Susskind's theoretical minimum

[43arcsec]

If anyone out there has worked through Susskind's book, I have two questions on the Lagrangian to Hamiltonian section, any help would be greatly appreciated:

1) In Lecture 8 exercise 2, he wants you to calculate take the Lagrangian of

L=1/2ω d/dt q - ω/2 q^2 as a Hamiltonian and says it equals (ω=sqrt(k/m) )

H=ω/2 ( p^2 + q^2)

From what I can tell from his book, the Lagrangian is kinetic energy - potential energy, while the Hamiltonian is kinetic energy plus the potential energy.

I've tried making this work every which way but couldn't come up with it.

Also, on the next page (158) he says the Lagrangian is (d/dt)^2 q = - ω q

This is just the equation of motion for a harmonic oscillator; how does this pass for a Lagragian that is supposed to be the K.E - P.E.?

Sorry if I'm missing something easy, but thanks for taking a look.

-Marc

 

[maajdl]

First, I think L is maybe given by:

L=1/2ω (d/dt q)^2 - ω/2 q^2

From there, you can begin to calculate p = dL/d(qp) where qp = d/dt q .
Result follows immediately.

 

[43arcsec]

Canonical momentum, of course. Thanks Maajdl, I am in your debt.