woodssnoop
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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:
T = \frac{1}{2} m v^{2}
V = \frac{1}{2} k q^{2}
L(q,\dot{q},t) = T + V
H(p,q,t) = p \dot{q} - L(q,\dot{q},t)
\dot{q} =\frac{\partial H}{\partial p}
\dot{p} = - \frac{\partial H}{\partial q}
I have been replacing v with \dot{q}, 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 q = a \sin(2\pi f t), H could be expressed as just a function of just t.
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:
T = \frac{1}{2} m v^{2}
V = \frac{1}{2} k q^{2}
L(q,\dot{q},t) = T + V
H(p,q,t) = p \dot{q} - L(q,\dot{q},t)
\dot{q} =\frac{\partial H}{\partial p}
\dot{p} = - \frac{\partial H}{\partial q}
I have been replacing v with \dot{q}, but I don't believe I am getting the right answer. So my first questions are:
1a. Are the terms p and \dot{q} the same thing, and if not why?
1b. Are the \dot{q}, \dot{p} and other dotted terms I see in many texts referring to the time derivative of that term? If so, why is \dot{p} not referred to as \ddot{q}?
1b. Are the \dot{q}, \dot{p} and other dotted terms I see in many texts referring to the time derivative of that term? If so, why is \dot{p} not referred to as \ddot{q}?
While I was working though the problem I tried above, I noticed that given q = a \sin(2\pi f t), H could be expressed as just a function of just t.
2. So can the hamiltonian be function of only t?
Thank for your help in advance,
Dan