Transformations Definition and 863 Threads

I just realized that given a hamiltonian H, the set generated by its canonical transformations under composition just might be a category. Just checking the axioms: Given canonical transformations f:(H,q,p)\rightarrow (K,Q,P) and g:(K,Q,P)\rightarrow (H^\prime, q^\prime,p^\prime), g\circ f...

Homework Statement Let L denote the line through the origin in R2 that that makes angle -∏ < theta ≤ ∏ with the positive x-axis. The reflection operator that reflects points about L in R2 is the matrix transformation R2 --> R2 with standard matrix [cos 2(theta) sin 2(theta); sin...

Homework Statement Find the matrix representations [T]\alpha and [T]β of the following linear transformation T on ℝ3 with respect to the standard basis: \alpha = {e1, e2, e3} and β={e3, e2, e1} T(x,y,z)=(2x-3y+4z, 5x-y+2z, 4x+7y) Also, find the matrix representation of...

I'm studying in a class about phase transformations, and have a couple of questions needing answers/clarification: N0.1 A metallic sample can show faceing when heated, but a plastic sample does not because_______? NO.2 a)What are the general energies associated with a small...

Hi, i was thinking about the metric tensor transformation law: g_{cd}(x) = \frac{{dx'}^a}{{dx}^c} \frac{{dx'}^b}{{dx}^d} g'_{ab}(x') and, in view of this definition, the differences between Poincare transformations and reparametrization-like transformation (f.e. various conformal...

Homework Statement For each of the following, identify the base function and describe the transformation(s): f(x)=-4(3x)^2 + 5 Homework Equations f(x) = -4(3x)^2 + 5 The Attempt at a Solution Alright so my attempt at figuring out the requested answers are: Base function = x^2...

I am reading Grove and Benson's book on Finite Reflection Groups and am struggling with some of the basic linear algebra. Some terminology from Grove and Benson: V is a real Euclidean vector space A transformation of V is understood to be a linear transformation The group...

Homework Statement Given the generating function F=\sum _i f_i (q_j,t)P_i, 1)Find the corresponding canonical transformations. 2)Show that the transformations of generalized coordinates are canonical transformations. 3)What meaning does the canonical transformation originated by the generating...

Homework Statement My instructor wants me to only solve for the case m=2. The Attempt at a Solution So I thought I should discover what T does to the standard basis for matrices of size 2x2: T \left| \begin{array}{cc} 1 &0 \\ 0&0 \end{array} \right| = \left| \begin{array}{cc}...

Consider Minkowski spacetime with signature (-+++) and coordinates (ct,x,y,z) with respect to the standard orthogonal basis. I'm looking for the smallest set of matrices that can generate any Lorentz transformation with respect to this basis. I came up with 8 matrices (see below). Am I missing...

I was looking at this table here: http://en.wikipedia.org/wiki/Tensor#Examples And i didn't understand why a (1,1) tensor is a linear transformation, I was wondering if someone could explain why this is. A (1,1) tensor takes a vector and a one-form to a scalar. But a linear transformation...

Hi,all, I m an undergrades and I am suffering on understanding the different representation transformations, namely from schrodinger picture to interaction picture tupically, my lecturer didn't state which representation he was using and I m so confused, any helps would be great. Shall I bring...

Homework Statement The Attempt at a Solution So I observed: T(B) = λB T-1(B) = λ'B Also, T-1(T(B)) = λ'λB = B This implies, λ'λ = 1 And so, there should be a relation λ = \frac{1}{λ'}. Is that right?

Hi, Two questions: 1) Compute the matrix product corresponding to the composition of the transformations. Let U = P4(R) [polynomial degree 4], V = P3(R) , and W = P2, and let S = d/dx (derivative) and T = d/dx (derivative). Then the composition TS = d^2/dx^2 (second deriv) Attempt...

I want to understand how to make a Mobius Transformation.If someone can help me with an example that will be great. Let's say we have f(0) = i, f(1) = 1, f(−1) = −1 for instance ...how should I proceed in finding one?Thank you

Homework Statement Some matrix transformations f have the property that f(u) = f(v), when u ≠v . That is, the images of different vectors can be the same. For each of the following matrix transformations f : R^{2} → R^{2} defined by f(u) = Au , find two different vectors u and v such...

What is the most tangible way to introduce linear transformations in a linear algebra course? Most books tend to take a very abstract approach to this topic.

