Bracket Definition and 83 Threads

Hi, I've been doing a course on Tensor calculus by Eigenchris and I've come across this problem where depending on the way I compute/expand the Lie bracket the Torsion tensor always goes to zero. If you have any suggestions please reply, I've had this problem for months and I'm desperate to...

As an aside, fresh_42 commented and I made an error in my post that is now fixed. His comment, below, is not valid (my fault), in that THIS post is now fixed.Assume s and w are components of vectors, both in the same frame Assume S and W are skew symmetric matrices formed from the vector...

Let me define ##L_{x;v}## as the operator that produce a Lorentz boost in the ##x##-direction with a speed of ##v##. This operator acts on the components of the 4-position as follows $$L_{x;v}(x) =\gamma_{v}(x-vt),$$ $$L_{x;v}(y) =y,$$ $$L_{x;v}(z) =z,$$ $$L_{x;v}(t)...

I've recently been starting to get really confused with the meaning of equality in multivariable calculus in general. When we say that the poisson bracket is invariant under a canonical transformation ##q, p \rightarrow Q,P##, what does it actually mean? If the poisson bracket ##[u,v]_{q,p}##...

I'm trying to make some simple rails for a makeshift server rack. I already have some aluminum angle (6063-T52) that has 2" legs and .125" thick. I have been trying to figure out how much weight could be safely held and can't find anything that makes sense. The weight load is fairly evenly...

I am wondering why the two methods below give different answers. If I multiply z through the second bracket I get $$(\frac{d}{dx} +x)(-\frac{dz}{dx} + xz)$$which, on expansion, yields $$-\frac{d}{dx}\frac{dz}{dx} -x\frac{dz}{dx} + \frac{d(xz)}{dx} + x^{2}z = -\frac{d^{2}z}{dx^{2}} + x^{2}z +...

Prove that for a 2 sphere in R3 the Lie bracket is the same as the cross product using the vector: X = (y,-x,0); Y = (0,z-y) [X,Y] = JYX - JXY where the J's are the Jacobean matrices. I computed JYX - JXY to get (-z,0,x). I computed (y,-x,0) ^ (0,z,-y) and obtained (xy,y2,yz) = (z,0,x)...

Thanks for taking interest but I studied basic statics about three decades ago and I wasn't quite sure what the definitive answer was...can anybody tell me what the preferred arrangement of this reinforcement bracket should be? Should it be as shown in the picture to achieve maximum stiffness or...

Hi all, The photo is a reverse L-shaped bracket installed on the wall, L or reverse L is of more supporting force if all things being equal? Why? Thank you very much!

I have been reading about Rings and Modules. I am trying reconcile my understanding with Lie groups. Let G be a Matrix Lie group. The group acts on itself by left multiplication, i.e, Lgh = gh where g,h ∈ G Which corresponds to a translation by g. Is this an example of a module over a ring...

Hi I'm on to the last question for my HNC and having a nightmare trying to work out my best options! Any help would be greatly appreciated! The bracket show in figure 1 is to be used to mount an outboard motor onto the transom of a boat. a. suggest two appropriate materials which will require...

Homework Statement Determine the Lie bracket for 2 elements of SU(3). Homework Equations [X,Y] = JXY - JYX where J are the Jacobean matrices The Attempt at a Solution I exponentiated λ1 and λ2 to get X and Y which are 3 x 3 matrices.. If the group elements are interpreted as vector...

Hello! So I have 2 vector fields on a manifold ##X=X^\mu\frac{\partial}{\partial x^\mu}## and ##Y=Y^\mu\frac{\partial}{\partial x^\mu}## and this statement: "Neither XY nor YX is a vector field since they are second-order derivatives, however ##[X, Y]## is a vector field". Intuitively makes...

I am installing an L-shaped bracket into a brickwork chimney to support some large pipework. I am trying to work out the forces in each fixing to ensure that I won't be pulling the bracket off the wall once the pipes are filled with water. I've tried drawing the free body diagram and to resolve...

In some sources QM is explained using bracket notation. I quite understand algebra of bracket notation, but I do not understand how is this notation related with physically meaningful things? How is bracket notation related to wavefunction notation? Could you tell me whether following is true...

