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The context comes from trying to understand that Hamiltonian flow of a Homogeneous Hamiltonian. Let $p(x,\xi) \in C^\infty\big(\mathbb{R}^{2n}\big)$ be homogeneous of degree $2$ in $\xi$ and let $X(t,x,\xi)$ and $\Xi(t,x,\xi)$ solve the Hamilton flow equations

\begin{align*} \partial_t X &= (\partial_\xi p)(X, \Xi), \quad \ \ \ X(0,x,\xi) = x \\ \partial_t \Xi &= -(\partial_xp)(X, \Xi), \quad \Xi(0,x,\xi) = \xi \end{align*}

How do we show that from the degree $2$ homogeneity of $p$ in $\xi$ that for $\lambda > 0$ we have \begin{align} X(\lambda t, x , \xi/\lambda) = X(t,x,\xi), \quad \Xi(\lambda t,x,\xi/\lambda) = \lambda^{-1}\Xi(t,z,\xi) \end{align}

The first set of equations tell us that $X$ is homogeneous of degree $1$ in $\xi$ and that $\Xi$ is homogeneous of degree $2$ in $\xi$. I've tried to exploit this fact and the chain rule to differentiate the rescaled equations in $t$ and show they solve the same equations as $X(t,x,\xi), \lambda^{-1}\Xi(t,x,\xi)$ but I can't seem to get exactly what I want.

Edit: The bolded statement is not necessarily correct. The answer to the initial question is immediate once you noticed the same system is solved with the same initial conditions.

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  • $\begingroup$ Oh, I think you can show that $X$ and $\Xi$ are homogeneous of degree $1$ in $t$ and then this is immediate? $\endgroup$ Aug 12 '19 at 13:16
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I don't think my statement about any homogeneity of $Z$ and $\Xi$ is necessarily correct. In any case this is immediate. Let $Z_\lambda$ and $\Xi_\lambda$ denote $Z(\lambda t,x, \xi/\lambda), \ \Xi(\lambda t,x, \xi/\lambda)$ respectively. By definition \begin{align} \partial_t Z_\lambda &= \lambda (\partial_\xi p)(Z_\lambda, \Xi_\lambda)\\ \partial_t \Xi_\lambda &= - \lambda (\partial_zp)(Z_\lambda, \Xi_\lambda) \end{align}

Additionally we see using the Homogeneity of $p$ that \begin{align} \partial_t Z &= \lambda (\partial_\xi p)(Z, \frac{\Xi}{\lambda})\\ \partial_t \frac{\Xi}{\lambda} &= -\lambda(\partial_zp)(Z, \frac{\Xi}{\lambda}) \end{align}

Therefore these solve the same system of equations with the same initial conditions and it follows that $Z_\lambda = Z$ and $\Xi_\lambda = \frac{\Xi}{\lambda}$

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