I tried to prove that a hermitian matrix remains hermitian under a unitary similarity transformation.I just could do it to he point shown below.Any ideas? [ ( U A U ^ {\dagger}) B ] ^ {\dagger} = B ^ {\dagger} (U A U ^ {\dagger}) ^ {\dagger} = B (U A^ {\dagger} U ^ {\dagger}) thanks

Okay so I've done very well in college so far, and I thought I was at least decent at math, but I just started this precalculus class and I'm having an issue. I basically don't know, and can't get a straight answer about how to handle functions that have multiple transformations going on...

Hi, I hope this is the right forum to post. My question is, which is the math implied for transforming an objectA local space, to anothers objects local space, and then transforming it to world space? For example, Lets say we want to know if objectA is infront of objectB, So the...

In relation to thermodynamics or statistical mechanics, it is often briefly mentioned that the partition function is the Laplace transform of the microcanonical \Omega, or that the Helmholtz free energy is the Legendre transformation of entropy, etc. But in the courses I've had, it stayed to...

Homework Statement http://s2.ipicture.ru/uploads/20120109/dT4m6rNG.jpg The attempt at a solution x=\frac{u}{1+v} and y=\frac{uv}{1+v} Transforming the integrand: \frac{x+y}{x^2}e^{x+y}=\frac{(1+v)^2 e^u}{u} dxdy=J.dudv J=\frac{v(1+v)^2 +1+uv}{(1+v)^3} The double integral becomes: \int\int...

Homework Statement http://s2.ipicture.ru/uploads/20120107/vVVkUT7f.jpg The attempt at a solution I plotted the graph x-y: http://s2.ipicture.ru/uploads/20120107/ja3V9aSV.jpg y=\frac{1}{2}(u+v) and x=\frac{1}{2}(u-v) So, after finding the Jacobian, the double integral becomes: \int\int...

Homework Statement A charge q is released from rest at the origin, in the presence of a uniform electric field and a uniform magnetic field \underline{E} = E_0 \hat{z} and \underline{B} = B_0 \hat{x} in frame S. In another frame S', moving with velocity along the y-axis with respect...

Homework Statement Q is an invertible self-adjoint linear transformation on an inner product space V. Suppose Q is positive definite. I have already shown that inv(Q) is self-adjoint, that all eigenvalues of Q are positive, so there exists S s.t. S^2 = Q. Now suppose P is any self-adjoint...

I was struck with the following question: Is there a linear map that's injective, but not surjective? I know full well the difference between the concepts, but I'll explain why I have this question. Given two finite spaces V and W and a transformation T: V→W represented by a matrix \textbf{A}...

Homework Statement Given the following defined transformation T(a + bt+ct^{2}) = (a+c) - (c+b)t + (a+b+c)t^{2} find the matrix with respect to the standard basis From my understanding, the standard basis for a 3 element vector would be (0,0,1)^{T} (0,1,0)^{T}...

I'm trying to model D2 rotational symmetry in protein quaternary structure using my CoordTransformer code. A CoordTransformer is composed of a pre and post translation, and a quaternion rotation: def transform(self,point): point -= self.pre self.rotate(point) point += self.post...

From Peskin and Schroeder: The finite-dimentional representations of the rotation group correspond precisely to the allowed values for the angular momentum: integers or half integers. From the Lorentz commutation relations: \left[J^{\mu \nu},J^{\rho \sigma}\right]=i \left(g^{\nu \rho}J^{\mu...

A quasar is moving away from the Earth with a speed of 0.850C. It emits a proton that eventually reaches earth, and is traveling at a speed of 0.519C relative to the earth. How fast is the proton moving relative to the quasar? Is this answer as simple as it seems? is the answer simply...

Homework Statement Define the linear transformation T: R^{3} → R^{3} by T(v)= the projection of v onto the vector w=(1,2,1) Find the (standard matrix of T) Homework Equations T: V → W is a function from V to W (which means that for each v in V, there is a T9v) in W such that...

Homework Statement An excited nucleus of krypton-80 emits a gamma ray that travels at the speed of light relative to the nucleus. The nucleus itself has a speed of 0.60c relative to the sun. Use a relativistic velocity transformation to determine the speed of the gamma ray relative to the...

Hi all, I came up with the following problem myself and am trying to solve myself. I haven't seen it in any txtbook, grad or undergrad. Suppose you have the ground frame (Earth). Earth sees ship1 start at t=0, v=vo1, at x=xo1. Earth sees ship2 start at t=0, v=vo2, at x=xo2 All...

