Hamilton's equations in terms of initial conditions

Question

I'm trying to understand the way that Hamilton's equations have been written in this paper. It looks very similar to the usual vector/matrix form of Hamilton's equations, but there is a difference.

$$\frac{{\bf dZ}(t)}{dt} ~=~ J \frac{\partial H({\bf Z(t)})}{\partial z}, \qquad\qquad{\bf Z(0)} ~=~ z,$$

where $J$ is the block matrix $((0,1),(-1,0))$. $\bf{Z}(t)$ represents the point in phase space (positions and momenta). The part I don't understand is that the derivative of $H$ is taken with respect to the initial value (lowercase $z$) of ${\bf Z(t)}$. How does this form follow from the usual way of writing Hamilton's equations in vector form?

Note that my question is not about the vector notation for Hamilton's equations, which can be found in any introductory textbook on classical mechanics. I am specifically asking about the derivatives with respect to $z$, the initial values of ${\bf Z(t)}$.

Edit: I could not find an arxiv version of the paper, but it seems to be available as the first publication on one of the authors' websites, under the section "Coarse-graining with proper dynamics." The equation in question is the first one in section II of the paper.

Answer 1

The choice of letters in the paper is sloppy, and by abuse of notation, the author uses $z$ rather than $z_0$ to denote the initial condition. (Thus you need to replace $z$ everywhere by $z_0$ where it denotes the initial condition to get a onsistent reading. Alternatively, replace the $z$ in $\partial z$ by a capital $Z$.) Primarily, $z$ is the generic argument of $H(z)$; then everything makes sense:

$\partial H/\partial z = [\partial H/\partial p, \partial H/\partial q]$ (as a column vector in the right order). The factor $J$ changes the sign of the $q$ part and thus gives the standard conditions.

Thus the derivatives are not to be taken with respect to the inital values.

This kind of intelligent interpretation of a formula to make sense of it is often needed when reading papers by others. You know that you have the right interpretation once you have one that makes sense. Misprint correction in formulas works by the same principle.

Answer 2

From $$ \dot{Z} = J \frac{\partial H}{\partial z}$$

we can represent Z as (q,p) explicitly

$$ \left(\begin{array}{c} \dot{q} \\ \dot{p} \end{array} \right) = \left[\begin{array}{cc} 0 & 1 \\ -1 & 0 \end{array} \right] \left( \begin{array}{c} \partial H /\partial q \\ \partial H/\partial p \end{array}\right) = \left( \begin{array}{c} \partial H /\partial p \\ -\partial H/\partial q \end{array}\right) $$

Edit: Here is an explicit example to support Arnold's argument

Consider a harmonic oscillator (mass = 1)

$$x = x_0 \cos wt + p_0 \sin wt$$ $$p = p_0 \cos wt + - wx_0\sin wt$$ with $$H = \frac12(p^2 + w^2x^2) = \frac12 (p_0^2+w^2x_0^2)$$ Now, you can try

$$\partial H(x,p) / \partial p_0 = p_0$$

which is clearly not $\dot{x}$