generalized Definition and 217 Threads

Yes, that is a serious title for the thread. Could someone please define GENERALIZED COORDINATES? In other words (and with a thread title like that, I damn well better be sure there are other words ) I understand variational methods, Lagrange, Hamilton, (and all that). I understand the...

(Note that the title of this thread might be incorrect - I'm just drawing on the vocabulary people use when discussing Lagrangian Mechanics...) Hi, I'm trying to set up a coordinate system to represent points in space where one of the coordinates is the distance along a parametric curve, one is...

It just seems odd that the force on an object goes based on the second time derivative of its position vector. I understand that this is what is observed through experiment, but is this only for certain types of situations? Is the acceleration only some kind of low-order approximation of a...

Hello, I am trying to generalize the work equation and understand the very definition of it. From what I understand, Work is the energy required to displace an object with a force in the direction of the displacement. (also the change in kinetic energy but I'm not going to worry about that yet)...

Homework Statement Lecturer says that: Generalized Coordinate: θ = q1 Generalized Force: Q1 = 0 I don't understand why Q1 equals to zero. Homework Equations Q_i = \sum_j \underline F_j \frac{\delta \underline r_j}{\delta q_i} The Attempt at a Solution First i need to find partial...

Hi, i now studying vector calculus, and for sheer curiosity i would like know if there exist a direct fashion to generalize the rotor operator, to more than 3 dimensions! On wiki there exist a voice https://en.wikipedia.org/wiki/Curl_(mathematics)#Generalizations , but I do not know how you...

I have seen both rk and qj both used to represent generalized coordinates in the Lagrange equations. Are these both the same things? Does it matter which you use? Thanks!

Homework Statement a ladder of length L and mass m is leaning against a wall as shown in the figure below. assuming the wall and the floor are frictionless, the ladder will slide down the floor and until the left end loses contact with the wall. Before the ladder loses contact with the wall...

If suppose only if the velocities are determined for all N particles can the system be completely determined, can we not extend and say that only if acceleration for all particles are provided can the system be completely determined? For instance can there not be two systems of N particles with...

How is the generalized triangle inequality in b-metric spaces ? I find something...But I wonder your opinion...Thank you for your attention... Especially if you write for n,m>0 m>n $d({x}_{n},{x}_{m})$$\le$..... I will be happy...

Not sure if this will be of interest to others, but, as an exercise, I decided to derive formulas for SR velocity addition for any angular relationship, and similar aberration of angle for any object speed and direction and observer relative velocity - using pure algebra/geometry. That is, no...

Homework Statement Identify the intervals of increase/decrease of ##f(x) = \sin x + \cos x## Homework Equations ##f(x) = \sin x + \cos x## ##f'(x) = \cos x - \sin x = \sqrt 2 \cos(x+\frac {\pi}{4})## The Attempt at a Solution ##f## is increasing when ##f'(x) > 0## ##\sqrt 2 \cos(x+\frac...

Homework Statement Solve ∂v/∂θ and ∂v/∂r. (refer to attached image for equations) Homework Equations Refer to attached image. note that the velocity is expressed in cylindrical coordinates and attention must be paid to the directional unit vectors eθ and eρ.[/B] The Attempt at a Solution...

Homework Statement Prelim: my question is about a very specific part of a question whereby the student is asked to derive the final formula for the general solution in two vars, but I will post the entire question for clarify. Newton's Method for approximating the roots of an equation f(x)=0...

In Lagrangian Dynamics, I assume that generalized forces of constraint are applied at the location of the corresponding generalized coordinate. I don't recall seeing anything explicit about the point of application in the text.

I can start explaining the problem but a more quicker way would be to open this link: http://onlinelibrary.wiley.com/doi/10.1002/9783527627486.app2/pdf and check the paragraph resulting in expression (B.5). Note that I don't really care about the kinetic energy they talk about in this link...

Homework Statement Evaluate the integrals in the attached image Homework EquationsThe Attempt at a Solution

The original Dirac Equation was for the electron, a particle of spin 1/2. Is there a "Generalized Dirac Equation" that has been experimentally proven to work for all fermions, not just those of spin 1/2? Thanks in advance.

Hello! (Wave) I am looking at the proposition: If $(A_n)_{n \in \omega}$ is a sequence of sets and $(f_n)_{n \in \omega}$ is a sequence of functions then: for all $n \in \omega, f_n: \omega \overset{\text{ surjective }}{\rightarrow} A_n$ then there is a function $f: \omega \overset{\text{...

