Euler Definition and 387 Threads

If metric is $$ds^2 = -f(x)dt^2 + g(x)dx^2 + 2l(x)dxdt $$ Then we have this Lagrangian: $$L= \frac{1}{2}(-f(x)\dot{t}^2 + g(x)\dot{x}^2 + 2l(x)\dot{x}\dot{t}).$$ The Euler-Lagrange equation for $$t$$ is: since $$t$$ is not there in the Lagrangian then $$\partial L/ \partial t=0$$ This implies...

Homework Statement Write a program to simulate motion of simple harmonic oscillator. Initial conditions: Let ω = 1, x(t=0) = 1, v(t=0) = 0. Integrate over 30 seconds in intervals of 0.05s. Homework Equations δ2x / δt2 = -ω2x As set of 2 coupled ODE's; x' = v, v' = -w2x The...

"A rigid lamina (i.e. a two dimensional object) has principal moments of inertia about the centre of mass given by ##I_1=u^2-1##, ##I_2=u^2+1##, ##I_3=2u^2## Choose the initial angular velocity to be ##ω = µN \hat{e_1} + N \hat{e_2}##. Define tan α = ω2/ω1, which is the angle the component of ω...

I'm trying to solve a(x-x_0)y''+b(x-x_0)y'+cy-c=0 So I let $$y=(x-x_0)^m$$ then $$y'=m(x-x_0)^{m-1}$$ and $$y''=m(m-1)(x-x_0)^{m-2}$$ plugging in gives a(x-x_0)m(m-1)(x-x_0)^{m-2}+b(x-x_0)m(x-x_0)^{m-1}+c((x-x_0)^m-1)=0 now I want to find the values of m that make the equation 0, but factoring...

Does anyone know of a derivation or justification of Euler's substitution formulas for evaluating irrational expressions? In other words, to evaluate integrals of the form: \int R(x,\sqrt{ax^2+bx+c}) You can use Euler's substitutions: 1. \sqrt{ax^2+bx+c} = t \pm \sqrt{a}x, a>0 2...

Hi All, The Euler class of the tangent bundle of a compact, oriented manifold agrees with the evaluation of the top homology class on the fundamental class (which is represented by the manifold itself), and maybe also figure out how to do actual computations using Poincare duality (to figure out...

Homework Statement The system can pivot at point O and I am taking small angle approximations. I am trying to determine the Lagrangian, ##\mathcal{L} = T - U## for the following system: Homework Equations E-L equation with dissipation: ##\frac{\partial\mathcal{L}}{\partial q_i} -...

Hi, I'm having trouble with this one. Homework Statement Find a particular solution of the second-order homogeneous lineal differential equation x^2y'' + xy' - y = 0 taking in account that x = 0 is a regular singular point and performing a power series expansion. Homework...

Homework Statement Just like my title says, we are to prove the trig identity sin^2x+cos^2x=1 using the Euler identity. Homework Equations Euler - e^(ix) = cosx + isinx trig identity - sin^2x + cos^2x = 1 The Attempt at a Solution I tried solving the Euler for sinx and cosx...

Is momentum conserved? I am considering the Euler equations in conservative form and solving the Sod shock tube problem I have written a Godunov finite volume type solver. It solves for density ρ, momentum ρu, and total energy E; therefore, I would expect all of these quantities to be...

I like this book very much. Euler is a brilliant mathematician no doubt. He explains everything very well without holding back significant information with his exposition. However, I bump into an unfamiliar topic. I do believe it has something to do with advanced mathematics. I googled it, and I...

After running into the dilemma of having nothing to do a little while ago, I decided to try working on a Project Euler problem with a mathematical approach. Not being a mathematician, I soon found myself in a rut. Now, what I have so far is this: Let T(n) be the nth Triangle Number, n > 0...

Hey I don't understand this backward euler solution, in particular why the f(y,t+h) is equal to y(t+2)

I am trying to diagonalize the following matrix: M = \left( \begin{array}{cccc} 0 & 0 & 0 & a \\ 0 & 0 & -a & 0 \\ 0 & -a & 0 & -A \\ a & 0 & -A & 0 \end{array} \right) a and A are themselves 2x2 symmetric matrices: a = \left( \begin{array}{cc} a_{11} & a_{12}\\ a_{12} & a_{22}...

Homework Statement I would like to solve a 2nd Order Differential Equation using the Improved Euler Method. The 2nd ODE is a Mass-Spring-Damper equation. I tried coming up with an solution for the Improved Euler Method, but not entirely sure. Can you help me and have a look if this is correct...

I have a system with one generalized coordinate, x. In the potential energy part of the lagrangian, I have some constants multiplied by the absolute value of x. That is the only x dependence the lagrangian has, so when I take the partial derivative of the lagrangian with respect to x (to get the...

