Calculate the Lagrangian in a rotating coordinate system in terms of r and r\' w

Question

Calculate the Lagrangian in a rotating coordinate system in terms of r and r' where the system rotates with angular velocity and the preserved energy E(v,r), which is preserved in the equations of motion. We assume a potential V and a resulting force F=V/m.
Calculate the generalised momentum p in terms of r and r'.
Calculate Hamilton's function H(p,r).
We define the angular momentum J=r×p.
Calculate H(p,r) solely in terms of p and J and .

There is a ratio between the angular momentum that can be expressed with Poisson bracket as for =0 becomes [J_i,J_j]=mkijkJk (not sure how relevant this is).

Explanation / Answer

Solution:

The Lagrangian of a system is given by

L = T - V

Here, consider a particle of mass m is rotating in a coordinate system which is rotating by an angular velocity. If at any instant position of paricle is r' and r according to fixed and rotating frame respectively. So

r' = R + r

Where R is the position of the origin of the rotating frame according to the fixed frame.

So velocity dr’/dt = dr/dt + x r

Now Lagrangian, L = ½ m[dr/dt + x r]2 _ V

Generalised momentum pr = dL/dr = r + x r

Hamilton, H = pr  r - L or T + V

Related Questions

Navigate

• Browse (All)
• Browse C
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.