Is it possible, in general, to have a one-to-one transformation from Rn to Rm for n>m? I'm thinking in the context of geometry, where you want to map a bounded region from a higher space to a lower space.

The derivation of the Lorentz transformations is based on the homogeneity[of space and time] and the isotropy of space. Could one derive the same transformations wrt space which is not homogeneous or[not] isotropic? You may consider a few chunks of dielectric strewn here and there. I am...

Homework Statement Space ships A and B, each having a proper length of 100m, pass each other moving in opposite directions. According to the clocks on ship A, the front end of B takes 1.5 x 10^(-6) s to pass the entire length of A. a) what is the relative velocity of the two ships? b)...

For fun, I'm writing a simple special relativity simulator with a much smaller speed of light so that relativistic effects are clear even at low speeds. I already have time dilation and speed of light delay working. However, right now, the speed of light does NOT always appear to be the same for...

Homework Statement Find two linear operators T and U on R^2 such that TU = 0 but UT ≠ 0. The Attempt at a Solution Let T(x1,x2)=(0,x2) Let U(x1,x2)=(x2,0) TU(x1,x2)=T(x2,0)=(0,0) Am I right? 'Cause I can't remember if TU(x1,x2)=T[U(x1,x2)] Or TU(x1,x2)=U[T(x1,x2)]

Homework Statement Given: T is a linear transformation from V -> W and the dim(V) = n and dim(W) = m Prove: If β = {v1, ..., vm} is a basis of V, then { T(v1), ..., T(vm) } spans the image of T. NOTE: because of bad hand writing I can't tell if the bold is suppose to be an 'm' or an 'n'...

Homework Statement Homework Equations NoneThe Attempt at a Solution Well guys, this is a problem I've been having for the last 2 days and with my midterm tomorrow I have no time to fiddle around with it. So, I do not understand how (where it says b) how Im going to use a division symbol ( /...

Hello -- I have some reference object R (e.g. a protein), and I've got two transformations t1 and t2 (e.g. a transformation = quaternion + translation). In my case, t1 and t2 were obtained from symmetry operations. So after applying t1 to R I get object T1, and after applying t2 to R I get...

Homework Statement Show that if { v_1, ... , v_k} spans V then {T(v_1), ... , T(v_k)} spans T(v) Homework Equations The Attempt at a Solution So we know that every vector in V can be written as a linear combination of v_1,...v_k thus we only need to show that {T(v_1)...

Hello, I read somewhere that in 2D, the Möbius transformations do not represent all the possible conformal transformations, while according to Liouville's theorem, in spaces of dimension greater than 2 all the conformal transformation can be expressed as combinations of...

Hi. So if you have \frac{d p_{\alpha}}{ds} = \frac{q}{c} F^{\alpha \beta} u_{\beta} how could you possibly go on proving this its form is invariant under all coordinate transformations? Or any physical law of any form, really? I guess my point is how do you represent "all possible...

Homework Statement Homework Equations The Attempt at a Solution In the previous problem I was asked to identify if a polynomial, such as f(x)=2x was a linear transformation. In that case I checked to see if f(ax + by) = f(ax) + f(by). I figure I would be doing something...

Homework Statement I'm having trouble understanding coordinate transformations for vector fields. There are two 'coordinate pieces', the coordinates pieces of the vector at a point changes, and the function describing the field can also be rewritten in terms of the new coordinates. I'm...

Homework Statement Show that 2.1.1 is equivalent to the totality of 2.1.2 and 2.1.3.Homework Equations The Attempt at a Solution aTx + bTy = aT(x) + bT(y) = T(ax) + T(by) = T(ax + by) ?

Homework Statement Let T:ℝ^{2}→ℝ be defined by T\left(\begin{array}{c} x_{1} \\x_{2}\end{array}\right) = (0 if x_{2} = 0. \frac{x^{3}_{1}}{x^{2}_{2}} otherwise.) Show that T preserves scalar multiplication, i.e T(λx) = λT(x) for all λ \in ℝ and all x \in ℝ^{2} The Attempt at a Solution...

Homework Statement find the linear fractional transformations (bilinear transformations) which map the ponts: z_{1} = -1, z_{2} = 0, z_{3} = -1 into w_{1} = j, w_{2} = \infty, w_{3} = 1 Homework Equations N/A The Attempt at a Solution I really don't have anything. For every question of this...

I have seen in various locations different conventions regarding the location of a prime symbol denoting a tensor represented in a new frame. For example, if the position four-vector is x^{\mu} then this four-vector in a different frame is often written as either x'^{\mu} or...