Euler Definition and 387 Threads

import time start = time.perf_counter() def con(H): J = [] Q = [] W = [] for i in H: for j in range(len(i)-1): Cop = i[::] y = i[j]+i[j+1] del Cop[j:j+2] Cop.insert(j,y) J.append(Cop) for i in J: y...

For Initial Value problems I want to implement an ODE solver for implicit Euler method with adaptive time step and use step doubling to estimate error. I have found some reading stuff about adaptive time step and error estimation using step doubling but those are mostly related to RK methods. I...

This is a calculus of variations problem from Boas chapter 9. I seem to be misunderstanding something with differentiation. Given $$F=(1+yy')^2$$ then $$\frac {\partial F} {\partial y'}=2(1+yy')y$$ and $$\frac {\partial F} {\partial y}=2(1+yy')y' .$$ Now this one I am not so confident...

I have been studying the dynamics of free top from Morin's book. In his book when describing the dynamics,he writes down the equation of motion as shown in screenshot. However,I am not able to understand which term refers to which coordinate system. For eg: Here ##\omega## refers to angular...

I apologize in advance if any formatting is weird; this is my first time posting. If I am breaking any rules with the formatting or if I am not providing enough detail or if I am in the wrong sub-forum, please let me know. 1. Homework Statement Using Euler's formula : ejx = cos(x) + jsin(x)...

I'm trying to wrap my head around the concept. we use 3 rotations to transfer our regular cartesian coordinates (3 x,y,z unit vectors) to other 3 unit vectors. each rotation is associated with an angle. so far I'm good. but now I saw in Landau's and Lifshitz's "mechanics" book this thing...

This is not homework. I am reading a book: "The art of infinite: The Pleasure of Mathematics" and pages 119-120 give a proof of the Euler Line theoram: the circumcenter, centroild and orthocenter of a triangle are always colinear (see the attached files). 1. Homework Statement Page 119 shows...

Hi all: I really do not know what to ask here, so please be patient as I get a little too "spiritual" (for want of a better word). (This could be a stupid question...) I get this: eiθ=cosθ+isinθ And it is beautiful. I am struck by the fact that the trig functions manifest harmonic...

If the Euler equations have double roots as it's solution, second solution will be $y_2(x)=x^r\ln{x}$. what is its proof? or how it can be derived?

Hello Before I "phrase" my question (and that may be my problem), may I first state what I do know. I understand that a Rotation matrix (a member of SO(3)) has nine elements. I also understand that orthogonality imposes constraints, leaving only three free parameters (a sub-manifold) I also...

(I think I couldn't add the image) you can see my answer in link https://pasteboard.co/HPKZ6KD.jpg (Please first see my answer in the link) But in answer it is φ= Asin(kx) + Bcos(kx) I know that euler formula is eix = cosx +isinx But I can't get this answer can you help me?

Hello, I was struggling with solving a specific integral. I know that I can rewrite the exponential matrices and the range of the three Euler angles. However, I am not sure I should I write in terms those three Euler angles.

Homework Statement A mass m1 is located on a platform with mass M. The platfrom is located on springs with total constant k such that it can swing vertically in direction x. a) Write down the equations of motion assuming mass m1 will always be connected to the platform. Write it as x(t) b)...

Consider quadratic Diophantine equations of the form: x2 – Dy2 = 1 For example, when D=13, the minimal solution in x is 6492 – 13×1802 = 1. It can be assumed that there are no solutions in positive integers when D is square. By finding minimal solutions in x for D = {2, 3, 5, 6, 7}, we...

https://projecteuler.net/problem=60 I wrote a piece of code to solve the problem but it still takes too much time to solve it Any ideas how can I improve my code, like adjustments that I can make ? import itertools import time start = time.perf_counter() def prime(N): if N == 0 or N == 1...

Consider the following flow: x = (1+ct)x0. Let the density rho(t) = rho0/(1+ct) so that the flow conserves mass. Physically, this is just a bunch of fluid elements on the positive x0-axis each given initial velocities that are proportional to their initial positions. Each fluid element should...

Homework Statement https://projecteuler.net/problem=58 Homework EquationsThe Attempt at a Solution def prime(N): if N == 1: return False y = int(N**0.5) for i in range(2,y+1): if N%i == 0: return False return True def finder(N): L = len(N)...

Homework Statement By replacing the 1st digit of the 2-digit number *3, it turns out that six of the nine possible values: 13, 23, 43, 53, 73, and 83, are all prime. By replacing the 3rd and 4th digits of 56**3 with the same digit, this 5-digit number is the first example having seven primes...

