Recent content by lriuui0x0

Let's say we have a grid of contour lines defined by df and dg, consecutive contour lines have their function values changed by one. Then does each grid cell correspond to area one?

We can visualize 1-form by contour lines, since a 1-form / gradient sort of represents how fast the function changes. I wonder whether we can visualize 2-form df ^ dg by intersection of two sets of contour lines for f and g, or maybe something of a similar nature?

Hmmm... I fee it's weird that the old & new coordinates are described by different charts, since they are completely not comparable that way. The transformation between the coordinates would not reflect the symplectomorphism itself, but also a random chart transition function. So is the answer...

I'm just trying to understand in what sense symplectomorphism is canonical transformation. My understanding is the following. When you have a symplectomorphism, that's an active picture of the transformation which maps a point on the phase space to a different point. If we have a coordinate...

I have read that canonical transformation is basically a symplectomorphism which leaves the symplectic form invariant. My understanding is that the canonical transformation is a passive picture where we keep the point on the phase space fixed and change the coordinate chart, where...

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...

I just don't usually see coordinate independent formulation of the configuration space.

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...

Actually given the transformation is Galilean, shouldn't the ##\mathbf{a}_1## and ##\mathbf{a}_2## be the same, up to an orthogonal transformation?

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...