Naked Science Forum
Non Life Sciences => Physics, Astronomy & Cosmology => Topic started by: varsigma on 10/10/2023 03:40:52
-
I think I might have found a youtube vid that explains how polar coordinates are a Hilbert space, if you have vectors with (r,θ) these are orthonormal, hence geometrically perpendicular in the abstract Hilbert vector space.
The video describes how to visualise a transform from polar vector space to Cartesian space. The latter is where we usually define real angles and distances which are not orthogonal, like in the real world. Orthogonality is something that "falls out" of the vector algebra, if you have an inner product defined.
And it explains what Christoffel symbols are in an intuitive way. I'd recommend watching it at least twice; it really gives you a good idea about the metric and why there is one. This is the low-dimensional stuff so it's easy to visualise.
the vid: https://www.youtube.com/watch?v=TvFvL_sMg4g (https://www.youtube.com/watch?v=TvFvL_sMg4g)
-
Although the author of the vid doesn't mention Hilbert spaces (I do), these are ubiquitous in physics.
Anywhere you can find an orthonormal pair, like (r,θ) which are linearly independent, you have a basis.
For instance, the angular velocity and the precession of a spinning top are such a pair.
But that's only two dimensions . . .