Homework Statement I have designed a bracket and am interested in stress at a certain point. I have figured out the load needed to support. The area of interest carries an eccentric load so bending moment and axial load need to be figured out. Not sure which FBD is correct to use. Homework...

let say I have a vector |a> and |b> and a transformation matrix A What is the difference between <a|A|b> and <a|Ab>? And also, I don't quite understand why <a|Ab> = <A+a|b>. Where does this identity come from? Thanks!

I took a semester of QM as an undergrad engineering major, and I don't recall the motivation for replacing traditional vector notation with bracket notation. Can someone enlighten me? Thank you.

anyone can help me how to solve the following poisson bracket? {U(x,λ), U(y,µ)} = −(1/λµ) {S^{i} (x), S^{j} (y)} σ_{i} ⊗ σ_{j} where U(x, λ) = −(i/λ) S(x)

Just need someone to double check my work. So the bracket is going to be cut out from a steel plate in a weird, rounded out triangular shape, and then a hole will be drilled through it. For simplicity though, let's just say the bracket is a ring with one half of it welded onto a wall. OD of the...

I have a hydraulic cylinder that can lift up to 3 metric tons and I want to attach it to an arm. The bracket is in a rigid U shape (right angles only, no round corners), with a round pin going through it. At first I thought it'd be a simple bending stress calculation, but there is a giant hole...

Homework Statement See photo attachment Homework EquationsThe Attempt at a Solution Ive begun by defining an x and y axis, followed by find the sum of x-components of the forces, and y-components. Followed by the magnitude of the resultant force. The only method i could think of was to find...

I'm trying to show that the lie derivative of a tensor field ##t## along a lie bracket ##[X,Y]## is given by \mathcal{L}_{[X,Y]}t=\mathcal{L}_{X}\mathcal{L}_{Y}t-\mathcal{L}_{Y}\mathcal{L}_{X}t but I'm not having much luck so far. I've tried expanding ##t## on a coordinate basis, such that...

When proofing the poisson brackets algebraically, what is the tool of choice. Can one use the muti dimensionale chain rule or how is it usally done?

Member ABC is supported by a pin and bracket at C and by an inextensible cable of length L  7.8 m that is attached at A and B and passes over a frictionless pulley at D. Assuming P  450 N and   35 and neglecting the mass of ABC and the radius of the pulley, simplify the problem as much as...

Homework Statement To calculate a certain Dirac bracket I need to calculate this Poisson bracket (Weinberg QTF 1 p.349 first eq.) $$[F,\Pi_i(\mathbf{z})]_P$$ where F is any functional of matter fields and their conjugates and pi is the conjugate to the vector potential. It should be zero...

Hello In textbook by Kobayashi and Nomizu derivation of rank k in space of all differential forms on a manifold is defined to be operator that is linear, Leibnitz and maps r-forms into r+k-forms. By Leinbitz I mean, of course: D(\omega \wedge \eta)=(D \omega) \wedge \eta + \omega \wedge (D...

The Maurer-Cartan one-form ##\Theta = g^{-1} dg## is though of as a lie algebra valued form. It arises in connection with Yang-Mill's theory where the gauge potential transforms as $$A \mapsto g Ag^{-1} - g^{-1} dg.$$ However, one also defines for lie-algebra valued differential forms...

Definition/Summary In the Hamiltonian formulation of classical mechanics, equations of motion can be expressed very conveniently using Poisson brackets. They are also useful for expressing constraints on changed canonical variables. They are also related to commutators of operators in...

Hi, I am having awful trouble working this out and have been going round in circles. Can you help me please? Its: x/x+1 - 4/x - 2 = 2

In Hamiltonian formulation there is an expression df / dt = { f , H } + ∂f / ∂t where f is function of q, p and t. While expressing Hamiltons equations of motion in terms of Poisson Bracket, i.e if the function f = q of p then its partial time derivative ∂f / ∂t becomes zero.. Please explain why?

Hi all, I just started learning engineering mechanics this semester,so what I'm about ask may be a basic question in the subject.I came across a bracket that is used in a industrial equipment (i have attached a sketch as image).The bracket is used to mount a hydraulic cylinder.The cylinder has...

When (canonically) quantizing a classical system we promote the Poisson brackets to (anti-)commutators. Now I was wondering how much of Poisson bracket structure is preserved. For example for a classical (continuous) system we have $$ \lbrace \phi(z), f(\Pi(y)) \rbrace = \frac{\delta...

Proving Some Poisson Bracket identities -- a notational question I need some help just understanding notation, and while this might count as elementary it has to do with Hamiltonians and Lagrangians, so I posted this here. Homework Statement Prove the following properties of Poisson's...

