Lagrange Definition and 542 Threads

Homework Statement A ray of light enters a glass block of refractive index n and thickness d with angle of incidence θ1. Part of the ray refracts at some angle θ2 such that Snell's law is obeyed, and the rest undergoes specular reflection. The refracted ray reflects off the bottom of the block...

Homework Statement Two masses move in a plane restricted to concentric circles with radii R1 and R2. They are joined by a solid rod of length B. Use Lagrange first order equations to find the equilibrium point Homework Equations Constraint due to the solid bar: B = R12 + R22 -2R1R2cos(θ1...

Hi, To calculate the intersection of two straight lines the cross product of the line vectors can be used, i.e. when the lines start in points p and q, and have direction vectors r and s, then if the cross product r x s is nonzero, the intersection point is q+us, and can be found from...

Homework Statement Lagrange multipliers to find the maximum and minimum values of f(x,y) = 4x^3 + y^2 subject to the constraint 2x^2 + y^2 = 1. Find points of these extremum. Homework Equations The Attempt at a Solution g(x,y)= 2x^2 + y^2 - 1 f(x,y)= 4x^3 + y^2 Gradient F=...

Here is the question: I have posted a link there to this topic so the OP can view my work.

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...

I'm stuck on this problem for the past hour. I've tried solving for all variables and none of the steps I'm doing are getting me to the right answer. Find the point on sphere x^2+y^2+z^2=25 farthest from point (1,1,-1). My steps: http://i.imgur.com/c5kUj9g.png Correct Answer: x=...

Homework Statement A particle of mass m moves over the cylindrical surface of radius R. The particle is connected to the origin O, located on the central point of the cylindrical surface axis, by a spring with spring constant k and length R. Ignore force of gravity. a) State how many...

Homework Statement Show that the largest rectangular box having a fixed surface area must be a cube. Homework Equations ##V(x,y,z) = xyz## ##\sigma(x,y,z) = 2(xy + yz + zx) = C \in \mathbf{R}## The Attempt at a Solution As of this assignment, we haven't yet learned Lagrange...

I was checking the proof of this, when things came vague at one point. It goes as follows, how to prove that Lagrange's equations hold in any coordinate system? Answer: Let q_{a} = q_{a}(x_{1},..., x_{3N}, t) here the possibility of using a coordinate system that changes with time is...

Find the minimum value of \(\int_0^1y^{'2}dx\) subject to the conditions \(y(0) = y(1) = 0\) and \(\int_0^1y^2dx = 1\). Let \(f = y^{'2}\) and \(h = y^2\). Then \begin{align*} G[y(x)] &= \int_0^1[f - \lambda h]dx\\ &= \int_0^1\left[y^{'2} - \lambda y^2\right]dx \end{align*}...

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*}

Homework Statement The problem asks to design a cantilever beam of a minimum weight consisting of 2 steps. Given: total length (L), Force (F) at the end of the beam and allowable stress (σ) Need to find the diameters D and d, the length of the smaller shoulder of the beam (x)...

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...

If you have a craft parked at the Phobos-Mars Lagrange point, it's stable. I get that. But, what if you move, horizontally with respect to Mars? I assume you're no longer balanced out by Phobos' gravity, and will therefore fall towards Mars and end up in an elliptical orbit? If so, how far would...

Homework Statement We have a three mass two strings system with: m_1 string M string m_2 The end masses are not attached to anything but the springs, the system is at rest, and k is equal for both strings and m_1 and m_2 are equal. The distance between to m_1 and m_2, on both sides of M...

f(x,y)=x^2y with the constraint of x^2+2y^2=6 Use lagrange multipliers to find the extrema. Thanks!

Here is the question: Here is a link to the question: Please help with lagrange multipliers? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.

Hi we covered the Lagrange multiplier method in Lagrangian Mechanics and as far as I know, is the physical meaning behind this to be able to solve either some non-holonomic constraints or to get some information about the constraint forces. my problem is, i do not know the physical meaning of...

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...

