What is Transformations: Definition and 861 Discussions
In linear algebra, linear transformations can be represented by matrices. If
T
{\displaystyle T}
is a linear transformation mapping
R
n
{\displaystyle \mathbb {R} ^{n}}
to
R
m
{\displaystyle \mathbb {R} ^{m}}
and
x
{\displaystyle \mathbf {x} }
is a column vector with
n
{\displaystyle n}
entries, then
T
(
x
)
=
A
x
{\displaystyle T(\mathbf {x} )=A\mathbf {x} }
for some
m
×
n
{\displaystyle m\times n}
matrix
A
{\displaystyle A}
, called the transformation matrix of
T
{\displaystyle T}
. Note that
A
{\displaystyle A}
has
m
{\displaystyle m}
rows and
n
{\displaystyle n}
columns, whereas the transformation
T
{\displaystyle T}
is from
R
n
{\displaystyle \mathbb {R} ^{n}}
to
R
m
{\displaystyle \mathbb {R} ^{m}}
. There are alternative expressions of transformation matrices involving row vectors that are preferred by some authors.
In a classical example, for a system consisting of a mass attached to a spring mounted on a massless carriage which moves with uniform velocity U, as in the image below, the Hamiltonian, using coordinate q, has two terms with U in it.
But if we use coordinate Q, ##Q=q−Ut##, which moves with the...
We know that all actions are invariant under their gauge transformations. Are the equations of motion also invariant under the gauge transformations?
If yes, can you show a mathematical proof (instead of just saying in words)?
I feel if we have the matrix equation X = AB, where X,A and B are matrices of the same order, then if we apply an elementary row operation to X on LHS, then we must apply the same elementary row operation to the matrix C = AB on the RHS and this makes sense to me. But the book says, that we...
Suppose we have an action ##S=S(a,b,c)## which is a functional of the fields ##a,\, b,\,## and ##c##. We denote the variation of ##S## wrt to a given field, say ##a##, i.e. ##\frac{\delta S}{\delta a}##, by ##E_a##.
Then ##S## is gauge invariant when
$$\delta S = \delta a E_a + \delta b E_b...
Question:
Eq. 12.109:
My solution:
We’ll first use the configuration from figure 12.35 in the book Griffiths. Where the only difference is
that v_0 is in the z-direction. The electric field in the y-direction will be the same.
$$E_y = \frac{\sigma}{\epsilon _0}$$
Now we're going to derive the...
Hi, I am reading through my lecture notes - I haven't formally covered killing vectors but it was introduced briefly in lectures.
Reading through the notes has highlighted something I am not sure about when it comes to co-ordinate transformations.
Q1.Can someone explain how to go from...
Hi,
unfortunately, I have problems with the following task
I first tried to calculate ##JIJ^T##.
$$\left( \begin{array}{rrr}
\frac{\partial q'_i}{\partial q_j} & \frac{\partial q'_i}{\partial P_j} \\\frac{\partial P'_i}{\partial q_j} & \frac{\partial P'_i}{\partial P_j} \\...
Source: Scully and Zubairy, Quantum Optics, Section 1.1.2 Quantization
Questions:
1. Why are the destruction and creation operators considered a canonical transformations?
2. If these are canonical transformations, does it suggest that we are also canonically transforming the Hamiltonian...
In describing the Galelian or Lorentz transformations, All books I've read keep talking about clocks and meter sticks, but I don't see how an event happening away from the observer could be instantaneously described by a set of coordinates and a point in time; information conveying the event...
I'm running raw data and although, visually, the trends are promising, none of it is statistically significant. I was just going to leave it at that because the data was obtained after only 1 year of the experiment and I was just going to say that if treatment continued for a longer period of...
Consider the phase space of a one degree of freedom mechanical system. We can pass from one phase space coordinates to another phase space coordinates via a canonical transformation. I want to focus on 1-parameter canonical transformations,
$$(q_{0},p_{0})\rightarrow(q_{\lambda},p_{\lambda})$$...
In the frame of Observer C standing by the side of the road, the speed of Car A with respect to Car B = v1 + v2. (Galilean Transformation).
In the frame of Car A, the speed of Car B < v1 + v2 (Lorentz Transformation).
Please tell me if this understanding is correct.
##\bar{\mathcal{O}}## is moving with a velocity ##v## relative to ##\mathcal{O}## along ##x^{1}##
The Lorentz transformations between a Frame ##\mathcal{O}## and ##\bar{\mathcal{O}}## is given by:
$$\Delta x^{\bar{0}} = \gamma\left(\Delta x^0 - v\Delta x^1\right)$$
$$\Delta x^{\bar{1}} =...
A π+ meson is an elementary particle with a mean lifetime, defined in its rest frame, of τ = 2.60×10−8 s. The meson decays to a muon (µ+) and a neutrino (νµ) via the reaction π+ → µ+ + νµ. A π+ traveling in the laboratory decays so that the µ+ travels in the same direction as the original π+ and...
