TITLE: What's phase curve according to Arnold at Mathematical methods of classical mechanics?
QUESTION [0 upvotes]: While trying to cleaning rust I read Arnold's Mathematical methods of classical mechanics edition 2. 
I find its phase curve and phase space definitions a bit vague.  Here is a snapshot of the relevant part

$\phi$ isn't dimension consistent. Once it's used as a scalar and once as a vector valued function.

REPLY [2 votes]: Start with the system
\begin{equation}
\begin{cases}
\dot{x}=y\\
\dot{y}=f(x)
\end{cases} \, ,
\end{equation}
where each point in phase space is indexed by $\{x,y\}$. This means the system above defines a velocity field in phase space. This velocity field has an integral curve, which in this context is called phase flow, given by $\psi:R \to R^2$. As an example, take the harmonic oscillator, where
\begin{equation}
\begin{cases}
\dot{x}=y \\
\dot{y}=-x
\end{cases} \, ;
\end{equation}
this means $\psi(t) = (q_0 \cos t + p_0 \sin t, -q_0 \sin t + p_0 \cos t)$, which is parametrized curve in $R^2$. This system can also be written as a second order differential equation, given by $\ddot{x}+x=0$. This equation has the solution $\phi(t) = q_0 \cos t + \dot{q}_0 \sin t$, which means $\phi:R \to R$. The thing is that as we transformed our first order system in a second order equation, we also transformed our vector solution $\psi(t)$ in a scalar solution $\phi(t)$. Notice, however, that $\psi(t) = (\phi(t),\dot{\phi}(t))$, making the identification $\dot{q}_0= p_0$. Arnol'd called the vector $\psi$ by $\boldsymbol{\phi}$, that is, bold $\phi$. This fits perfectly within the fragment you quoted.
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