Lagrange Definition and 542 Threads

I encountered this beautiful theorem and then I tried hard to prove it using ordinary algebraic methods and my understanding of calculus without involving real analysis in it but I didn't succeed. The theorem states that if f is an analytical function at some point x=a then f-1 has the following...

Homework Statement A cannonball is heated with with temperature distribution T(x,y,z)=60(y2+z2-x2). The cannonball is a sphere of 1 ft with it's center at the origin a) Where are the max and min temperatures in the cannonball, and where do they occur?Homework Equations \nablaf=λ\nablag Where...

Hello there, I am interested in the following matter. Given an ODE, can one always find a functional F such that the ODE is its Euler Lagrange equation? I am thinking at the following concrete case. I have the ODE y' = a y I would like a functional given by the intergral over a...

Homework Statement I'm given that the function f(x) is n times differentiable over an interval I and that there exists a polynomial Q(x) of degree less than or equal to n s.t. \left|f(x) - Q(x)\right| \leq K\left|x - a\right|^{n+1} for a constant K and for a \in I I am to show that Q(x)...

Homework Statement Prove I=T1+T2+...+Tk Where Ti=pi(T) Homework Equations T is kxk pi(x)=(x-c1)...(x-ck) is the minimal polynomial of T. pi=\pii(x)/\pii(ci) \pii=\pi(x)/(x-ci) To evaluate these functions at a matrix, simply let ci=ciI The Attempt at a Solution From lagrange interpolation...

My question is simple is every classical mechanics problem which is solvable by Lagrangian & Hamiltonian methods also solvable by Newtonian methods of forces and torques? And why does it seem that LH make solutions to be a lot more easier than Newtonian methods, and is it always this way?

In Lagrangian mechanics, can anyone show how to find the extrema of he action functional if you have more constraints than degrees of freedom (for example if the constraints are nonholonomic) using Lagrange Multipliers? I've looked everywhere for this (books, papers, websites etc.) but none...

Homework Statement Find the minimum and maximum values of the function subject to the given constraint f(x,y) = x^2 + y^2, 2x + 3y = 6 Homework Equations \nablaf, \nablag The Attempt at a Solution After doing all the calculation, x value and y value came out to be...

Hi Homework Statement Look at the drawing. Furthermore I have a constant acceleration \vec g = -g \hat y I shall find the Lagrange function and find the equation of motion afterwards.Homework Equations Lagrange/ Euler function and eqauation The Attempt at a Solution I found out the...

I'm in a bit of a hurry, so this isn't going to be very pretty. Homework Statement Maximize: V(l,d) = pi * (0.5*d)^2 * l Subject to: l + 3.5d = 84 -> C(l, d) = l + 3.5d - 84Homework Equations ∇V(l,d) = λ ∇C(l,d) The Attempt at a Solution ∇V(l,d) = 0.5*pi*d ∇C(l,d) = 0How do I find the...

Homework Statement Find the maximum and minimum values of f(x,y) = x5y3 on the circle defined by x2 + y2 = 10. Do the same for the disc x2 + y2 ≤ 10. The Attempt at a Solution for the first part, if I call the circle g(x,y) defined by x2 + y2 = 10 I need to now define some F(x,y,λ) =...

Hello Can any1 recommend a book that will show the derivation of the Euler-Lagrange equation. (I am learning in the context of cosmology ie. to extremise the interval). Ideally the derivation would be as simple/fundamental as possible - my maths is not up to scratch!

Homework Statement "Vary the following actions and write down the Euler-Lagrange equations of motion." Homework Equations S =\int dt q The Attempt at a Solution Someone said there is a weird trick required to solve this but he couldn't remember. If you just vary normally you get \delta...

Homework Statement Use the method of Lagrange multipliers to find the maximum and minimum values of the function f(x, y) = x + y2 subject to the constraint g(x,y) = 2x2 + y2 - 1 Homework Equations none The Attempt at a Solution We need to find \nablaf = λ\nablag Hence...

Hello everyone, i have 2 problems in my multivariable calculus homework that i can't solve . Please help me out, thank you so much 1/f(x,y)= [(x^2) -2y]^(0.5) a) Find directional derivatives of f at (2,-6) in the direction of <-4,3> b) Find equation of the tangent plane to the function...

Homework Statement Use Lagrange multipliers to find the max and min values of the function subject to the given constraints: f(x1,x2,...,xn) = x1 + x2 + ... + xn constraint: (x1)^2 + (x2)^2 + ... (xn)^2 = 1 The Attempt at a Solution fo x1 to xn values, x must equal 1/sqrt(n) in...

Homework Statement Use Lagrange multipliers to find the max and min values of the function subject to the given constraints: f(x,y,z)= x2y2z2 constraint: x2 + y2 + z2 = 1 Homework Equations ∇f = ∇g * λ fx = gx * λ fy = gy * λ fz = gz * λ The Attempt at a Solution i can't solve...