For a complex null tetrad ##(\boldsymbol{m}, \overline{\boldsymbol{m}}, \boldsymbol{l}, \boldsymbol{k})##, how to arrive at formulae (3.14), (3.15) and (3.17)? The equation (3.16) is clear as is. (I checked already that they work i.e. that ##\boldsymbol{e}_a' \cdot \boldsymbol{e}_b' = 2m'_{(a}...
Hi
The Hamiltonian for a harmonic oscillator is H = 1/(2m) ( p2+m2ω2q2). A canonical transformation is then made to a new Hamiltonian K( P , Q )
It is said that K ( P , Q ) = H ( p , q ) but K ( P , Q ) = ωP ( cos2Q +sin2Q ) = ωP
I don't understand how K ( P , Q ) = H ( p , q ) when they...
Conformal field theory is way over my head at the moment, but I decided to "dip my toes into it," and I watched a little video talking about conformal transformations. Now, I know that in a conformal transformation, $$x^\mu \to x'^\mu ,$$ the metric must satisfy $$\Lambda (x) g_{\mu \nu} =...
A drop of fuel is ignited in an engine cylinder, that produces heat, light and sound energies from the chemical energy stored in the drop of oil.
What I am not clear about is how heat energy gets transformed into mechanical work? I think the heat energy produced from ignition flows from burnt...
A matrix of dimension nxm
a. transforms a vector of dimension n to a vector of dimension m
b. transforms a vector of dimension m to a vector of dimension n
c. a vector of dimension n+m to a vector of dimension m
d. a vector of dimension n+m to a vector of dimension n
I'm watching this minutephysics video on Lorentz transformations (part starting from 2:13 and ending at 4:10). In my spacetime diagram, my worldline will be along the ##ct## axis and the worldline of an observer moving relative to me will be at some angle w.r.t. the ##y## axis.
When we switch...
Hello,
I have a question regarding the contravarient transformation of vectors.
So the formula:
V'n = dx'n / dxm Vm
So in words, the nth basis vector in the ' frame of reference over the mth (where m is the summation term) basis vector in the original frame of reference times the mth...
I consider three material points O, O', M, in uniform rectilinear motion in a common direction, so that in relation to the point O, the points O' and M move in the same direction with the constant velocities v and u (u>v>0). Assuming that at the initial moment (t0=0), the points O, O', M were in...
Hi,
I was looking at this derivation
https://en.wikipedia.org/wiki/Derivations_of_the_Lorentz_transformations#From_group_postulates
and I was wondering
1- where does the group structure come from? The principle of relativity? or viceversa? or what?
2- why only linear transformations? I remember...
Hey, so I've been studying some math on my own and I'm really confused by this one bit. I understand what contravariant components of a vector are, but I don't understand the ways in which they transform under a change of coordinate system.
For instance, let's say we have two coordinate...
Hello, why time is the fourth dimention and not another quantity or variable? General relativity has as a special case the special relativity, so Lorentz transformations are contained in general relativity but are they in a more general form than that of special relativity generally? If they...
ok we are supposed to go to here
Find 3 different matrices that reflect the following transformations, report the matrix, the determinant, and the eigenvalues.
1. Rotation by $\dfrac{\pi}{4}$
2. Shear along $x$ by a factor of $k$
3. Reflection by the line $\theta$
there are some more but the...
[Mentors' note: This thead was forked from another thread - hence the reference to "these replies" in the first post]
I am wondering why all these replies only discuss Lorentz transformations in 1+1 spacetime dimensions. That is the easy bit. The problems in understanding arise in 2+1...
The Lorentz tranformations are:
##x' = \gamma (x-vt) ##
##t' = \gamma(t - \frac{vx}{c^2})##
Consider an event (x,t) happening in S frame. Let S' frame be moving w.r.t. S frame along x direction with speed v whose origins coincide at t=0.
We find that the new coordinates of this event are...
I am stuck on this problem and keep going in a cycle coming back to the same state and would like to get hints on how to proceed. \( A \) is a \(R^{m*n} \) matrix and \( B \) is a \( R^{n*p} \) matrix. \( I_{n} \) is the \( n*n \) identity matrix.
Use elementary row and column operations to...
Dear Everybody,
I am in the process of relearning high school geometry through Khan Academy. I am current an graduated undergraduate student in mathematics. I am doing this because geometry is one of my weakest subject in mathematics. Second reason is that I want to reason out a problem...
Good evening, I'm trying to solve this exercise that apparently seems trivial, but I wouldn't want to make mistakes, just trivial. Proceeding with the first point I wonder if my approach can be correct in describing this situation.
From the assumptions, the two fields are in this...
