Defining Euclidean Norm in Phase Space: A Differential Geometry Analysis

[Rael]

Hi to everynoe!
I have a bit of trouble in understanding the following thing :
Suppose we have a phase space, in which a dynamical system evolves: for example a two dimensional vector space: temperature and time.
Now, does it make a sense to define the euclidean norm of a vector in such space ?
It just seems a bit strange to me to sum square seconds with square degrees and than extracting the square root. From the physical point of view it sould not have any sense...
Well, perhaps some differential geometry topics are involved...

Thanks four your help and attention.

 

[lippyka]

Yes, it does make sense to define the Euclidean norm of a vector in a phase space. The Euclidean norm is simply a measure of the 'length' or 'magnitude' of a vector, regardless of its components. So, you can measure the distance between two points in a phase space (e.g. temperature and time) using the Euclidean norm. This can be useful for studying the dynamics of the system, such as understanding how the system changes over time. Differential geometry is indeed involved here, as you can use the Euclidean norm to calculate curvature in the phase space.