Find the fixed points of the implicit Euler scheme \begin{equation} y_{n+1}-y_{n}= hf(t_{n+1},y_{n+1}) \end{equation} when applied to the differential equation y'=y(1-y) and investigate their stability? => implicit Euler scheme \begin{equation} y_{n+1}-y_{n}= hf(t_{n+1},y_{n+1})...

Hi! I am having trouble following the derivation from Euler's Equation to Bernoulli's Equation. The trouble lies in the math, not the physics part. Especially the step when partial derivatives are being integrated. I have attached the relevant part as a screenshot. How does the...

This is a very well-known portrait of Leonhard Euler, which I'm sure most people have seen before, but does anyone know what the heck he's wearing on his head? http://upload.wikimedia.org/wikipedia/commons/d/d7/Leonhard_Euler.jpg It looks like someone threw a dirty old rag on him - does anyone...

Homework Statement The Euler equations for ideal compressible flow are given by \partial_t v + (v\cdot \nabla)v = g-\frac{1}{\rho}\nabla p \\ \partial_t \rho + \nabla \cdot(\rho v) = 0 In my book these are written in terms of the small-value expansions \rho = \rho_0 + \delta \rho, p = p_0 +...

Homework Statement Solve the following system for 0<t<5 u^\prime = u-e^{-2t} v, u(0) = 1 v^\prime = u+3v, v(0) = -2 using Forward Euler method and implement the numerical scheme into a MATLAB code. Homework Equations Forward Euler : \vec x^(\prime)_{n+1} = \vec F(t,\vec x)...

I recently was looking for programming challenges and it was suggested to me that I check out Project Euler. http://projecteuler.net/problems These all seem straight-forward to me. Or there's something I'm not understanding about them. For instance, problem 1: Is there supposed to be...

Homework Statement The problem is attached as TheProblemAndSolution.png, and everything is typewritten, so it should be easily legible (but you will likely need to zoom into read the text since the image's height is significantly larger than its width). Homework Equations Differential...

The Euler-Maclaurin summation formula and the Riemann zeta function The Euler-Maclaurin summation formula states that if $f(x)$ has $(2p+1)$ continuous derivatives on the interval $[m,n]$ (where $m$ and $n$ are natural numbers), then $$ \sum_{k=m}^{n-1} f(k) = \int_{m}^{n} f(x) \ dx -...

I am trying to understand an example from my textbook "applied finite element analysis" and in the variational calculus, Euler lagrange equation example I can't seem to understand the following derivation in one of its examples ∫((dT/dx)(d(δT)/dx))dx= ∫((dT/dx)δ(dT/dx))dx= ∫((1/2)δ(dT/dx)^2)dx...

Hi everybody, I am programming a new code for a problem. The problem is numerically solving the Simple Harmonic Motion using the Euler method. This approach is just an approximate solution and not a exact solution, however when I run the code successfully and plot my data, it comes up as an...

What is most motivating way of introducing this function? Does it in itself have any real life applications that have an impact. I can only think of a^phi(n)=1 (mod n) which is powerful result but is this function used elsewhere.

What is most motivating and tangible way of introducing this function? Does it in itself have any real life applications that have an impact. I can only think of a^phi(n)=1 (mod n) which is powerful result but is this function used elsewhere.

I found what looks like an interesting document on a solution to the Basel Problem by Euler. http://eulerarchive.maa.org/pages/E063.html What stinks is that the document is in German and French(I think) so I can't read it apart from the math. I can't seem to find a translated copy of this...

I am trying to run a program with fortran. The program is about solving the Oscillator using Euler Method. I am trying to run this code and applying array arguments (as I want to extend it to 3 dimensions afterwards). When I try to compile, it comes up with an error "Unclassifiable statement at...

Prove the following Euler sum $$\sum_{k\geq 1}\left(1+\frac{1}{3}+\cdots +\frac{1}{2k-1} \right) \frac{x^{2k}}{k}=\frac{1}{4}\ln^2\left( \frac{1+x}{1-x}\right)$$

Homework Statement Use the Euler method with h=0.05 to find approximate values of y'=3+t-y, y(0)=1 at t=0.1, 0.2, 0.3, and 0.4. Homework Equations I don't even know what formula and how to use it. The Attempt at a Solution The answers are 1.1975, 1.38549, 1.56491, 1.73658.

Given \(F = A(x)u_1^{'2} + B(x)u'_1u'_2 + C(x)u_2^{'2}\). \[ \frac{\partial F}{\partial u_i} - \frac{d}{dx}\left[\frac{\partial F}{\partial u_i'}\right] = 0 \] From the E-L equations, I found \begin{align*} \frac{d}{dx}\left[2Au_1' + Bu_2'\right] &= 0\\ \frac{d}{dx}\left[2Cu_2' + Bu_1'\right] &=...

