Geometry of phase space and extended phase space

In summary, the content of the conversation is about the geometrical interpretation of objects in the basic theory of ODEs. The example of a free-falling particle in classical mechanics is used to discuss the concept of phase space and extended phase space. It is mentioned that the phase curves in phase space are parabolas, and it is questioned whether the phase curves in extended phase space are simply the projection of integral curves onto the position-velocity plane. It is also asked if phase velocity vector fields are the projection of a direction field in extended phase space onto the position-velocity plane. The response clarifies that for autonomous systems, there is no extended phase space, but for non-autonomous systems, the dimension of the phase space can be
  • #1
Coto
307
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
 
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  • #2
Hi Coto,

Coto said:
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
 

1. What is phase space in geometry?

Phase space in geometry is a mathematical concept used to describe the possible states of a physical system. It is a multi-dimensional space where each dimension represents a different variable of the system, such as position and momentum. In phase space, the movement of a system can be represented as a trajectory.

2. What is the difference between phase space and extended phase space?

The main difference between phase space and extended phase space is the number of dimensions. Phase space has the minimum number of dimensions required to fully describe a system, while extended phase space has additional dimensions that are not directly related to the system but can still influence its behavior.

3. How is phase space related to chaos theory?

Phase space is closely related to chaos theory, as it is used to study the behavior of chaotic systems. In chaos theory, phase space is used to visualize the complex and unpredictable behavior of a chaotic system, as well as to identify patterns and structures in the system's dynamics.

4. Can phase space be applied to all physical systems?

Phase space can be applied to a wide range of physical systems, from simple mechanical systems to complex biological and chemical systems. As long as the system's behavior can be described by a set of variables, phase space can be used to analyze its dynamics.

5. How is phase space used in practical applications?

Phase space has many practical applications in fields such as physics, engineering, and biology. For example, it is used in the design of control systems, the study of dynamical systems, and the analysis of biological networks. Phase space also has applications in data analysis and pattern recognition.

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