Lagrange Multiplier----to find out the dimensions when metal used min. Homework Statement I have a rectangular tank with a capacity of 1.0m^3. The tank is closed and the cover is made of metal half as thick as the sides and base. Find the dimensions of the tank for the total amount of metal...

Homework Statement Consider a bead of mass m sliding without friction on a wire that is bent in the shape of a parabola and is being spun with constant angular velocity ω about its vertical axis. Use cylindrical polar coordinates and let the equation of the parabola be ##z = kρ^{2}##. Write...

Homework Statement Maximize C_{t} for any given expenditure level \int_{0}^{1}P_{t}(i)C_{t}(i)di\equiv Z_{t} The Attempt at a Solution The Lagrangian is given by: L = \left(\int_{0}^{1}C_{t}(i)^{1-(1/\varepsilon)}di\right)^{\varepsilon/(\varepsilon-1)} - \lambda...

Homework Statement This is not a homework problem but I would like to clarify my concern. It is stated that a function can be written as such: f(x) = \lim_{n \rightarrow ∞} \sum^{∞}_{k=0} f^{(k)} \frac{(x-x_{0})^k}{k!} R_{n}=\int^{x}_{x_{0}} f^{(n+1)} (t) \frac{(x-t)^n}{n!} dt They...

Homework Statement Find max/min of f subject to constraint: x^2+y^2+z^1 = 1 Homework Equations f(x,y,z) = 1/4*x^2 + 1/9*y^2 + z^2 g(x,y,z) = x^2 + y^2 + z^2 - 1 The Attempt at a Solution L = 1/4*x^2 + 1/9*y^2 + z^2 - λ(x^2 + y^2 + z^2 - 1) Lx = 2/4*x - λ*x*2 Ly = 2/9*y -...

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...

Here is the question: Here is a link to the question: Calc 3 Lagrange multiplier question? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.

Folks, I am puzzled how the linear interpolation functions (see attached) were determined based on the following equation below ##\displaystyle...

Homework Statement Hi guys. http://img189.imageshack.us/img189/5123/systemn.jpg The image shows the situation. A pointlike particle of mass m is free to move without friction along a horizontal line. It is connected to a spring of constant k, which is connected to the origin O. A...

Problem stated: Let \(a_1, a_2, ... , a_n\) be \(n\) positive numbers. Find the maximum of $$\sum_{i=1}^{n}a_ix_i$$ subject to the constraint $$\sum_{i=1}^{n}x_i^2=1$$. I honestly have not much of an idea of how to go about solving this. If I use lagrange multipliers which I think I am supposed...

The problem given is to find the local extreme values of \(f(x,y)=x^2y\) on the line \(x+y=3\). I went through the system of equations with the partial derivatives of \(x\), \(y\), and \(\lambda\), and found two extreme points \((0,3)\) and \((2,1)\). Plugging that into the original function I...

Homework Statement Consider the intersection of the elliptic paraboloid Z = X2+4Y2 , and the cylinder X2+Y2= 1. Use Lagrange multipliers to find the highest, and lowest points on the curve of intersection.Homework Equations The gradient equations of both functions.The Attempt at a Solution I...

Hello, I have been trying to follow the start of these lecture notes I found online and I have having trouble understanding what is happening between two steps. The notes I am looking at are located: http://pillowlab.cps.utexas.edu/teaching/CompNeuro10/slides/slides16_EntropyMethods.pdf...

Homework Statement Minimize f(X, Y, Z) = 2XY + 6YZ + 8XZ subject to the constraint XYZ = 12. Homework Equations The gradients of the equations, and XYZ = 12. The Attempt at a Solution I have the gradients for both of the equations. ∇f = <2Y + 8Z, 2X + 6Z, 6Y + 8X> ∇g = <...

Homework Statement Here is the problem, the solution and my question (in red): I'm guessing it was rejected because for the volume function, the dimensions cannot be negative? What if it was not volume and instead was just an arbitrary function. In that case you would not reject...

