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...
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...
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...
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...
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##...
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...
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...
Let's say we have a function ##M(f(x))## where ##M: \mathbb{R}^2 \to \mathbb{R}^2## is a multivariable linear function, and ##f: \mathbb{R} \to \mathbb{R}^2## is a single variable function. Now I'm getting confused with evaluating the following second derivative of the function:
$$
[M(f(x))]''...
I know we can prove that a Galilean transformation sends one inertial frame to another inertial frame, by proving ##\frac{d^2 f(\vec{r})}{d(f(t))^2} = \frac{d^2 \vec{r}}{dt^2}##, but can we prove the reverse? Can we prove that if the acceleration seen in two frames are the same, then the...
I have tried to Fourier transform in ##x## and get the result in the transformed coordinates, please check my result:
$$
\tilde{u}(k, y) = \frac{1-e^{-ik}}{ik}e^{-ky}
$$
However, I'm having some problems with the inverse transform:
$$
\frac{1}{2\pi}\int_{-\infty}^\infty...
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...
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...
I have attempted a solution using conservation of momentum. Could people help check if this solution is correct (the result looks weird), as the problem doesn't have solution with it.
$$
\begin{aligned}
\begin{pmatrix}Mc \\ 0\end{pmatrix} &= \begin{pmatrix}E_R/c \\ \mathbf{p}_R\end{pmatrix} +...
According to this link here https://en.wikipedia.org/wiki/Relativistic_mechanics#Force , we can inverse the relation of force in terms of velocity and acceleration:
$$
\mathbf{F} = \frac{m\gamma^3}{c^2}(\mathbf{v} \cdot \mathbf{a})\mathbf{v} + m\gamma\mathbf{a}
$$
to get:
$$
\mathbf{a} =...
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...
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...
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...
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:
$$
\begin{aligned}
y_1 &= M(x, a, b) \\
y_2 &= x^{1-b}M(x, a-b+1, 2-b) \\
\end{aligned}
$$...
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...
I'm trying to solve the Goldstein classical mechanics exercises 1.7. The problem is to prove:
$$\frac{\partial \dot T}{\partial \dot q} - 2\frac{\partial T}{\partial q} = Q$$
Below is my progress, and I got stuck at one of the step.
Now since we have langrange equation:
$$\frac{d}{dt}...
In the first two chapters of Goldstein mechanics, the Lagrange equations are derived from both D'Alembert's principle and Hamilton's principle. I want to know what're the applicability of these two approaches to Lagrangian mechanics? Is one more powerful than the other or are they completely...
As far as I understand, when we want to differentiate a vector field along the direction of another vector field, we need to define either further structure affine connection, or Lie derivative through flow. However, I don't understand why they are needed. If we want to differentiate ##Y## in...
For a function ##f: \mathbb{R}^n \to \mathbb{R}##, the following proposition holds:
$$
df = \sum^n \frac{\partial f}{\partial x_i} dx_i
$$
If I understand right, in the theory of manifold ##(df)_p## is interpreted as a cotangent vector, and ##(dx_i)_p## is the basis in the cotangent space at...