Lagrange Definition and 510 Threads

I know full well the proof using Lagrange's thm. But is there a direct way to do this without using the fact that the order of an element divides the order of the group? I was thinking there might be a way to set up an isomorphism directly between G and Z/pZ. Clearly all non-zero elements...

Homework Statement Seems straightforward enough, Lagrangian optimization Homework Equations Find the max of x^-1 + y^-1 subject to the constraint m=x+y The Attempt at a Solution At first I thought no problems, x*=y*=m/2, however: Using the Lagrangian formula yields...

Homework Statement Let G be group, H<G , K<G, if gcd(lHl,lKl)=1, prove that H\bigcapK={1} Homework Equations The Attempt at a Solution so Lagrange theorem says that lHl l lGl, lKl l lGl, and of course 1 is inside both H and K, but how when they are coprime, the element are all...

Homework Statement See figure Homework Equations N/A The Attempt at a Solution Alright we'll this is my first shot at a question like this, so in all honesty I don't know what concepts this question is testing. It mentions finding absolute max/min of a function inside a...

Hi, Dear Math forum users, I was practicing with my optimization course problem and encountered one type of Lagrange multiplier question which I have trouble with. I am wondering if anyone could enlighten me for the following Lagrange problem. function f = x*y*z subject to 4xy+3yz+2xz...

Hi, I'm trying to do a constrained optimization problem. I shall omit the details as I don't think they're important to my issue. Let f:\mathbb R^n \to \mathbb R and c:\mathbb R^n \to \mathbb R^+\cup\{0\} be differentiable functions, where \mathbb R^+ = \left\{ x \in \mathbb R : x> 0...

Homework Statement Using Lagrange multipliers, find the maximum and minimum values of f(x,y)=x^3y with the constraint 3x^4+y^4=1.Homework Equations The Attempt at a Solution Here is my complete solution. I just wanted to make sure there are no errors and I did it correctly. Thanks for any...

I have been reading about Lagrange Multipliers, my book along with wiki and other resources I have read use an intuitive argument on why the max/min contour lines end up tangent to the constraint equation. I don't really understand it, especially considering the obvious flaw as shown by the...

Homework Statement Find the points on the ellipse x2 + 2y2 = 1 where f(x,y) = xy has its extreme values. Homework Equations The Attempt at a Solution f(x,y,z) = x2 + y2 + z2 -- constraint g(x,y,z) = x2 + 2y2 -1 = 0 gradient of f = \lambda * gradient of g 2xi + 2yj + 2zk =...

f(x,y,z)=4x^2+4y^2+z^2 subject to x^2+y^2+z^z=1 So I have: F(x,y,z,c) = 4x^2+4y^2+z^2+L(x^2+y^2+z^2-1) dF/dx = 8x+2xL dF/dy = 8y+2yL dF/dz=2z+2zL Either x=y=0 and L=-1 OR z=0 and L=-4 For first case, z^2=1 therefore z=+/- 1 giving f(0,0,1)=1 For second case, x^2+y^2=1 2x^2=1...

Hello, I am interested in what would happen if a photon became nested inside the Lagrangian point of a binary black hole system that was already far into the process of merging. It seems that the photon would be "frozen."

Homework Statement find the points on the surface x^2-z^2 = 1 which are in minimum distance from (0,0) i should find the points using d = x^2+y^2+z^2 first of all gradf = λ gradg where f = d and g = x^2-z^2 so we have (2x,2y,2z) = λ (2x,0,2z) now 2x = λ2x 2y = 0 => y = 0 2z = λ2z so...

Homework Statement I am doing this lagrange multiplier problem with 2 constraints. I have completely solved it as shown in the image below. I have found that for lambda = 1 and mu = +/- 1/2 I have that x=+/- [sqrt(2)] y=+/- [1/sqrt(2)] and z=+/- [1/sqrt(2)]. So I am trying to figure...

Homework Statement Find the extrema of the given function subject to the given constraint: f(x,y)=x2-2xy+2y2, subject to x2+y2=1Homework Equations Lagrange Multipliers The Attempt at a Solution First, I defined the constraint to be g(x,y)=0, that is, g(x,y)=x2+y2-1 I then set up the usual...

Hey folks. :smile: I have some more or less qualitative questions regarding optimization problems via Lagrange multipliers. I am following the http://en.wikipedia.org/wiki/Lagrange_multipliers" on this one and I am just a little confused by their wording. In the first section titled...

Homework Statement Find the points on the level surface xy2z4=1 that are closest to the origin. Homework Equations Lagrange's method for finding extrema The Attempt at a Solution If I have a level surface F(x,y,z)=c, it's points closest to the origin will be the ones in which...