Let $a_i \in \mathbb R^n$ with $a_i = (a_{i}^j)_{j = 1 ... n} = (a_{i}^1, ... ,a_{i}^n)$ for $i = 1, ... , k$ and let $p_1,...,p_k \in \mathbb R_{>1}$ with $\frac1{p_1}+ ... + \frac1{p_k} = 1$ Then show the following inequality by assuming that there are for every $i = 1, ... ,k$ one $N \in...

I am looking for more information (e.g., reference, the CDF and descriptive stats) about a four-parameter skewed generalized Gaussian (SGG) distribution. I have come across the PDF for this distribution, but with no reference and not a lot of other information. Here is a snippet... On...

This is problem 20b from chapter I 4.10 of Apostol's Calculus I. The geometric mean G of n positive real numbers x_1,\ldots, x_n is defined by the formula G=(x_1x_2\ldots x_n)^{1/n}. Let p and q be integers, q<0<p. From part (a) deduce that M_q<G<M_p when x_1,x_2,\ldots, x_n are not all...

It is a nonsense use the generalized Stokes' theorem in right side of Faraday's Law? we know this is true...\displaystyle \oint_{\partial \Sigma}\vec{E}\cdot dl=-\int_{\Sigma} \frac{d\vec{B}}{dt}\cdot d\vec{A}\Rightarrow \int_{\Sigma}\vec{\nabla}\times\vec{E}\cdot d\vec{A}=-\int_{\Sigma}...

The problem states that a particle moves in a plane under the influence of the following central force: F = \frac{1}{r^2}\Big(1 - \frac{\dot{r}^2 - 2\ddot{r}r}{c^2}\Big) and I am asked to find the generalized potential that results in such a force. Goldstein gives the following equation...

When you have a general coordinate chart on spacetime you have a lot of freedom to pick your coordinates, but you are always going to have 4 coordinates and each 4-tuple uniquely (in that chart) identifies one event in the manifold. When you are choosing generalized coordinates for a...

Just over two years ago, I was introduced to the process of completing the square as a way to solve the roots in a quadratic equation. More recently, I've thought about how I could go about extending this to completing the cube. The following short story/proof is the result of during just that...

Homework Statement Solve the system: ##x' = 5x - y## ##y' = 4x + y## Homework Equations ##t## is transpose. The Attempt at a Solution I'm a bit rusty with these and I had a small question. I put the system into the form ##x' = Ax## and proceeded to solve for the...

I red griffiths many times but even now there is something I can't understand. It's about statistical interpretation. In his book chapter 3.4 he says "If you measure an observable Q(x,p) on a particle in the state ψ(x,t), you are certain to get one of the eigenvalues of the hermitian operator...

Hi all, Generally boundary condition (Dirichlet and Neumann) are applied on the Load Vector, in FEM formulation. The equation i solved, is Generalized eigenvalue equation for Scalar Helmholtz equation in homogeneous wave guide with perfectly conducting wall ( Kψ =λMψ ), and found, doesn't...

What do you think, might be generalized the helix in the manner that I propose in the attached material?

Homework Statement see the pictures Homework EquationsThe Attempt at a Solution one. In the third picture. 3rd line and 4th line. i don't understand why 1-e^(-t^2/n^2) become (1+t^2)/n^2 two. I'm not familiar with proving equality by showing difference going to zero. for example, I prove...

This equation (see attachment) appears in one of Prof. Susskinds's lectures on Quantum Mechanics: in trying to differentiate the coefficients of the eigenvectors of a wave function with respect to time, an exponential e^(-iEt) is introduced for alpha. I can see that d/dt e^(-iEt) = -iE...

Recall that the fibonacci sequence is defined as { f0=0; f1 = 1 and {fn = f n - 1 + fn -2 for n 2 Prove by generalized mathematical induction that fn = 1/sqrt(5)[ϕn - (-ϕ)-n] where ϕ = [1+sqrt(5)]/2 is the golden ratio.. (This is known as de Moivre's formula.) So...

I have been researching on a generalization of Erdos-Moser, which asks for ordered tuple of consecutive integers with first $n-1$ integers, summed and exponentiation by $n$, equals the $n$-th power of the last and the greatest. The generalization can be observed as $$3^2 + 4^2 = 5^2$$ $$3^3 +...

This thread is dedicated to the study of Log-Trig series of the form: \mathscr{S}_{(m, n)} (z) = \sum_{k=1}^{\infty}\frac{(\log k)^m}{k^n}\, \sin 2\pi k z\mathscr{C}_{(m, n)} (z) = \sum_{k=1}^{\infty}\frac{(\log k)^m}{k^n}\, \cos 2\pi k zWhere m, n \in \mathbb{Z} \ge 1, and 0 < z < 1 \in...

