- #1
woodssnoop
- 10
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Hello:
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:
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].
Thank for your help in advance,
Dan
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]?
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