Hamiltonian Mechanics
An extremely flexible rope of uniform mass density, mass m and total length l lies on a frictionless table with a length x hanging over the edge of the table. Assume that only gravity acts on the rope.
(a) Calculate the kinetic energy of the rope.
(b) Calculate the potential energy of the rope.
[Hint: start with an infinitesimal potential dU then use that to get U ]
(c) Calculate the generalized momentum of the rope.
(d) Write down the Hamiltonian for the rope.
(e) Derive Hamilton's equations of motion for the rope.
(f) Is the Total Energy conserved? Reasoning needed for credit.
You do not need to solve the differential equations resulting from (e).
3.Hamiltonian Mechanics An extremely flexible rope of uniform mass density, mass m and total length l lies on a frictionless table with a length x hanging over the edge of the table. Assume that only gravity acts on the rope.
(a) Calculate the kinetic energy of the rope (b) Calculate the potential energy of the rope [Hint: start with an infinitesimal potential dU then use that to get U] (c) Calculate the generalized momentum of the rope. (d) Write down the Hamiltonian for the rope. (e) Derive Hamilton's equations of motion for the rope. (f) Is the Total Energy conserved? Reasoning needed for credit.
You do not need to solve the differential equations resulting from (e)