Lagrange Definition and 510 Threads

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*}

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...

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)) ?

Homework Statement Find the max and min values of f(x,y,z) = x3 - y3 + 6z2 on the sphere x2 + y2 + z2 = 25. Homework Equations I will use λ to denote my Lagrange multipliers. The Attempt at a Solution So clearly there is no interior to examine since we are on the boundary of the sphere...

Consider a rectilinear rod (A-B) with negligible mass that is attached, without friction, to a vertical OZ axis. The rod rotates about that axis with a constant angular velocity ω and it maintains a angle α with OZ. A particle of mass m moves about the rod and it is attracted by the...

Hi, I have problem with the calculation the error caused by the lagrange inversion. Hence, accroding to Lagrange theorem if f(w)=z it is possible to find w=g(z) where g(z) is given by a series. I wonder, if I consider up to N-th term in the Lagrange series, what will be the error caused by...

Homework Statement Find the maximum and minimum values of 2x2 + y2 on the curve x2 + y2 - 4x = 5 by the method of Lagrange Multipliers. Homework Equations I will express my Lagrange multipliers as λ. The Attempt at a Solution Okay so we want the max min of f(x,y) = 2x2 + y2 given...

Homework Statement If n is a fixed positive integer, compute the max and min values of the function (x-y)^n = f(x,y), under the constraint x^2 + 3y^2 = 1 The Attempt at a Solution I got the 4 critical points (±\frac{\sqrt{3}}{2}, ±\frac{1}{2\sqrt{3}})\,\,\text{and}\,\...

I understand that for Lagrange multipliers, ∇f = λ∇g And that you can use this to solve for extreme values. I have a set of questions because I don't understand these on a basic level. 1. How do you determine whether it is a max, min, or saddle point, especially when you only get one...

Homework Statement Use Lagrange multipliers to find the maximum and minimum values of the function subject to the given constraint. f(x,y) = exy; g(x,y) = x3 + y3 = 16 Homework Equations ∇f(x,y) = λ∇g(x,y) fx = λgx fy = λgy The Attempt at a Solution ∇f(x,y) = < yexy, xexy > ∇g(x,y) = <...

In Goldstein, the action is defined by I=\int L dt. However, when dealing with constraints that haven't been implicitly accounted for by the generalized coordinates, the action integral is redefined to I = \int \left( L + \sum\limits_{\alpha=1}^m \lambda_{\alpha} f_a \right) dt. f is...

Homework Statement A particle of mass m is placed on top of a vertical hoop of radius R and mass M. The particle is free slide on the outside of the hoop without friction while the hoop is free to roll in a vertical place without slipping. Use the method of Lagrange multipliers to determine...

Normally lagrange multipliers are used in the following sense. Suppose we are given a function f(x,y.z..,) and the constraint g(x,y,z,...,) = c Define a lagrange function: L = f - λ(g-c) And find the partial derivatives with respect to all variables and λ. This gives you the extrema...

I have been reading a little about calculus of variations. I understand the basic method and it's proof. I also understand Lagrange multipliers with regular functions, ie since you are moving orthogonal to one gradient due to the constraint, unless you are also moving orthogonal to the other...

Linearization of Lagrange EOMs of an Inverted Pendulum Hi Folks, I am modelling a state space model of an Inverted Pendulum mounted on a cart over a balancing seesaw. I developed the equations of motion using the Lagrangian approach an obtained 3 PDEs. I solved them using Mathematica 8...