My intuition about the Lie algebra is that it tries to capture how infinitestimal group generators fails to commute. This means ##[a, a] = 0## makes sense naturally. However the Jacobi identity ##[a,[b,c]]+[b,[c,a]]+[c,[a,b]] = 0## makes less sense. After some search, I found this article...
Sorry, the question may sound a bit weird. I will try to give a more detailed explanation on what structure I have in mind before rasing the question.
So first, we have a 4d spacetime, there are n worldlines in the spacetime representing n particles forming a system with certain interation...
We can formulate the spacetime in an observer/coordinate independent way, i.e. a particle becomes a worldline in the 4d space. Then relative to each observer, the worldline can be casted to a function in R^3. However, I haven't found any reference on formulating configuration space in a...
I think I started to understand why it's an unnatural thing to think about Galilean transformation on non-inertial observers, because the Galilean group naturally applies to the set of all inertial observers and is in one-to-one relationship to it (not sure what's the right math word here?).
I...
What I originally had in mind is below:
So we have an affine spacetime ##N^4##, the associated vector space ##V^4##, with the simultaneity subspace ##V^3##. An oberserver in my mind is the combination of a smooth worldline ##\gamma: \mathbb{R} \to N^4## and a smooth orthonormal basis assignment...
Hmmmm...there must be something I understood wrongly on this (why Galilean transformation is not applicable to non-inertial frame). I will try to be a bit more explicit:
Let's say the world is the 4 dimensional space with Galilean structure defined in the Arnold's. A Galilean transformation is...
If we take the definition from Arnold's
The galilean group is the group of all transformations of a galilean space which preserve its structure. The elements of this group are called galilean transformations. Thus, galilean transformations are affine transformations of A^4...
It's frequently discussed Galilean transformation brings one inertial frame to another inertial frame, and such a transformation leaves Newton's second law invariant (of the same form). I wonder what happens for non-inertial frame? If we start with a non-inertial frame, and Galilean transform...
That's a really good recommendation! I wonder if there're classical mechanics books that follows a similar philosophy? E.g. starting from a mathematical formulism in coordinate free langauge.
Thanks! I watched Susskind's lectures before but I'm looking for something with a bit more mathematical...
I found some new reference on this. In book "Special Relativity in General Frame", the concept of "local frame" is defined, similarly the concept of observer, in section 3.4.
I wonder if this is a common definition, and does it generalise to GR well?