0)0, show that the potential energy of the particle is V(r,y,2)k( +22) 3 marks] ii. Solve Hamilton's equations for this particular case if the particle is initially located at (ro, Vo, 2o) with velocity (vo, Vvo, Vz0). [5 marks]

Answer 1

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