Solving Classical Mechanics #3d - Momentum p in Jaccobi Function

[Cosmossos]

Homework Statement

The problem is number 3d in the following file:
http://phstudy.technion.ac.il/~wn114101/hw/wn2010_hw06.pdf

The Attempt at a Solution

I think the difference comes from the using of the momentum p. In the Jaccobi function, we use only coordinate x and its derivatives.
Is it correct?

 

[Cosmossos]

And also number 4a ii.
How Do I do that?
thank you

 

[Cosmossos]

O.k
I made some progress.
In 4, i need only help with a)ii
thanks

 

[jdwood983]

The Euler-Lagrange equations of motion are second order differential equations while the canonical equations of motion are first order differential equations. So in the case of the 1D simple harmonic oscillator (with $m=k=1$),

$$H=\frac{p^2}{2}+\frac{x^2}{2}$$

the canonical equations come from:

$$\dot{p}=-\frac{\partial H}{\partial x}=-x$$

$$\dot{x}=\frac{\partial H}{\partial p}=p$$

so how can you relate these to the Euler-Lagrange equations of motion:

$$\frac{d}{dt}\left(\frac{\partial L}{\partial\dot{x}}\right)-\frac{\partial L}{\partial x}=0$$

for the same problem?EDIT: Fixed the E-L eom so that its coordinates match the Hamiltonian & canonical equations of motion.