Find an equation of the largest sphere that passes through the point (-1,1,4) and is such that each of the points (x,y,z) inside the sphere satisfies the condition x^2 + y^2 + z^2 < 136 + 2(x + 2y + 3z) I know this problem requires Lagrange multipliers. I assume that x^2 + y^2 + z^2 is...

Homework Statement Assume that the surface temperature distribution of an ellipsoid shaped object given by 4x2 + y2 + 4z2 = 16 is T(x,y,z) = 8x2 + 4yz - 16z + 600.Homework Equations The Attempt at a Solution I'm assuming we just have to find the maximum value of this function using the lagrange...

I'm trying to figure out how much force, over what period of time, is necessary to reach an earth-moon Lagrange point. L1 is about 323110 kilometers from earth, and an object there could remain (more or less) stationary relative to the Earth and the moon. Earth gravity is working against the...

Homework Statement Why is \nabla f = \lambda \nabla g where f is the function you want to find the extrema of and g is the contraint? Also how would you identify the above in the following Determine the least real number M such that the inequality |ab(a^2-b^2) +...

Homework Statement y = xf(y') + g(y') Let y' = P taking d/dx and rearranging gives dx/dP - xf'(P)/{P - f(P)} = g'(P)/(P - f(P)) a 1st order linear differential equation in standard form. Homework Equations When I attempt to solve by the suggested standard method, I end up...

Homework Statement Estimate sin4 accurate to five decimal places (using maclaurin series of sin) Homework Equations The Attempt at a Solution Lagrange error bound to estimate sin4° to five decimal places( maclaurin series) 4°=pi/45 radians |Rn(pi/45)<1*(pi/45)^n+1/(n+1)...

Homework Statement Find the optimal value of the function f (x,y) = 3.5x^2+y^2-42x-28y+5xy+190 subject to 6x+5y = 37 Homework Equations Use the second order condition to determine if the optimal point is maximum or minimum The Attempt at a Solution

Homework Statement f(x,y,z)=exy and x5+y5=64 Find Max and MinHomework Equations ∇F = <yexy, xexy> λ∇G = <5x4λ, 5y4λ> The Attempt at a Solution yexy = 5x4λ xexy = 5y4λ x5+y5=64 No idea where to go from here...

Homework Statement if you have dl/dx= -2 +0.002x-lagrange function(backword L) dl/dy=0.012y-5-lagrange function dl/dl= -(x+y-2000) How do you solve for x, y and backword l? Homework Equations The Attempt at a Solution

Homework Statement Use lagrange multipliers to find the shortest distance between a point on the elliptic paraboloid z=x^2 +y^2 Homework Equations The Attempt at a Solution http://img716.imageshack.us/img716/7272/cci1902201000000.jpg I'm not that good with using the equation...

I am attempting for my own curiosity to find out at what point during a geodesic path from the Earth to the Moon one would reach a gravitationally neutral point. This is essentially the L1, but without adjustments for centripetal force of a moving system, and ignoring all other gravitational...

Dear everyone can anyone help me with the euler lagrange equation which is stated in d'inverno chapter 11? in equation (11.26) it is said that when we use the hilbert-einstein lagrangian we can have: ∂L/(∂g_(ab,cd) )=(g^(-1/2) )[(1/2)(g^ac g^bd+g^ad g^bc )-g^ab g^cd ] haw can we derive...

I have a problem where I'd like to minimize a certain function subject to the constraint that a related function is at a maximum, that is I have a function F(a,b) I would like to know what its minimum is when G(a,b) is at a maximum. I'm not sure how to set this problem up, I know that for the...

SOS .. Problem with lagrange derivation! Homework Statement Im having a hard time with the problem illustrated in the following figure: http://img199.imageshack.us/img199/812/20091224344.th.jpg it said that solve the equations of motion for a coupled oscillators system consists of a...

hello everyone:smile: for i=1,2,...,(n+1) let P_{i}(X)=\frac{\prod_{1\leq j\leq n+1,j\neq i}(X-a_j)}{\prod_{1\leq j\leq n+1,j\neq i}(a_i-a_j)} prove that (P_1,P_2,...P_{n+1}) is basis of \mathbb{R}_{n}[X] . i already have an answer but i don't understand some of it. ... we have...

Homework Statement Find the point on 2x + 3y + z - 11 = 0 for which 4x^2 +y^2 +z^2 is a minimum Homework Equations The Attempt at a Solution Using lagrange multipliers I find: F = 4x^2 + y^2 + z^2 + l(2x + 3y + z) Finding the partial derivatives I get the three equations...

The hamilton function of a particle in two dimensions is given by H = (p\stackrel{2}{x})/2m + (p\stackrel{2}{y})/2m + apxpy + U(x,y) Obtain the Hamiltonian equations of motion. Find the corresponding Lagrange function and Lagrange equations. Would it be px = dH/dpy (of course it...