I messed up somewhere, but don't know why! We consider this sequence of infinitesimal transformations,$$U = e^{i\varepsilon K_{\mu}}e^{i\varepsilon K_{\nu}}e^{-i\varepsilon K_{\mu}}e^{-i\varepsilon K_{\nu}}$$with ##K_{\mu}## and ##K_{\nu}## being two generators. I said, this simplifies...
I decided to return to my favourite topic (heavy sarcasm implied...), because somehow this active/passive stuff still trips me up. Let's say we have some operator ##A \in L(\mathcal{H}) : \mathcal{H} \rightarrow \mathcal{H}##, and also some unitary transformation ##U## between two sets of basis...
a. y=x^2 undergoes transformation 1 to become y=(x+2)^2
y=x^2+2 undergoes transformation 2 to become y=3(x+2)^2
y=3(x+2)^2 undergoes transformation 3 to become y=3(x+2)^2+4
So would the equation of the resulting curve be y=3(x+2)^2+4? I am very uncertain when it comes to performing...
I have introduced the Lorentz gauge on my perturbed metric ## \gamma_{\alpha\beta} ## given by ##\partial^{a}\gamma_{\alpha\beta}##. However, there remains the freedom to make further gauge transformations $$\gamma_{\alpha\beta} \rightarrow \gamma_{\alpha\beta} + \partial_{\alpha}\xi_{\beta} +...
Hello everyone,
I looked for the best physics forum to ask this question because the subject interests me a lot. The authors of this preprint (https://arxiv.org/abs/1804.10053) seem to claim that their approach (using linear canonical transformations) is a new alternative way for the...
Given that
H_{1} = reflection about the line y = x + 1;
H_{2} = counterclockwise rotation of pi/2 about the point (1,0);
H_{3} = translation by 1 - i.
What is the image of the triangle and arrow under the map H = H_{1} *H_{2} * H_{3} ?
I need help visualising the above transformation H. I...
We take an arbitrary spacetime point ##(x,t)## in any observer's reference frame ##A##.
Let ##(x(v),t(v))## be the co-ordinates of this same event as seen from a frame ##B## moving at a velocity ##v## wrt ##A##. As ##v## varies, the set of points ##(x(v),t(v))## constitute some curve ##C##.
So...
Please note that the transformed quantities will be indicated by ##'##.
Let me give some context first.
Let us assume here that the general approximate form of the potential energy ##V## and the kinetic energy ##T## are given to be
$$V^{app} = q^T V q \tag 1$$
$$T^{app} = \dot q^T V \dot q...
I'm watching Sean Carroll's video on symmetry [relevant section at around 8:05]
He talks about 120 degree rotations of triangles that leave them invariant. Then he proceeds to talk about flipping them with an interesting (at least to me) remark - "there's nothing that says I'm confined to...
I am totally new to the theory of Special Relativity, but find it very facinating. As a young man I saw a few documentaries on how Einstein saw a clock's movement reaching noon, and how he, traveling in a tram heard the gong only later. He then thought about what if he traveled at the speed of...
I'm trying my hand at deriving Lorentz transformations using 3 postulates - it's a linear transformation, the frames are equivalent, so they see the same speed of each other's origins and that the speed of light is the same. Let's say frame ##S## is moving at velocity ##v## in the...
Let ##V## be a real vectorspace of finite dimension ##n##. Let ##L, K:V \rightarrow \Re## be linear transformations so that ##ker(L) \subset ker(K)##. Then there's a parameter ##\lambda \in \Re## so that ##K=\lambda L##
a) Show that ##K=\lambda L## holds when ##K=0##.
b) Suppose that ##K...
Evening,
The reason for this post is because as the title suggests, I have a question concerning matrix transformation. These are essentially test prep problems and I am quite stuck to be honest.
Here are the [questions](https://prnt.sc/riq7m0) and here are the...
I am reading Tong's lecture notes and I found an example in which there are several aspects I do not understand.
This example is aimed at:
- Understanding what is the analogy in field theory to the fact that, in classical mechanics, rotational invariance gives rise to conservation of angular...
I am reading Kristopher Tapp's book: Matrix Groups for Undergraduates.
I am currently focused on and studying Section 1 in Chapter2, namely:
"1. Complex Matrices as Real Matrices".I need help in fully understanding Tapp's Proposition 2.5.
Proposition 2.5 and some comments following it read...
I am reading Kristopher Tapp's book: Matrix Groups for Undergraduates.
I am currently focused on and studying Section 1 in Chapter2, namely:
"1. Complex Matrices as Real Matrices".
I need help in fully understanding how to prove an assertion related to Tapp's Proposition 2.4.
Proposition 2.4...
The example is about the transformation between the cartesian coordinates and polar coordinates using the definition
In lewis Ryder's solution, I got confused in this specific line
I really can't see how is that straightforward to find?