Homework Statement Use Lagange Multipliers to find the max and min values of the function subject to the given constraint(s). f(x,y)=exp(xy) ; constraint: x^3 + y^3 = 16 Homework Equations \nablaf = \nablag * \lambda fx = gx * \lambda fy = gy * \lambda The Attempt at a Solution...

Homework Statement I am trying to find the min and max values of f(x,y)=2x^2 + 3y^2 subject to xy=5. Homework Equations f(x,y)=2x^2 + 3y^2 subject to xy=5 \mathbf\nablaf=(4x, 6y) \mathbf\nablag=(y,x) The Attempt at a Solution When I go through the calculations, I end up with two critical...

Homework Statement Use Lagrange multipliers to ¯nd the maximum and mini- mum value(s), if they exist, of f(x; y; z) = x^2 -2y + 2z^2 subject to the constraint x^2+y^2+z^2 Homework Equations The Attempt at a Solution Basically after I find the gradient of the functions I get this. 2x=2x lamda...

Use lagrange multipliers to find max/min of f(x,y)=x^2+6y subject to x^2-y^2=5 grad f =λgrad g 2x=2xλ, λ=1 6=-2yλ, λ=-3/y 1=-3/y, y=-3 x^2-(-3)^2=x^2-9=5 x^2=14 x=+/-√14 two points are √14, -3 and -√14, -3 plugging both points into f(x,y) gives me the same answer. now what?

Homework Statement Find the polynomial p(x) of degree 20 satisfying: p(-10) =p(-9) = p(-8) = ...=p(-1) = 0 p(0) = 1 p(1) = p(2) = p(3) = ...p(10) = 0 Homework Equations L(x) := \sum_{j=0}^{k} y_j \ell_j(x) The Attempt at a Solution i tried using the formula above: a =...

My textbook is using Lagrange multipliers in a way I'm not familiar with. F(w,λ)=wCwT-λ(wuT-1) Why is the first order necessary condition?: 2wC-λu=0 Is it because: \nablaF=2wC-λu Why does \nablaF equal this? Many thanks! Edit: C is a covariance matrix

Homework Statement In Classical mechanics 2 i have an assignment based on the Euler Lagrange method and i cannot seem to grasp the concept, even with all the internet resources i can find as well as my two textbooks which have a chapter on it. (Boass (Mathematical methods in teh physical...

Homework Statement Use Charpits equations to solve 4u\frac{\partial u}{\partial x} = (\frac{\partial u}{\partial x})^2 where u=1 on the line x+2y=2 Homework Equations The Attempt at a Solution from the charpit equations i get \frac{dx}{dt} = 4u \frac{dy}{dt} = -1...

Homework Statement I made this up, so I am not even sure if there is a solution Let's say I have to find values for which these two inequality hold x^2 + y^5 + z = 6 and 8xy + z^9 \sin(x) + 2yx \leq 200And by Lagrange Multipliers that \nabla f = \mu \nabla g So can I let f = 8xy + z^9 \sin(x)...

Dear all, I have an optimization problem with boundary conditions, the type that is usually solved with Lagrange multipliers. But the (many) variables my function depends on can take only the values 0 and 1. Does anyone know how to apply Lagrange multipliers in this case? I am a...

so i know for example that d/dt (∂L/∂x*i) = ∂L/∂xi for cartesian coordinates, where xi is the ith coordinate in Rn and x*i is the derivative of the ith coordinate xi with respect to time. L represents the lagrangian. so using an arbitrary change of coordinates, qi = qi(x1, x2, ..., xn) i...

1. Assume we have function V(x,y,z) = 2x2y2z = 8xyz and we wish to maximise this function subject to the constraint x^2+Y^2+z^2=9. Find the value of V at which the max occurs 2. Function: V(x,y,z) = 2x2y2z = 8xyz Constraint: x^2+Y^2+z^2=9 3. So far I have gone Φ= 8xyz +...

A friend and I were debating the solution to this problem, seen below, and cannot solve it without using Lagrange equations but it is suppose to have a solution that is super simple; but we didn't see it. Anyway, it is a old qualifier question from the Univ. of Wisconsin (open record so...

So I need to find the min and max values of f(x,y,z) = x^2 + 2y^2 + 3z^2 given the constraints x + y + z = 1 and x - y + 2z =2. I've gotten as far as (2x, 4y, 6z) = (u,u,u) + (m,-m,2m). I'm stuck trying to solve this system of equations. Any hints?

Hello, i have done the following code in c++ which gives me a result but : 1) I compute the product "prod*=(y-x[i])/(x[k]-x[i])" which has variable "y" inside ,but the result i take is just a number.(i want the result to be for example in this form "p(x)=1.02*x^3+2*x^2..." I can't...

Homework Statement L = - \Sigma x,y (P(x,y) log P(x,y)) + \lambda \Sigmay (P(x,y) - q(x)) This is the Lagrangian. I need to maximize the first term in the sum with respect to P(x,y), subject to the constraint in the second term. The first term is a sum over all possible values of x,y...

