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Can anyone help me make the connection here

  1. Mar 30, 2008 #1
    if F is any function and H is the hamiltonian, then the Poisson Bracket of F and H, is given by:

    [F,H] = dF/dt - [tex]\partial[/tex]F/[tex]\partial[/tex]t

    Can someone show me how the right side of this equation comes about?
    Also how can the normal derivative of F w.r.t t be different from the partial of F w.r.t t?
     
  2. jcsd
  3. Mar 30, 2008 #2

    cristo

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    In this case, where F is any function of coordinates and momenta (i.e. F=F(t,q,p), where the position and momentum depend upon time) the total derivative of F with respect to time will be [tex]\frac{dF}{dt}=\frac{\partial f}{\partial t}+\frac{\partial F}{\partial q}\frac{d q}{dt}+\frac{\partial F}{\partial p}\frac{dp}{dt}[/tex].

    Now, use the Hamilton-Jacobi equations, to get the equation in terms of H, and use the definition of the Poisson Bracket of H and F to get the equation in the required form.
     
  4. Mar 30, 2008 #3

    George Jones

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    This is a standard abuse of notation.

    Suppose, for example, [itex]F = F \left( q, p, t \right)[/itex] ([itex]F:\mathbb{R}^3 \leftarrow \mathbb{R}[/itex]), and [itex]q = q \left( t \right)[/itex] and [itex]p = p \left( t \right)[/itex] (i.e., each is a map from [itex]\mathbb{R}[/itex] to [itex]\mathbb{R}[/itex]).

    Define a new function [tex]\tilde{F} \left( t \right) = F\left( q\left(t) , p\left(t)\right), t \right)[/tex] (another map from [itex]\mathbb{R}[/itex] to [itex]\mathbb{R}[/itex]) by composition of functions. The multivariable chain rule then gives

    [tex]
    \frac{d \tilde{F}}{dt} = \frac{\partial F}{\partial q} \frac{dq}{dt} + \frac{\partial F}{\partial p} \frac{dp}{dt} + \frac{\partial F}{\partial t}.
    [/tex]

    Even though [itex]F[/itex] and [tex]\tilde{F}[/tex] are different functions, as they have different domains, it is standard to omit the twiddle.
     
    Last edited: Mar 30, 2008
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