1-D Lagrange and Hamilton equation gives different results.

[13Nike]

Homework Statement

This was supposed to be an easy question. I have a question here that wants you to describe a yoyo's acceleration (in one dimension) using Lagrangian mechanics. I did and got the right answer. Now I want to use Hamilton's equations of motion but I get a wrong number. Here is the excerpt

"A yoyo attached to a mass less string is suspended vertically from a fixed point and the other end is wrapped several times around a uniform cylinder of mass, m and radius R. When the cylinder is released it moves vertically down, rotating as the string unwinds. Write down the Lagrangian equation using the distance x as your generalized coordinate. Find the Lagrange equation of motion and show that the cylinder accelerates downward with #\ddot{x}# = 2g/3"

Homework Equations

I used Hamilton equation xdot = partial H/partial P
To help,
L = (3/4) m xdot^2 + mgx
and
H = (3p^2)/(4m) - mgx

The Attempt at a Solution

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I used Hamilton's equation and took the derivative
d/dt (xdot) = xddot = d/dt(3p/4m)
xddot = (3 xddot)/2

 

[TSny]

The momentum, $p$, that enters into the Hamiltonian should be the "canonical momentum" which is determined from the Lagrangian. You should find that $p \neq m \dot{x}$.

 

[13Nike]

The momentum, $p$, that enters into the Hamiltonian should be the "canonical momentum" which is determined from the Lagrangian. You should find that $p \neq m \dot{x}$.

I get $p = \partial L / \partial \dot{q} = (6/4) m \dot{x}$ but I don't know how/where to replace this $p$ into. Back into L and then re equate the Hamiltonian then use the Hamiltonian equation?

 

[TSny]

OK for your result for $p$.

I don't understand what you mean when you say "re equate the Hamiltonian". There is a specific prescription for constructing the Hamiltonian from the Lagrangian and the canonical momentum.

 

[13Nike]

OK for your result for $p$.

I don't understand what you mean when you say "re equate the Hamiltonian". There is a specific prescription for constructing the Hamiltonian from the Lagrangian and the canonical momentum.

We I am stuck at $p = (6/4)m \dot{x}$ do I take this result and somehow put $p$ back into the lagrange?

 

[TSny]

See https://en.wikipedia.org/wiki/Hamiltonian_mechanics#Basic_physical_interpretation

 

[13Nike]

See https://en.wikipedia.org/wiki/Hamiltonian_mechanics#Basic_physical_interpretation

Thank You for the help I think I got it now,
and for anybody who sees this thread here's another great section.
http://www.people.fas.harvard.edu/~djmorin/chap15.pdf (section 15.2.4)