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- 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?

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?