Geometry of phase space and extended phase space

[Coto]

I just want to clarify the geometrical interpretation of these objects as encountered in the basic theory of ODEs.

For discussion let's use the simple set of differential equations found in classical mechanics for a free falling particle:

$$\dot{x} = v;\ \ \dot{v} = -g;$$

Now in phase space the phase curves are simply parabolas (as can easily be seen). How about extended phase space then?

Are the phase curves simply the projection of integral curves in the extended phase space onto the position-velocity plane?

Are phase velocity vector fields the projection of a direction field in extended phase space onto the position-velocity plane?

Thanks in advance,
Coto

 

[mathador]

Hi Coto,

Now in phase space the phase curves are simply parabolas (as can easily be seen). How about extended phase space then?

Your example is an autonomous system (does not depend on time explicitly), therefore
the dimension of its phase space is 2 (i.e. there is no extended phase space for this problem).
When time enters the problem explicitly (for example, as in a forced system), the non-autonomous system
x'=f(x,t)
can be rewritten as an autonomous one with an increase in the dimension of the phase space (this is the so-called extended phase space)
x'=f(x,y)
y'=1

Mathador