In this tutorial we will be exploring the Nielsen Polylogarithm, and it's close relative, the Generalized Logsine Integral. Both of these functions have numerous applications in maths and physics, not least of all in Quantum Theory. The main purpose here, however, will be employing them in the...

This is NOT a tutorial, so any and all contributions are very much welcome... :DI've recently been working on the Barnes' function - see tutorial in Math Notes board - and been trying to generalize some of my results to higher order Barnes' functions (intimately connected with the Multiple Gamma...

This thread will be dedicated to find a general formula for the integral I(a,t) = \int^t_0 x \log|\sin(a x )| \, dx \,\,\,\,\, a,t>0 This is not a tutorial . Any comments or attempts are always be welcomed .

Homework Statement Hi we have Lorentz operators J^{\mu\nu} = i(x^{\mu}\partial^{\nu} - x^{\nu}\partial^{\mu}) and these have [J^{\mu\nu}, J^{\rho\sigma}] = i(\eta^{\nu\rho}J^{\mu\sigma} + \eta^{\mu\sigma}J^{\nu\rho} - \eta^{\mu\rho}J^{\nu\sigma} - \eta^{\nu\sigma}J^{\mu\rho}) Now define...

Let A be an 3x3 matrix such that A\mathbf{v_1}=\mathbf{v_1}+\mathbf{v_2}, A\mathbf{v_2}=\mathbf{v_2}+\mathbf{v_3}, A\mathbf{v_3}=\mathbf{v_3} where \mathbf{v_3} \neq \mathbb{0}. Let B=S^{-1}AS where S is another 3x3 matrix. (i) Find the general solution of \dot{\mathbf{x}}=B\mathbf{x}. (ii)...

This is NOT a tutorial, so by all means, if you've a mind to, the please DO very much feel free to contribute...Preamble:As a consequence of various families of definite integrals I've been studying recently, I've been led to consider what I've come to call the q-shifted Inverse Tangent Integral...

I've recently been working on a number of integrals related to the loggamma function, so I thought I'd share my results here. I'll have to post as and when I have time, and there will be a fair bit of preliminary work before we get to the final results, but - loosely speaking - the main aim here...

In this thread we consider the integrals of the form \beta(a,b,c;s) = \int^1_0 \frac{1}{(1-x)^a(s-x)^b x^c}\, dx \,\,\,\,\,\,-1<a,b,c<1 \,\,\,s\geq 0 This is NOT a tutorial , all suggestions are encouraged.

Homework Statement The generalized Green function is $$ G_g(x, x') = \sum_{n\neq m}\frac{u_n(x)u_n(x')}{k_m^2 - k_n^2}. $$ Show G_g satisfies the equation $$ (\mathcal{L} + k_m^2)G_g(x, x') = \delta(x - x') - u_m(x)u_m(x') $$ where \delta(x - x') = \frac{2}{\ell}\sum_{n =...

The generalized Green function is \[ G_g(x, x') = \sum_{n\neq m}\frac{u_n(x)u_n(x')}{k_m^2 - k_n^2}. \] Show \(G_g\) satisfies the equation \[...

Hello, I am currently reading about the topic alluded to in the topic of this thread. In Taylor's Classical Mechanics, the author appears to be making a requirement about any arbitrary coordinate system you employ in solving some particular problem. He says, "Instead of the Cartesian...

I've read an article in which a more general version of gauss-codazzi equations are presented namely a version where one has a connection $$\nabla_X Y = '\nabla_X Y + B(X,Y)u$$ where ##'\nabla## is the projection of (the symmetric and metric compatible connection)##\nabla## orthogonal to...

Homework Statement I need some help understanding a derivation in a textbook. It involves the Lagrangian in generalized coordinates. Homework Equations The text states that generalized coordinates {q_1, ..., q_3N} are related to original Cartesian coordinates q_\alpha = f_\alpha(\mathbf r_1...

Homework Statement A spin-half particle is in a known eigenstate of Sz. Show that the product <S^2_x> < S^2_y> is consistent with the Uncertainty principle Homework Equations The Attempt at a Solution I know that the generalized uncertainty principle gives ΔS_x ΔS_y ≥ |<[S_x...

There is a lot of information on the web about how to calculate the probability that, in an arbitrarily-sized group, 2 people will share a birthday. However, I am trying to determine the probability that a larger number of people are born on a specific day (e.g., a group of people have a...