- #1
Coto
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- 3
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
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