The gravitational potential energy of two massic points ##P_1## and ##P_2## with respective masses ##m_1## and ##m_2## is given by
$$U = -G \frac{m_1 m_2}{|| P_2 - P_1 ||}$$
Now I was wondering how this formula could be applied to continuous matter. Let us imagine a very simple case where we...
Does that mean that now that I've made explicit that what I'm looking is in the end some mathematical structures/framework (that would account for the differences between potential and kinetic energy) nobody has a clue?
Noether's Theorem is indeed something I only have a superficial understanding (basically the vague idea that geometrical spacio-temporal invariances of a system imply the conservation of some related physical quantities) and I had already promised myself to investigate. I don't know if that can...
This is correct but missing the relevant point: that the presentation contains a false statement. The fact that you can indeed find counter examples where the direct sum is bigger than the tensor product does not makes the insight presentation any more correct.
I don't see how this answers my questions. Furthermore you seem to miss that a zero kinetic energy in a given frame means immobility. So suggesting that only the change of Kinetic Energy is physically relevant seems incorrect to me. On the contrary it is indeed correct that only "the Change in...
If I interpret correctly this sentence says the underlying set of U⊗V is U×V. This is not correct. This works for the direct sum U⊕V but the tensor product is a "bigger" set than U×V. One usual way to encode it is to quotient ##\mathbb{R}^{(U \times V)}## (the set of finitely-supported functions...
The more I think about it the less clear the respective natures of kinetic energy and potential energy (and of their sum, the so-called total energy) become. The thing is I have the impression that once you try to go a bit further than the usual description of "scalar values assigned to systems...
I think there are still important pitfalls that might deserve deeper explanations and details:
The opposition variable/constant is not a mathematical one but a meta-mathematical/logical one. Variables and constants are not mathematical objects but notation tools to describes mathematical...
You don't need coordinates to define a gradient. Really people rely too much on coordinates. Take a differential manifold ##M##. If ##M## is equipped with a (not necessarily Riemanian) metric tensor ##g## one can naturally define a unique inverse metric tensor ##\tilde{g}##. Using abstract index...
This is typically the kind of statement that is confusing. If ## A \wedge B ## is what we should call a pseudo-vector, ## A \times B ## should not. This may be a pure question of terminology of course but I think calling ## A \times B ## a pseudo-vector is misleading. The word pseudo-vector...
The problem is that to give a good answer to that question one need to know the domain of f. Saying that it is ##\mathbb{R}^n## (or an open subset) is not sufficient because it depends on the structure implicitly implied. If it is just a finite dimensional vector (or affine) space with no inner...
No I didn't but as far as I understand this is equivalent to stating that the electric constant is a number. So I think this example and the debate about ##c## can be summed up with a single question: To what extent can we consider fundamental physical constants like ##c, \varepsilon_0, h, G, e...
I do not agree. Statements "physical units are mere conventions" and "physical quantities are numbers" are not equivalent. "c=1" implies at least that speeds can be identified absolutely to number which is certainly stronger than just asserting that "meter/second" is no more special than...