Homework Statement Calculate the magnitude of the force supported by the point at A under the action of the 2.5-kN load applied to the bracket. Neglect friction in the slot. AB = .17 m B to force = .16 m angle between vertical and BtoForce = 24The Attempt at a Solution I found the x and y...

In order to understand how related are the theories of General Relativity and Electromagnetism, I am looking at the electric and magnetic parts of the Weyl tensor (in the ADM formalism) and compare them with the ones from Maxwell's theory. I have tried to look at the Poisson bracket, but the...

Hi to all, I'm new to the forum. I'm designing a bracket for a front caliper of a motorcycle. I'm learning to use solidworks just to design a bracket that will be safe to use in the real world :-).. but I'm stuck on calculating the force that will be generated on the bracket (or better...

Homework Statement The Lie bracket of the fundamental vector fields of two Lie algebra elements is the fundamental vector field of the Lie bracket of the two elements: [\sigma(X),\sigma(Y)]=\sigma([X,Y]) Homework Equations Let \mathcal{G} a Lie algebra, the fundamental vector field of an...

Hi all, Can anyone help me with (what I think is) an equation relating to moment. I want to know what the point load will be on a wall with a TV screen and articulated bracket. The screen weighs 25kg The bracket weighs 20kg The bracket puts the screen out at 700mm when fully extended...

hello everyone. I am in desperate need of help from some mechanical design/materials engineers/students. I am an Electrical/Electronic Engineering student currently completing a project and require assistance on some mechanical design issues. I have searched numerous websites and books...

Hi, Suppose you have a collection of fields \phi^i (t,x) depending on time and on 1 space variable, for i=1,...,N. Its dynamics is defined by the Lagrangian L=\frac{1}{2} g_{ij}(\phi) (\dot{\phi}^i \dot{\phi}^j - \phi ' ^i \phi ' ^j ) + b_{ij}(\phi) \dot{\phi}^i \phi ' ^j where...

Is it okay to mix the types of brackets you use when writing out a solution to help make it a bit clearer? For example: If I was completeing the square with: $$ 3x^{2} + 5x -2 = 0 $$ I would factor out the 3 onto square brackets like this: 3 \left[x^{2} + \frac{5}{3}x - \frac{2}{3} \right]...

Homework Statement Consider the Klein-Gordan action. Show that the Noether charges of the Poincare Group generate the Poincare Algebra in the Poisson brackets. There will be 10 generators.Homework Equations \{ A,B \}=\frac{\delta A}{\delta \phi}\frac{\delta B}{\delta \pi}-\frac{\delta...

I have a cantilever bracket, fixed at one end, free at the other. One vertical element AB, 330mm long with a fixing at either end to a wall. One horizontal element CD, 375mm long. CD is fixed at 90deg to AB, 130mm from the top. A constant, downward load W=30kg, is applied to the end of CD...

Hi everyone, i came across this forum a while ago and have been lurking here for a while. Any ways, i have a simple problem but can't seem to find a solution to it the photo of the problem is the link just here http://i1053.photobucket.com/albums/s463/Popa911/1cc85138.jpg Homework...

Hi friends I am trying to drive constraints of a Lagrangian density by Dirac Hamiltonian method. But I encountered a problem with calculating one type of Poisson Bracket: {\varphi,\partial_x\pi}=? where \pi is conjugate momentum of \varphi. I do not know for this type Poisson Bracket I can...

Hello, If you have two observables f and g both of which start off as: f =0 and g =0 and you evaluate their possion bracket: {f,g}, will it necessarily be equal to zero? Also, if just f=0 and g wasn't zero, would {f,g} =0? Thanks!

In book http://www.phy.uct.ac.za/people/horowitz/Teaching/lecturenotes.pdf in section 2 it is described transition from Poisson bracket into Canonical Commutation Relations. But it is written The experimentally observed phenomenon of incompatible measurements suggests that position and...

Homework Statement I have to create a design to mount a steel (yield=250MPa) weight 100kg and thickness of 15 mm to aluminum plate (Yield = 100MPa) with a tap hole of 15mm in vertical direction with 6 of M6 screws. Homework Equations σ (bearing load) =F/A τ (shear load) = F/A The...

This question may not come in the bracket of quantum mechanics but here's the question- If most of the atom is empty space what gives the illusion of solidity?