Homework Statement I seem to be struggling a bit to understand how my prof solved this problem...I think it might be my diminishing system of equation skills, so forgive me if this doesn't belong in the calc section. Use Lagrange multipliers to find all extrema of the function subject to...

Homework Statement Prove that for every pair of numbers x and h, \left|sin\left(x+h\right)-\left(sinx+hcosx\right)\right|\leq\frac{h^{2}}{2} The Attempt at a Solution Let f(x)= \left|sin\left(x+h\right)-\left(sinx-hcosx\right)\right|? and then to center the taylor polynomial around 0...

In a exercise says: Find max a min of f=-x^2+y^2 abaut the ellipse x^2+4y^2=4 i tried -2x=\lambda 2x 2y=\lambda 8y x^2+4y^2-4=0 then \lambda =-1 or \lambda =\frac{1}{4} , but, ¿how i find x,y?

URGENT - Lagrange Multipliers Homework Statement :confused: Using the method of lagrange multipliers prove the formula for the distance from a point (a,b,c) to a plane Ax + By + Cz = DThe Attempt at a Solution Using the equation of the form; H(x,y,z,L) = (x-a)^2 + (y -b)^2 +(z-c)^2 + L(Ax...

Homework Statement Maximize f(x,y,z)=x^{2}+y^{2}+z^{2} with constraint x^{4}+y^{4}+z^{4}=1 using Lagrange multipliers The Attempt at a Solution I've got the setup as: \Lambda(x,y,z,\lambda)=x^{2}+y^{2}+z^{2}+\lambdax^{4}+\lambday^{4}+\lambdaz^{4}+\lambda I solve for all partials nice...

Physics gurus: I understood from Newton's Law that a 2 bodies would rotate around their common center of mass. Should one body disappear (Harry Potter invoked here), the other would go flying off at a tangent... like a 'David's Sling" releasing a missile. The mass of the bodies was crucial to...

Let f(x,y)= -2x^2-2xy+y^2+2 Use Lagrange multipliers to find the minimum of f subject to the constraint 4x-y = 6 ∂F / ∂x =..... i got -4x-2y+2y but i coming out as wrong what am i missing ∂F/ ∂Y= ... The function f achieves its minimum, subject to the given constraint, where x =...

Homework Statement Im supposed to use the lagrange error bound to find a bound for the error when approximating ln(1.5) with a third degree taylor polynomial about x=0, where f(x)=ln(1+x) Homework Equations Lagrange error bound m/(n+1)! abs(x-a)^n+1, m=f(n+1)(c) The...

A rectangular solid of maximum volume is to be cut from a solid sphere of radius r. Determine the dimension of the solid so formed and its volume? I have defined my function F(l,b,h) as lbh, but i don't know how to define my constraint condition from my question

1. Problem Statement: Use Lagrange multipliers to find the volume of the largest box with faces parallel to the coordinate system that can be inscribed in the ellipsoid: 6x2 + y2 + 3z2 = 2 2. Homework Equations : f(x,y,z) = \lambdag(x,y,z) 3. Attempt at a solution f(x,y,z) is the...

Homework Statement Find the extreme values of the function f(x,y,z) = xy + z^2 in the set S:= { y\geq x, x^2+y^2+z^2=4 } Homework Equations The Attempt at a Solution Ok, so This is clearly a lagrange multiplier question. Geometrically, I can see that the region that is the constraint is...

Find the maximum value of f(x,y,z) = 5xyz subject to the constraint of [PLAIN]http://www3.wolframalpha.com/Calculate/MSP/MSP9619f6019f3fia87i60000567g3gb3dhi833if?MSPStoreType=image/gif&s=6&w=126&h=20. I know to find the partial derivatives of the function and the constraint. Then, set up...

1. Use the method of Lagrange multipliers to nd the minimum value of the function: f(x,y,z) = xy + 2xz + 2yz subject to the constraint xyz = 32. I understand the method how Lagranges Multipliers is donw done but seem to have got stuck solving the Simultaneous Equations involving the...

Homework Statement Find the point closest to the origin on the line of intersection of the planes y + 2z = 12 and x + y = 6Homework Equations \nuf = \lambda\nug1 +\mu\nug2 f = x2+y2+z2 g1: y + 2z = 12 g2: x + y = 6 There are supposed to be gradients on all of those, whether or not LaTeX...

Hey all, this is my first post, so I apologize in advance if data are missing/format is strange/etc. I'm working with lagrange multipliers, and I can get to the answer about half the time. The problem is, I'm not really sure how to deal with things when the multiplier equation becomes...

Hello there, I was wondering if anybody could indicate me a reference with regards to the following problem. In general, the Euler - Lagrange equation can be used to find a necessary condition for a smooth function to be a minimizer. Can the Euler - Lagrange approach be enriched to cover...

Homework Statement Find the minimum of f(x,y) = x^2 + y^2 subject to the constraint g(x,y) = xy-3 = 0 Homework Equations delF = lambda * delG The Attempt at a Solution Okay, after lecture, reviewing the chapter and looking at some online information, this is what I have so far...