Find the equation of motion of a particle of mass m constrained to move on the s

Question

Find the equation of motion of a particle of mass m constrained to move
on the surface of a cylinder denoted by x^2 + y^2 = R^2. The particle is subject
to a force directed toward the origin and proportional to the distance of the
particle from the origin F = -kr with |r| the distance from the origin and k a constant

Hint: The potential energy of the particle is U = (1/2)kr^2. Your
answer will be graded according to these steps:

(a) Write the Lagrangian (3 points).
(b) Write and use the constraint equation (1 point).
(c) Write Lagrange's equations (3 points).
(d) Solve the equations of motion (3 points).

Thanks!

Explanation / Answer

The problem is easiest in cylindrical coordinates. Let R = . (That is the constraint equation). The particle has two degrees of freedom, and z.

U = (1/2)kr2 = (1/2)k(R2 + z2) is the potential energy.

T = (1/2)mv2 = (1/2)m('2R2 + z'2) where ' and z' are the time derivatives (can't put dots over the variables the way you normally do)

So L = T - U = (1/2)m('2R2 + z'2) - (1/2)k(R2 + z2)         (That is the Lagrangian)

The z equation of motion is dL/dz = d/dt (dL/dz') which is

-kz = d/dt (mz') = mz''                       (that is SHM)

The equation of motion is dL/d = d/dt (dL/d') which is

0 = d/dt (mR2') = mR2'' which tells us that ' is constant; in other words whatever rotational velocity around the cylinder the particle had stays constant.

The z equation tells us that the particle executes simple harmonic motion up and down.

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