Soft question about connection between flow and group actions I am learning about flow and came across the group action formal definition of flow on Wikipedia. First of all, why is it a group action of the real numbers on the set of particle positions? Is this because it corresponds to linear movement, or multiplication by a scalar? Does this definition break down if the underlying vector field is nonlinear?
Also, is it correct to say that the orbit of this group action is the set of all integral curves passing through the initial conditions by which the flow is defined and that this is the path relating to the flow, i.e. it passes from the point to the end point of the flow? The conditions on flow seem pretty weak. I assume the flow function has to be at least piecewise smooth, but other than that, is the only other condition that the flow function matches the solution to the ode at the initial point. 
 A: Here is the intuition:
Think of time as $(\mathbb{R},+)$, the group of additive real numbers. If the "system" is deterministic and reversible (for a loose interpretation of these words), then $(\mathbb{R},+)$ acts on the "system" by moving forward and backwards in time. (You can and should convince yourself of the associativity property for this action, and also that the unit acts by the identity.)
You can track the orbits of individual particles through space, as well as the system as a whole through some parameter space.
Here is a toy model:
$(\mathbb{R},+)$ acts on a vector space $V$ by $f(t)(w) = w + tv$, for some fixed vector $v$. $f(t)(w)$ denotes the position of the particle $w$ (represented by a vector) at time $t$. The vector field inducing this here is a copy of $v$ at every point.
A more interesting example: the vector field $(-y,x)$ on the plane. What does it do? I.e. what is the induced flow?
Here is a caveat:
It is not going to be the case that $(\mathbb{R},+)$ always acts - some systems "blow up" in finite time, or stop making sense in less dramatic ways. You can take the previous example and remove a point from the vector space. Some of the particles will be unaffected, but other particles will reach that point at some finite time $t$. Where can you put them at time $t$? 
This counter-example is less artificial than it seems, as in fact when the space on which your flow lives is missing all holes - even holes at infinity, whatever this means - a flow generated by instructing particles to follow some sufficiently differentiable vector field is guaranteed to exist for all time. (The technical words are smooth vector field on a compact smooth manifold.)
Things to think about:
What is the role of the existence and uniqueness theorem for ODES? (This has to do with how you instruct particles to follow the vector field. You are right in that you need some hypothesis.)
Can you construct some examples that are not time reversible (which I am loosely taking to mean that you can make sense of what the system looked like for negative times?), but exist for all future times? 
Is it possible to build examples that are not associative? 
You can also construct discrete examples of this stuff, where for instance the group $\mathbb{Z}$ acts, or maybe $\mathbb{N}$ acts if the construction isn't reversible (like the map on polynomials sending each polynomial to its square). This shows up in stuff having to do with random walks and dynamical systems.
Hope that is helpful.
