Transformations Definition and 863 Threads

Show that when basic rotations are combined to find composite rota- tional transformations, if the rotation is about one of the principal axes of OXYZ (the fixed frame) the previous resultant rotation matrix is premulti- pled by the new rotational transformation, and if the rotation is about...

Hi everybody I've got a problem related to canonical transformations that I can`t solve: Given the expression of the canonical transformation Q=3q\cdot\big[ \exp\big((p+q)^5\big)+1\big] +3p\cdot \big[\exp((p+q)^5)+1\big]+p P=p+q I have to calculate an associated canonical transformation...

Anyone help. I know I must be doing this wrong somehow Lightning hits both a tree and a pole. The spacetime coordinates for each is (x=0, t=10us) for the tree and (x=30000m, t=10us) for the pole relative to the ground. Therefore they occur simultaneously relative to the ground. A rocket comes...

I want to make sure of my understanding of coordinate transformations. First of all, is it true that if \[{x_i}\] is a coordinate system on a manifold, then \[{q_j} = {q_j}({x_i})\] is a coordinate transform from "x" space to "q" space? If so can "x" be a flat space and "q" a curved...

1. Problem Horizontal rod of length x traveling along the positive y-direction at velocity u. Determine the orientation of the rod in frame S', which is moving at velocity v in positive x-direction. 2. Homework Equations Lorentz Transformation for length contraction, x' =...

Hello, suppose one has a classical canonical transformation between two sets of canonical variables such that the new (primed) positions and momenta can be written as functions of the old (unprimed) ones. {\cal K}: x_i \to x_i^\prime(x); \quad p_i \to p_i^\prime(p) Using these relations one...

Hey I'm new here. Well we're currently doing Laplace in our Maths lectures. Now the Teacher has set us a project on Laplace and we need to find some applications of Laplace Transformations. Can anyone tell me some specific areas where Laplace is applied. I remember reading somewhere it's...

There's something about the lorentz transformations which is somewhat confusing to me, and that is how to treat the "x" coordinate. Supposing I have some spaceship which is moving from Earth to some other planet located at a distance "D" (from earth) with a velocity v. Now, the spacetime...

Homework Statement which of the following are linear transformations. a) L(x,y,z) = (0,0) b) L(x,y,z) = (1 ,2, -1) c) L(x,y,z) = (x^2 + y, y - z) The Attempt at a Solution I know that L is a linear transformation if L(u + v) = L(u) + L(v) and L(ku) = kL(u). I am not sure how to...

I read that the form of a galilean transformation on the point (t,x) is the following: constant velocity transform by velocity v: (t,x) ---> (t,x+vt) translation transform by (t0,x0): (t,x)--->(t+t0,x+x0) rotation transformation by rotation matrix R: (t,x)--->(t,Rx) and that it is based...

Homework Statement The system S' moves in relation to the system S with velocity \upsilon along the -x- axis. At the time when the beginnings of the coordinate system are in the same point, clocks in both system shows t=t'=0. Which coordinates will have a reference point during the motion in...

Homework Statement I have a question regarding how to compose 2 transformations, a rotation and a translation, of a linear algebra problem. Suppose we have a quadratic curve like the following one: (i) x^2 + y^4 - 6xy +2x -3y +6 = 0 We want to transform the above into its standard...

I've spent a large portion of my day trying to figure this out and I figured my best answer is likely to come from here. Forgive me if I'm wildly wrong about anything, I'm somewhat basic with physics, largely due to the fact that I'm 15 and my maths is limited to a GCSE level. My dilemma is...

Hello, I would like to check if my understanding of this linear algebra problem dealing with transformations is correct: Part (1) we have the following coordinates systems: \tilde{x} = \begin{pmatrix} x_1 \\ x_2 \\ 1 \end{pmatrix} and x = \begin{pmatrix} x_1 \\ x_2 \end{pmatrix}...

I am wondering about the order of operations concerning the Lorentz transformation of fields and the superposition of fields. I was given a problem: Two positively charged electrons start at the origin and then travel along the x-axis at a constant speed v in opposite directions. Calculate...

Hello, I had a question regarding unitary transformations. The most common definition I see for unitary transformations is defined as a transformation between Hilbert spaces that preserves inner products. I was wondering if all unitary transformations between Hilbert spaces (according to this...

I'm a teacher. I need to teach this subject to a group of smart 9th grade students that I am prepping to begin a "Green Physics" project. What are the energy transformations associated with a hydroelectic dam? I think I have it up to a certain point, and then I'm less than sure. Here's...

Iv just been reading a physics textbook and i feel iv completely missed something. It may help to draw a diagram and to read the thread slowly. Sorry if it is a little thick. My understanding of Special Relativity is that it allows two seemingly conflicting principles to co-exist, these being...

