I Interpretations of phase space in Dynamical Systems Theory

[Stephen Tashi]

TL;DR Summary

Can some phase spaces be interpreted as fields as well as a set of non-simultaneous states of a single system?

In Dynamical Systems Theory, a point in phase space is interpreted as the state of some system and the system does not exist in two states simultaneously. Can some phase spaces be given an additional interpretation as describing a field of values at different locations that exist simultaenously in a different space?

If $p$ is a point in a phase space and $p$ is a vector or more complicated mathematical object then there can be cases where there exists a 1-to-1 mapping that maps $p$ to a location in some other space (e.g. 3D space or a general metric space). Denote that location by $x(p)$. Denote the other information contained in $p$ by $x^c(p)$.

Let the dynamics of a system be given by the set of transformations $\mathbb{T}$ and use the notation that $T_{\Delta t} \in \mathbb{T}$ is the transformation that transforms $p$ to a new state $T_{\Delta t}(p)$ after an interval of time $\Delta T$ passes.

To regard the phase space as a field, consider the values $( x(p), x^c(p))$ to give information about the values of a field at locations $x(p)$ at time $t = 0$. At time $\Delta t$, define the value at location $x(p)$ to be $(x(p), x^c( T_{\Delta t}(p))$.

Is this a mathematically sound definition that describes a field changing in time?

 

[Fred Wright]

If $p$ is a point in a phase space and $p$ is a vector or more complicated mathematical object then there can be cases where there exists a 1-to-1 mapping that maps $p$ to a location in some other space (e.g. 3D space or a general metric space). Denote that location by $x(p)$. Denote the other information contained in $p$ by $x^c(p)$.

For classical mechanics, isn't $x^c(p)$ additional dimension of the phase space? Isn't it just additional dimensions to the cotangent bundle of configuration space?

 

[Stephen Tashi]

For classical mechanics, isn't $x^c(p)$ additional dimension of the phase space?

The definition doesn't specify a unique way to define $x(p)$ and $x^c(p)$ in terms of the phase space used by classical mechanics.

A "natural" way to do this for the phase space of a single particle would be to let $x(p)$ be the 3-D position information in $p$ at time $t=0$ and let $x^c(p)$ be the remaining components of $p$. However, for this to define a 1-to-1 mapping between a point $p$ and 3-D space, we can't have a phase space where the particle can be at the same location in 3-D space and have two possible states $p_1, p_2$ at that location. If such a thing is possible in a phase space, then finding spaces where $x(p)$ can defined will require an abstract viewpoint of what a space is.

Isn't it just additional dimensions to the cotangent bundle of configuration space?

I don't know because I haven't studied the physical interpretation of a "cotangent bundle".

 

[atyy]

You can put a probability distribution over phase space.
https://en.wikipedia.org/wiki/Liouville's_theorem_(Hamiltonian)
https://hepweb.ucsd.edu/ph110b/110b_notes/node93.html

 

[weirdoguy]

I don't know because I haven't studied the physical interpretation of a "cotangent bundle".

Cotangent bundle is the phase space in mechanics.