Register to reply

Geometry of phase space and extended phase space

by Coto
Tags: extended, geometry, phase, space
Share this thread:
Coto
#1
Jun8-09, 01:32 PM
P: 308
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:

[tex]\dot{x} = v;\ \ \dot{v} = -g;[/tex]

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
Phys.Org News Partner Science news on Phys.org
Hoverbike drone project for air transport takes off
Earlier Stone Age artifacts found in Northern Cape of South Africa
Study reveals new characteristics of complex oxide surfaces
mathador
#2
Aug7-09, 04:52 PM
P: 7
Hi Coto,

Quote Quote by Coto View Post
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

Best, Mathador


Register to reply

Related Discussions
Geometry of phase space and extended phase space Differential Equations 0
Phase space Calculus & Beyond Homework 0
Phase space for GR Special & General Relativity 0
SHO in phase space Advanced Physics Homework 0
Phase space geometry for a deterministic quantum mechanics General Physics 2