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1. I Jacobi identity of Lie algebra intuition

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...
2. I Can configuration space be observer independent?

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...
3. I Galilean transformation of non-inertial frame

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...
4. I Anyone knows why musical isomorphism is called so?

Anyone knows why musical isomorphism is called so? Why is it musical? https://en.wikipedia.org/wiki/Musical_isomorphism
5. Relativity Special relativity in Lagrangian and Hamiltonian language

Some introduction books on Lagrangian and Hamiltonian mechanics use classical mechanics as the theoretical framework, and when it come to special relativity it goes back to the basics and force language again. I would like to ask for some recommendations on good books that introduces Lagrangian...
6. I Understanding of special relativity and coordinates

I'd like to get some help on checking my understanding of special relativity, specifically I'm trying to clarify the idea of coordinates. Any comment is really appreciated! The spacetime is an affine space ##M^4##, which is associated with a 4 dimensional real vector space ##\mathbb{R}^4##...
7. I Benefits of Lagrangian mechanics with generalised coordinates

I have sometimes seen the claim that one advantage of Lagrangian mechanics is that it works in any frame of reference, instead of like Newtonian mechanics which will hold only in the inertial frame of reference. However isn't this gain only at the sacrifice that the Lagrangian will need to take...
8. I Reference frame vs coordinate system

Just want to clarify some concepts. There seems to be difference between reference frame and coordinate system. See https://en.wikipedia.org/wiki/Frame_of_reference#Definition . A reference frame is something has physical meaning and is related to physical laws, whereas coordinate system...

12. I Understanding relationship between heat equation & Green's function

Given a 1D heat equation on the entire real line, with initial condition ##u(x, 0) = f(x)##. The general solution to this is: $$u(x, t) = \int \phi(x-y, t)f(y)dy$$ where ##\phi(x, t)## is the heat kernel. The integral looks a lot similar to using Green's function to solve differential...
13. Prove eigenvalues of the derivatives of Legendre polynomials >= 0

The problem has a hint about finding a relationship between ##\int_{-1}^1 (P^{(k+1)}(x))^2 f(x) dx## and ##\int_{-1}^1 (P^{(k)}(x))^2 g(x) dx## for suitable ##f, g##. It looks they're the weighting functions in the Sturm-Liouville theory and we may be able to make use of Parseval's identity...

16. Central force field derivation

The total energy of the particle is ##u^2 / 2 - k/R##. When ##u^2 \gg 2k/R##, we take the total energy to be ##u^2/2## only. By the conservation of energy, we have: $$\frac{u^2}{2} = \frac{w^2}{2} - \frac{k}{p}$$ Take the angular momentum expression ##l = bu##, we can replace ##u## with...
17. Stokes' theorem gives different results

Given surface ##S## in ##\mathbb{R}^3##: $$z = 5-x^2-y^2, 1<z<4$$ For a vector field ##\mathbf{A} = (3y, -xz, yz^2)##. I'm trying to calculate the surface flux of the curl of the vector field ##\int \nabla \times \mathbf{A} \cdot d\mathbf{S}##. By Stokes's theorem, this should be equal the...
18. I General solution of heat equation?

We know $$K(x,t) = \frac{1}{\sqrt{4\pi t}}\exp(-\frac{x^2}{4t})$$ is a solution to the heat equation: $$\frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2}$$ I would like to ask how to prove: $$u(x,t) = \int_{-\infty}^{\infty} K(x-y,t)f(y)dy$$ is also the solution to...
19. I Solve second order linear differential equation

Consider the second order linear ODE with parameters ##a, b##: $$xy'' + (b-x)y' - ay = 0$$ By considering the series solution ##y=\sum c_mx^m##, I have obtained two solutions of the following form: \begin{aligned} y_1 &= M(x, a, b) \\ y_2 &= x^{1-b}M(x, a-b+1, 2-b) \\ \end{aligned}...
20. I Prove that the limit of this matrix expression is 0

Given a singular matrix ##A##, let ##B = A - tI## for small positive ##t## such that ##B## is non-singular. Prove that: $$\lim_{t\to 0} (\chi_A(B) + \det(B)I)B^{-1} = 0$$ where ##\chi_A## is the characteristic polynomial of ##A##. Note that ##\lim_{t\to 0} \chi_A(B) = \chi_A(A) = 0## by...