I am not sure how to do this one. Nothing I try goes anywhere. Consider the two-body equation of motion in vector form $$ \ddot{\mathbf{r}} = -\mu\frac{\mathbf{r}}{r^3}. $$ Show that the $f$ and $g$ functions defined by $$ \mathbf{r} = f\mathbf{r}_0 + g\mathbf{v}_0 $$ satisfy $$ \ddot{f} =...

I think I solved it a week ago, but I didn't write down all the things and I want to be sure of doing the things right, plus the excersise of writing it here in latex helps me a loot (I wrote about 3 threads and didn't submited it because writing it here clarified me enough to find the answer...

Advanced Calculus of Several Variables, Edwards, problem II.4.1: Find the shortest distance from the point (1, 0) to a point of the parabola y^{2} = 4x. This is the Lagrange multipliers chapter. There might be another way to solve this, but the only way I'm interested in right now is the...

Homework Statement Find the geodesics on a sphere g(x,y,z)=x^{2}+y^{2}+z^{2}-1=0 arclength element ds=\sqrt{dx^{2}+dy^{2}+dz^{2}} Homework Equations f(x,y,z)=\sqrt{x'^{2}+y'^{2}+z'^{2}} where x'^{2} \text{means} \frac{dx^{2}}{ds^{2}} and not d^{2}x/ds^{2} The Attempt at a...

Given the equations $$ rv\cos\gamma - h = 0,\quad \frac{v^2}{2} - \frac{\mu}{r} + \frac{\mu}{2a} = 0 $$ I want to maximize gamma. Do I have to solve for gamma in the first equation to use the method of Lagrange multipliers, or if not, how would I do this in the current form?

Homework Statement Find extrema for f\left( x,y,z \right) ={ x }^{ 3 }+{ y }^{ 3 }+{ z }^{ 3 } under the constraint g\left( x,y,z \right) ={ x }^{ 2 }+{ y }^{ 2 }+{ z }^{ 2 }=16Homework Equations (1) \nabla f=\lambda \nabla g (2) g\left( x,y,z \right) ={ x }^{ 2 }+{ y }^{ 2 }+{ z }^{ 2 }=16...

Hi, Homework Statement I am trying to limit Lagrange's remainder on taylor expansion of ln(4/5) to be ≤ 1/1000. Homework Equations The Attempt at a Solution I have tried using both ln(1+x), where x=-1/5 and x0(the center)=0, and ln(x), where x=4/5 and x0=1. Every time I keep...

Homework Statement Being a>0 and f:[a,b]--->R continuos and differentiable in (a,b), show that there exists a t ##\in## (a,b) such that: ## \frac{bf(a)-af(b)}{b-a}=f(t)-tf'(f)## The Attempt at a Solution For lagrange's theorem, we have: ## \frac{f(a)-f(b)}{b-a}= -f'(t) ## thought i could...

My first post and I am new to forum etiquette, please go easy on me. I have an exam in a few days and have a good grasp on the content. Historically the professor has been putting harder Lagrange questions on the exam than are available in my textbook and supplement books or that I can find...

Speaking of theorems, I have another question. I need to show, using Lagrange's theorem, that: 1.71<\sqrt{3}<1.75 By Lagrange's theorem I mean the one of: f ' (c)=(f(b)-f(a)) / (b-a) thanks !

Homework Statement Homework Equations Lagrance Multipliers. The Attempt at a Solution This is a pretty dumb question, and I feel a little embarassed asking but.. I know how to do the Lagrange part (I think). I'm assuming you maximize/minimize the distance, \sqrt{x^{2} +...

Homework Statement Use Lagrange multipliers to find the eigenvalues and eigenvectors of the matrix A=\begin{bmatrix}2 & 4\\4 & 8\end{bmatrix} Homework Equations ... The Attempt at a Solution The book deals with this as an exercise. From what I understand, it says to consider...

I was wondering how they got all the different combinations of points? Why can't they just put (+-√2,+-1,+-√(2/3)) ?