Hi all, I've been struggling with this for a couple of days now and am positing a question in the hope that someone can help me out. I have a global cartesian coordinate system X, Y, Z and a cube with it's centre at (0,0,0) and dimension 1. Hence it's corners are: (0.5, -0.5, 0.5), (-0.5...

Hi, I just have a quick question, I understand that (T\vec{u},T\vec{v})=(\vec{u},\vec{v}) defines a unitary transformation T , but how does one go from this relation to (T\vec{u},\vec{v})=(\vec{u},T^{-1}\vec{v}) as the other way to define a unitary transformation? Thanks

Homework Statement Theorem. The change of variables is reversible near (u0,v0) (with continuous partial derivatives for the reverse functions) if and only if the Jacobian of the transformation is nonzero at (u0,v0). 1. Consider the change of variables x=x(u,v)=uv and y=y(u,v)=u2-v2. (a)...

Homework Statement http://img338.imageshack.us/img338/9682/spaceships.gif V1, V2, U, alpha are given. The red spaceship is moving left in Velocity V1 (in the lab system) and a ball is thrown in angle alpha and speed U(in the red spaceship system). What is Alpha in the Blue system...

1. Whenever we perform canonical transformations in Hamiltonian mechanics, we look for those generating functions which leave the form of the canonical equations unchanged. Why do we restrict ourselves to those transformations which leave the equations unchanged? Can I not do some...

For the linear transformation, T: R^2\rightarrow R^2, T(x,y) = (x^2,y) find the preimage of.. f(x)= 2x+1 I have no trouble with these types of problems when it comes to vectors that aren't functions. Any help would be appreciated! Thanks! ~Matt

Below is a HW problem which I believe is correct. Can you guys take a look and advise? Which of the following are linear transformations A) L(x,y,z)= (0,0) B) L(x,y,z)= = (1,2,-1) C) L(x,y,z)= ( x^2 +y , y-z) To prove these relationships are linear transformations, they must satisfy...

Homework Statement describe the energy transformations required in the following Fax/Modem Radio And Television thanks

L: R^2=>R^2 is defined by L(x,y)= (x+2y), (2x-y) let S be the natural basis for R^2 and T=(-1,2), (2,0). T is another basis for R^2. Find the matrix representing L with respect to A) S B) S and T C) T and S D) T E) Compute L L(1,2) using the definition...

This is just to see if I remember? Please confirm, correct any errors, and answer the questions (q's in bold) Homework Statement What geometric transformations will "transform" f(t) = et \stackrel{transformations}{\rightarrow} \frac{mg}{b} * (1 - e-bt/m)? Homework Equations f(t) =...

Homework Statement Find the power dissipated by the 8 ohm resistor. This was an example for source transformations. Homework Equations P=VI=V^2/r=I^2*R V=IR The Attempt at a Solution First I transformed the 1 amp current source and 20 ohm in parallel to a 20 V voltage...

1. Find out which of the transformations are linear. For those that are linear, determine whether they are isomorphisms. T(f(t)) = f'(t) + t^2 from P2 to P2 2. To be linear, T(f(t)+g(t))=T(f(t)) + T(g(t)), kT(f(t))=T(f(kt)) 3. After testing for linearity, I am thinking that the...

Hi, In component form the transformation for the following tensor can be written as F^{\mu\nu}=\Lambda^{\mu}_{\alpha}\Lambda^{\nu}_{\beta}F^{\beta\alpha} or in matrix notation, apparently as F^{'}=LFL^{T} Here L is the Lorentz transformation matrix Im happy with the component form...

Homework Statement [1] A random variable X is distributed as fX(x) = 1/9*(1+x)^2 1{-1<= x<= 2}. a) Find the density function of Y = -X^2 + X + 2. b) Find the cummulative distribution function of Y = X1{-1<=X<=1} + 1{X>=1} [2] Find the function that transforms a variable X with fX(x) =...

Lorentz transformations ("synchronising" reference frames?) Homework Statement A particle moves from (x,y,z,t) = (0 m,0 m,0 m,0 s) to (1 m,1 m,0 m,10 ns). i. What is the speed of the particle in this reference frame? ii. What is the speed of the particle in a reference frame moving...

Good evening, As an effort for trying to understand Lorentz transformations, I'm trying to use them to derive the "length contraction" result. Consider two reference frames, O (non-primed) and O' (primed), moving with respect to each other with a velocity v. Consider them to be under...