Homework Statement By using the Lagrange multipliers find the extrema of the following function: f(x,y)=x+y subject to the constraints: x2+y2+z2=1 y+z=12. The attempt at a solution Using lambda = 1/(2x) I got x=y-z and y=1-z plugging that into the first constraint, I got: 6y^2-6y+1=0 which...

Homework Statement Use the Lagrange Multiplier method to find the maximum and minimum values of x2 − 2xy + 7y2 on the ellipse x2 + 4y2 = 1. Homework Equations Lagrange multiplier method The Attempt at a Solution L(x,y,z,λ) = x2 − 2xy + 7y2 - λ(x2 + 4y2 - 1) Find Lx, Ly, Lλ Then, solve for x...

Homework Statement A window of fixed perimeter is in the shape of a rectangle surmounted by a semi-circle. Prove that its area is greatest when its breadth equals its greatest height. Homework Equations SA = lw + (pi*l^2)/4 <--- Thats what I got the surface area to be. Perimeter = 2w +...

Homework Statement Use Lagrange Multipliers to find the points closest to the origin on the curve defined implicitly by x2-xy+y2-z2 = 1 x2+y2=1 2. The attempt at a solution I know how to do this for regular curves, but I don't know where to start with implicitly defined ones. Any...

Basically I've got to design and develop a software for computing a polynomial function involving a set of data points. I've got to use an algorithm based on the lagrange interpolation method. I know it should involve two loops inside the code. What I've been told is that "The input to the...

Homework Statement Find max/min using L.M of the function : F(x,y) = x^2 + y^2 ; xy = 1 let G(x,y) = xy - 1 F_x = 2x F_y = 2y G_x = y G_y = x F_x = L*G_x F_y = L*G_y G(x,y) = 1 1) 2x = L * y 2) 2y = L * x 3 ) xy = 1 Now I need to solve those equations. so x =...

Dear all, The question I've been struggling with is supposed to be solved using the way Lagrange's thm was proven( with number of cosets and stuff). However, it remains a mystery how to do it: Let G be a finite group and H<G with |G|=m|H|. Proof that g^{m!} \in H, \forall g \in G

In the article "The Lattice Theory of Quark Confinement", by Claudio Rebbi (Scientific American) there is a graphic representing the chromoelectric field. The caption reads: "Chromoelectric field is a gauge field similar in principle to the electromagnetic field but more complicated...

Homework Statement A hoop of mass m and radius R rolls without slipping down an inclined plane of mass M, which makes an angle \alpha with the horizontal. Find the Lagrange equations and the integrals of the motion if the plane can slide without friction along a horizontal surface. Homework...

Homework Statement The plane 4x − 3y + 8z = 5 intersects the cone z^2 = x^2 + y^2 in an ellipse. Use LaGrange Multipliers to find the highest and lowest points on the ellipse. Homework Equations Lagrange Multiplier The Attempt at a Solution I guess I lack an understanding of...

Consider the linear model of a molecule with three atoms. The outer atoms are of mass m and the atom in the molecules center is of mass M . The outer atoms are connected to the center atom through springs of a constant k. (a) Find the Lagrange function of the system. Use as coordinates the...

I tried to use Lagrangian and Hamiltonian to solve 1-D elastic collision, but I got nothing but constant velocity motion. Is it because I miss some constraint? Such as the motion is colinear or something?But how to write a constraint like colinear? Or it's not actually solvable with Hamiltonian...

ok this is just an example so you can see where I am having problems with these(it isn't hw) i need to find the optimum values of X and Y U= XY m= Psuby(Y) + Psubx(X) the first order conditions are Y +u*Psubx X+ u*PsubY m= Psuby(Y) + Psubx(X) now , where I am having...

Can someone kind of give me a step by step as to how you get the equations of motion for this problem? http://www.enm.bris.ac.uk/teaching/projects/2002_03/ca9213/images/msp.jpg the answer is this: http://www.enm.bris.ac.uk/teaching/projects/2002_03/ca9213/msp.html Though I am not quite...

Homework Statement A particle of mass m moves in a force field whose potential in spherical coordinates is, U = \frac{-K \cos \theta}{r^3} where K is constant. Identify the two constants of motion of the system. The Attempt at a Solution L = T - V = \frac{1}{2} m (\dot{r}^2 + r^2...

Homework Statement Using Lagrange Multipliers, we are to find the maximum and minimum values of f(x,y) subject to the given constraint Homework Equations f(x,y,z) = x^2 - 2y + 2z^2, constraint: x^2 + y^2 + z^2 = 1 The Attempt at a Solution grad f = lambda*grad g (2x, -2, 4z) =...

Homework Statement Consider a pendulum with a spring as in the following diagram: Please note the 'rotated' coordinates. The bob has a mass m. The spring has a spring constant k and an unextended length \ell. We can not ignore air friction. Assume the initial velocity and horizontal...