Only recently started to understand Euler angles and rotation matrices, and I am reasonably comfortable with the concepts already posted here. I am pretty sure I am missing something obvious, but I cannot figure out the way to solve this problem: A body in 3D space with a orientation defined by...

to all members of the forum. In the attached image is equation numbered 3.23 which, by the application of Euler’s Identity - called De Moivre Theorem one line below - leads to equation 3.24. Above is an a textbook frought with errors - printing ones. I would be highly appreciative of a...

I'm trying to get a more intuitive understanding of Euler's identity, more specifically, what raising e to the power of i means and why additionally raising by an angle in radians rotates the real value into the imaginary plane. I understand you can derive Euler's formula from the cosx, sinx and...

So I am kind of lost... I don't really know how to ask this. Project Euler is a website that hosts multiple programming contests and I am interested in this problem https://projecteuler.net/problem=608 but my question isn't truly about this problem but a more solution. I know that the Divisor...

Hi, I'm trying to solve the following problem ##\max_{f(x)} \int_{f^{-1}(0)}^0 (kx- \int_0^x f(u)du) f'(x) dx##. I have only little experience with calculus of variations - the problem resembles something like ## I(x) = \int_0^1 F(t, x(t), x'(t),x''(t))dt## but I don't know about the...

I was trying to Extrapolate Eulers formula , after deriing the basic form I wanted to prove: ∂F/∂y - d(∂F/∂yx)/dx +d[SUP]2[/SUP](∂F/∂yxx)/dx2 = 0 Here is my attempt but I get different answers: J(y) = ∫abF(x,yx,y,yxx)dx δ(ε) = J(y+εη(x)) y = yt+εη(x) ∂y/∂ε = η(x) ∂yx/∂ε = η⋅(x)...

Say you have an un-damped harmonic oscillator (keep it simple) with a sine or cosine for the forcing function. We can exploit Euler's equation and solve for both possibilities (sine or cosine) at the same time. Then, once done, if the forcing function was cosine, we choose the real part as the...

I am using from the following Euler equations : $$\dfrac{\partial f}{\partial u^{i}}-\dfrac{\text{d}}{\text{d}s}\bigg(\dfrac{\partial f}{\partial u'^{i}}\bigg) =0$$ with function ##f## is equal to : $$f=g_{ij}\dfrac{\text{d}u^{i}}{\text{d}s}\dfrac{\text{d}u^{j}}{\text{d}s}$$ and we have...

Is calculating the Euler angles analitically possible? I am trying to obtain the angles to transform the body-fixed reference frame to the inertial reference frame. I can get them without problems with numerical methods. But I would to validate them analitically, if possible. I followed the...

Homework Statement Write an implicit Euler code to solve the system ##c'(x) = \epsilon c''(x)-kc(x)## subject to ##1-c(0)+\epsilon c'(0) = 0## and ##c'(1)=0##. Homework Equations Nothing out of the ordinary comes to mind. The Attempt at a Solution In the following code, there is central...

Homework Statement In the calculation in high temperatures of ##Z_{rot} = (\sum_{j=0}^\infty (2j+1)\exp{j(j+1)\theta_{rot}/T})^N##; they use Euler summation formula: $$\sum_{n=0}^\infty f(n) = \int_0^\infty f(x)dx+\frac{1}{2}f(0)-\frac{1}{12}f'(0)+\frac{1}{720}f^{(3)}(0)+\ldots$$ for ##f(x) =...

The Euler Characterist of the projective plane and sphere is given by V - E + F. V is vertices, E is edges, F is faces. A presentation of the projective plane is {a | aa} and a presentation of the sphere is {b | bb^1} Yet the Euler characteristic is 2 for the sphere and 2-n for the connected...

Homework Statement A mouse starts at the origin and runs up the y-axis with a speed a. At the same time, a cat running with speed b, starts at the point (c,0) and pursues the mouse. i. What is the path of the cat? ii. Assume a<b and solve for y(x). How far does the mouse run before being...

In the derivation given in Goldstein's book it is given I can't understand from where it comes. It's not at all trivial for me but it's presented as if it's trivial.

I've just started with project Euler. The problem I just finished is phrased as follows: "The prime factors of 13195 are 5, 7, 13 and 29. What is the largest prime factor of the number 600851475143 ?" The method i used was trial division. Here is my code in C: #include<stdio.h> int main() {...

Homework Statement 12.81[/B] A train traveling at 20m/s is brought to an emergency stop. During braking, the acceleration is a=-(7/4)+(t/16) m/s^2, where t is the time in seconds measured from when the brakes were applied. (a) Integrate the acceleration from t=0 to t=16s using Euler's method...

