Lagrangian hamiltonian mech COC Goldstein 8.27

[Liquidxlax]

Homework Statement

a) the lagrangian for a system of one degree of freedom can be written as.

L= (m/2) (dq/dt)²sin²(wt) +q(dq/dt)sin(2wt) +(qw)²

what is the hamiltonian? is it conserved?

b) introduce a new coordinate defined by

Q = qsin(wt)

find the lagrangian and hamiltonian with the new coordinate and is it conserved?

Homework Equations

qp-L = H

The Attempt at a Solution

Just wondering what the method is to solve b) or is it as simple as

q = Q/sin(wt)

(dq/dt) = (dQ/dt)/sin(wt) - Qwcos(wt)/(sin²(wt))

subbing that in and solving?

I'm assuming that this change of coordinate is supposed to make the Hamiltonian independent of time and therefore conserved.

Yet what i found is not conserved, so i assume i used the wrong method.

 

[fluidistic]

Homework Statement

a) the lagrangian for a system of one degree of freedom can be written as.

L= (m/2) (dq/dt)²sin²(wt) +q(dq/dt)sin(2wt) +(qw)²

what is the hamiltonian? is it conserved?

b) introduce a new coordinate defined by

Q = qsin(wt)

find the lagrangian and hamiltonian with the new coordinate and is it conserved?

Homework Equations

qp-L = H

The Attempt at a Solution

Just wondering what the method is to solve b) or is it as simple as

q = Q/sin(wt)

(dq/dt) = (dQ/dt)/sin(wt) - Qwcos(wt)/(sin²(wt))

subbing that in and solving?

I'm assuming that this change of coordinate is supposed to make the Hamiltonian independent of time and therefore conserved.

Yet what i found is not conserved, so i assume i used the wrong method.

Looks like a reasonable and straightforward method that you've used. How did you check out that the Hamiltonian wasn't conserved? I'd derivate it with respect to time and see if that equals 0.

 

[Liquidxlax]

Well isn't that always how you check if its conserved? Thanks for replying didnt think anyone would reply.