Given this \(F = p(x)y^{'2}-q(x)y^2+2f(x)y\). What would be the integral of \(f(x)\) and \(q(x)\)? \begin{align*} f(x) - q(x)y - \frac{d}{dx}\left[p(x)y'\right] &= 0\\ \frac{d}{dx}\left[p(x)y'\right] &= f(x) - q(x)y\\ y'p(x) &= \int f(x)dx - y\int q(x)dx \end{align*}

I'm doing a research project currently and basically what I have is a camera measuring a probe. I have designed the camera to give the orientation of the probe using euler angles in the camera's frame of reference. This was working for most of my data, but now I need a 3-D visualization of what...

Hi... first post here! Sorry if not in the right place. I am trying to decode the parameters for an xml file format and I would appreciate help in interpreting some parameters. I know the thing specified is a "transition curve" or clothoid curve, as the transition between a straight path and...

Hello I need to plot this simple system: x'' = -x using midpoint Euler. u1 = -x , u2 = -x' u1' = u2 u2' = -x u1(n+1) = u1(n) + h*? u2(n+1) = u2(n) + h*f((1/2)*(u1(n) + u1(n+1)) We don't know u1(n+1). I tried approximating it with u1(n+1) = u1(n) + h*u2(n) u2(1+i) =...

I apologize upfront, as I have no experience with math to begin with. However, I have a real life problem I am trying to figure out here at work and I would appreciate some help. Here is my situation: Homework Statement In a 3 dimensional plane, I have an artillery piece pointing at...

I didn't understand the algorithm explained in my textbook. ("Introduction to combinatorics" P.165) I would an alternative explanation with an example. Here is the algorithm: (i) from a new vertex, any edge may be taken; (ii) from an old vertex, if the edge just traversed was nes, then turn...

Create the Euler-Lagrange equation for the following questions (if it's necessary change the variables). Homework Statement $$\tag{1}\int _{y_{1}}^{y2}\dfrac {x^{'}{2}} {\sqrt {x^{'}{2}+x^{2}}}dy$$ $$\tag{2}\int _{x_{1}}^{x_{2}}y^{3/2}ds $$ $$\tag{3} \int \dfrac {y.y'} {1+yy{'}}dx...

Homework Statement Do the Euler-Lagrange equations set to zero for each of the 3 orthogonal coordinates or do you sum them all equal to zero. Do the coordinates have to be orthogonal in order to write separate E-L equations? Or is there no such thing as non-orthogonal coordinates to analyze a...

Hi! I want to write a code in Matlab for the Backward Euler Method for 2x2 systems, using the fixed point iteration to find the yn+1. y1n+1=y1n+h*f(tn+1,y1n+1,y2n+1) (1) y2n+1=y2n+h*g(tn+1,y1n+1,y2n+1) (2) Could you tell how I use the fixed point iteration?? At (1) the fixed point iteration...

I'm modeling a single 3D rigid body in preparation for some more complicated modeling in order to gain a better understanding of Euler angles, the angular velocity vector and the rotating coordinate system. The body is rotated in inertial frame by an intrinsic ZXZ rotation, with respective...

We all know that the Euler characteristic is a topological invariant. But let's suppose that we don't know this or anything else about algebraic topology for that matter. We are given only the Gauss-Bonnet theorem, which expresses the Euler characteristic in geometrical terms. In his string...

Hi I wanted to know for which cases the Euler Lagrange equations are applicable? 1.) Imagine that we have a kinetic Energie T(q,q') and a potential that also depends on velocity V(q,q'). As far as i know the Euler Lagrange equations for a particle still hold in this case, is that true...

My question is relatively breif: is it true that \displaystyle \lim_{n \rightarrow \infty}(\varphi(n))=\lim_{n \rightarrow \infty}(n) \cdot \prod_{i=1}^{\infty}(1-\frac{1}{p_i}) Where p is prime? Pehaps \varphi(n) is too discontinuous to take the limit of, but it would seem that as it increases...

prove that there is no positve integer n such that g(n) dividies n/5, where g is the euler totient function.

Hi. Since these equations are approaching three hundred years old I'm pretty sure someone must have solved them somewhere before. However I have not been able to find any text-books or papers that actually show me how to solve these equations. So I'm wondering if anyone here know where I can...

Homework Statement Consider the function f(y,y',x) = 2yy' + 3x2y where y(x) = 3x4 - 2x +1. Compute ∂f/∂x and df/dx. Write both solutions of the variable x only. Homework Equations Euler Equation: ∂f/∂y - d/dx * ∂f/∂y' = 0 The Attempt at a Solution Would I first just find...

I am trying to solve the following pde numerically using backward f.d. for time and central difference approximation for x, in MATLAB but i can't get correct results. \frac{\partial u}{\partial t}=\alpha\frac{\partial^{2}u}{\partial x^{2}},\qquad u(x,0)=f(x),\qquad u_{x}(0,t)=0,\qquad...