What is wrong with the simple localised geometric derivation of the Euler Lagrange equation. As opposed to the standard derivation that Lagrange provided. Sorry I haven't mastered writing mathematically using latex. I will have a look at this over the next few days. More clarification. I...

Homework Statement Homework EquationsThe Attempt at a Solution I attempted to use the formula zj = xj + iyj to substitute both z's. Further simplification gave me (x1 + x2)cosθ + (y2 - y1)sinθ or, Re(z2 + z1)cosθ + Im(z2 - z1)sinθ. Is this a valid answer? Or are there any other identities...

I am working on the integral representation of the Euler-Mascheroni constant and I can't seem to understand why the first of the two integrals is (1-exp(-u))lnu instead of just exp(-u)lnu. It is integrated over the interval from 1 to 0, as opposed to the second integral exp(-u)lnu which is...

Homework Statement Linear Algebra Problem: Solving for Euler between two ordered bases I've got a problem I need to solve, but I can't find a clean solution. Let me see if I can outline the problem somewhat clearly. Okay, all of this will be in 3D space. In this space, we can define some...

Homework Statement Solve the differential equation ##(2x+1)^2y'' + (4x+2)y' - 4y = x^2## Can someone verify whether my solution is correct? Homework EquationsThe Attempt at a Solution We perform the substitution ##t = \ln|2x+1|##. Then, ##e^t = |2x+1|## and ##x = \pm(e^t -1)/2## Without...

Hi, i have this problem..., giving two circle (example radius 1 = 500 units, radius 2 = 200 units, distance between centers = 275.73 units) find Euler Spiral (aka Cornu spiral, aka Clothoid https://en.wikipedia.org/wiki/Euler_spiral) tangent giving circle (unknown tangent points). For this...

Hello, Homework Statement I'm given the following exercise: "A rod with neglected thickness exists. What is the relation between the α,β angles to Euler angles of orientation? α is defined as the angle between the rod and its projection on the XY plane. β is defined as the angle between the...

I hope someone can explain this to me. In the definition of Euler angles, the body-fixed-azimuthal angel γ is measured from the nodal line that defines the intersection of the body-fixed-XY plane and the space fixed-xy plane to the body-fixed-Y axis. This is the green line in this image from...

I have to find a primitive function below using the Euler formulas for ##\sin x## and ## \cos x## The problem $$ \int e^{2x} \sin 3x \ dx $$ Relevant equations ## \cos x = \frac{e^{ix}+e^{-ix}}{2} \\ \sin x = \frac{e^{ix}-e^{-ix}}{2i} \\ \\ \int e^{ix} \ dx = \frac{e^{ix}}{i} ## The attempt...

Homework Statement For the following integral, find F and its partial derivatives and plug them into the Euler Lagrange equation $$F(y,x,x')=y\sqrt{1+x'^2}\\$$ Homework Equations Euler Lagrange equation : $$\frac{dF}{dx}-\frac{d}{dy}\frac{dF}{dx'}=0$$ The Attempt at a Solution...

Homework Statement I need to solve: x^2y''-4xy'+6y=x^3, x>0, y(1)=3, y'(1)=9 Homework EquationsThe Attempt at a Solution I know that the answer is: y=x^2+2x^3+x^3lnx Where did I go wrong. I was wondering if it's even logical to solve it as an Euler Cauchy and then use variation of parameters...

Hi So the problem I have is I want to get the equation of motion of a vibrating beam under a nonlinear "Point" force. The equation would be like this for a distributed load (Which is not the case) But I want the load to be at a point at x=L So I have to options. Add an impulse dirac function...

Well, that question just about states my issue. We have a body and we rotate about, say, the 3-axis of its body frame. Then, we must do the next rotation about the 1 or 2 axis. Let me say we choose the 1-axis Then we have a choice: continue on to the 2 axis or repeat the 3 axis. One set is...

So I want to read Euler's : Elements of Algebra. However, I see many editions on Amazon. What is the best version of this book?

Before I start, there are only really two pieces of information this concerns and that is the idea that 1x = 1 and that ei*π = -1 So it would follow that (ei*π)i = -1i And so that would mean that i2i = e-π which doesn't seem to be right at all. Where is the issue here as there must be one but I...

The well known Euler angles (αβγ) are defined as in the image It is easy to see that under inversion α → π+α β → π-β but I cannot figure out how γ transforms under inversion. actually I am stuck at the question whether I should measure it from the same intersection line ON (thence γ →π+γ) or...