Hello:(adsbygoogle = window.adsbygoogle || []).push({});

I am trying to understand how to build a hamiltonian for a general system and figure it is best to start with a simple system (e.g. a harmonic oscillator) first before moving on to a more abstract understanding. My end goal is to understand them enough so that I can move to symplectic transforms and then on to symplectic integration methods, but I plan on taking this one step at a time. From what I know and understand so far:

[itex]T = \frac{1}{2} m v^{2}[/itex]

[itex]V = \frac{1}{2} k q^{2}[/itex]

[itex]L(q,\dot{q},t) = T + V[/itex]

[itex]H(p,q,t) = p \dot{q} - L(q,\dot{q},t) [/itex]

[itex]\dot{q} =\frac{\partial H}{\partial p}[/itex]

[itex]\dot{p} = - \frac{\partial H}{\partial q}[/itex]

I have been replacing [itex] v [/itex] with [itex] \dot{q} [/itex], but I don't believe I am getting the right answer. So my first questions are:

1a. Are the terms [itex] p [/itex] and [itex] \dot{q} [/itex] the same thing, and if not why?

1b. Are the [itex] \dot{q}, \dot{p} [/itex] and other dotted terms I see in many texts referring to the time derivative of that term? If so, why is [itex] \dot{p} [/itex] not referred to as [itex] \ddot{q} [/itex]?

While I was working though the problem I tried above, I noticed that given [itex] q = a \sin(2\pi f t) [/itex], [itex] H [/itex] could be expressed as just a function of just [itex] t [/itex].

2. So can the hamiltonian be function of only [itex] t [/itex]?

Thank for your help in advance,

Dan

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Constructing a hamiltonian for a harmonic oscillator

**Physics Forums | Science Articles, Homework Help, Discussion**