Hi, this is a question from a practice paper I have. I can't think how to do this. As far as I'm aware this has to be assumed to derive the Lorents transforms, so it must be by definition true, making the question pointless. Does anyone have any thoughts or suggestions on this? Regards, Pete

The phase flow is the one-parameter group of transformations of phase space g^t:({\bf{p}(0),{\bf{q}(0))\longmapsto({\bf{p}(t),{\bf{q}(t)) , where {\bf{p}(t) and {\bf{q}}(t) are solutions of the Hamilton's system of equations corresponding to initial condition {\bf{p}}(0) and {\bf{q}}(0)...

Homework Statement Let B be a basis of R^2 consisting of the vectors <5,2> and <1,5> and let R be the basis consisting of <2,3> and <1,2> Find a matrix P such that [x]_r=P[x]_b for all x in R^2 Homework Equations Ax=B? The Attempt at a SolutionI attempted by using Ax=B as a format to solve...

Homework Statement The phase flow is the one-parameter group of transformations of phase space g^t:({\bf{p}(0),{\bf{q}(0))\longmapsto({\bf{p}(t),{\bf{q}(t)) , where {\bf{p}(t) and {\bf{q}}(t) are solutions of the Hamilton's system of equations corresponding to initial condition...

It seems that the common approach to obtain the equations for the Lorentz transformations is to guess at its form and then, by considering four separate situations, determining the values for the constants. From these equations, things like time dilation and length contraction can be worked out...

Homework Statement Determine if this is a linear transformation from R3 to R2 Homework Equations L(x) = (1 + x1, x2) The Attempt at a Solution Whenever I perform addition and scalar multiplication, I obtain this is closed under both. The book says this isn't a transformation though.

Homework Statement Derive a formula for T. T([1 1]^T)=[2 -1]^T and T([1 -1]^T)=[0 3]^T (...^T=transpose and T(...)=Linear Transformation Homework Equations T(c1v1+...+cnvn)=c1T(v1)+...+cnT(vn) The Attempt at a Solution The solutions manual's method and the method I am...

Let P,Q be self-adjoint linear transformations from V to V, Q is also positive-definite. Deduce that there exist scalars λ1 , . . . , λn and linearly independent vectors e1 , . . . , en in V such that, for i, j = 1, 2, . . . , n: (i) P ei = λi Qei ; (ii) <P ei , ej > = δi j λi ; (iii) <Qei ...

How can I convince myself of the following statement: If x2<0, there exists L orthochronous Lorentz tranformation such that: Lx = -x My concern is this: If for example, we take xµ=(1,0,0,0), then Lx in component form is: Λµβxβ=Λµ0x0 =(Λ00, Λ10, Λ20, Λ30). By definition, if it is an...

Hi guys Ok, let's say I have a matrix given by M = \sum_{ij}M_{ij}a_i^\dagger a_j, and I wish to transform it. Now in some books I have read they write the transformation as s = \sum_{j}U_{ij}a_j, while in some notes I have read they write it as s =...

Homework Statement T: V-> V, dimV = n, satisfies the condition that T2 = T 1. Show that if v \in V \ {0} then v \in kerT or Tv is an eigenvector for eigenvalue 1. 2. Show that T is diagonalisable. Homework Equations The Attempt at a Solution I have shown in an earlier part...

Homework Statement The matrix A =(1,2,3;4 5 6) defines a linear transformation T: R^3-->R^2 . Find the transformation matrix for T with respect to the basis (1,0,1),(0,2,0),(-1,0,1) for R^3 and the basis (0,1),(1,0) for R^2. Homework Equations - The Attempt at a Solution I have no...

Homework Statement Determine if the transformation T: R^{2}\rightarrow R^{2} is linear if T(x, y)= (x+1, 2y) Homework Equations 1. T(u + v) = T(u) + T(v) 2. T(c*u) = cT(u) 3. T(0) = 0 The Attempt at a Solution I believe I have to use the above provided equations to determine...

Homework Statement Let u = (1,2), v = (3,1) and T: R^{2}\rightarrow R be a linear transformation such that T(u)= 4 and T(v)= 5. What is T(-3, 4)? (Hint: Write (-3,4) as a linear combination of u and v.) Homework Equations T(x)= b The Attempt at a Solution I don't really know where...

Note: in the following, parentheses denote "subscript". "G" denotes "gamma". A round, uncharged current loop is at rest in the xz plane of IRF K. The loop is centered on the Origin. Negative charge circulates around the loop, positive charge remains at rest. There is a nonzero B(y) field...

Given linear transformations S: Rn --> Rm and T: Rn --> Rm, show the following: a) S+T is a linear transformation b) cS is a linear transformation I know that since both S and T are linear transformations on their own, they satisfy the properties for being a linear transformation, which is that...