Lagrange Definition and 542 Threads

Homework Statement I need to find a "Lagrange basis" corresponding to the function space spanned by the basis (1, x^2). Homework Equations I have been told the Lagrange polynomial is of the form (x-x_1)...(x-x_(k-1))(x-x_(k+1))..(x-x_n) / (x_k-x_1)...(x_k-x_(k-1))(x_k-x_(k+1))..(x_k-x_n)...

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...

Homework Statement from one thing in Lagrange mechanics (general coordinates: \phi,\dot\phi,s,\dot{s}) I got a equation system: \begin{cases}R\ddot\phi\sin\phi+R\dot\phi^2\cos\phi+\ddot{s}=0\\ g\sin\phi+R\ddot\phi+\ddot{s}\sin\phi=0\end{cases} The Attempt at a Solution Is it good idea to...

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 I am looking for some help in finding the Lagrange Remainder Theorem from the integral form of the remainder of a Taylor series Homework Equations Integral form of Taylor Series: Rn,a(x) = x∫a [f(n+1)(t)]/n! *(x-t)dt The Attempt at a Solution We are given the...

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...

Homework Statement A uniform circular cylinder of mass `m' (a yo-yo) has a light inextensible string wrapped around it so that it does not slip. The free end of the string is fastened to a support and the yo-yo moves in a vertical straight line with the straight part of the string also...

Homework Statement A uniform solid cylinder of mass `m' and radius `a' rolls on the rough outer surface of a fixed horizontal cylinder of radius `b'. Let `theta' be the angle between the plane containing the cylinder axes and the upward vertical (generalized coord.) Deduce that the cylinder...

Homework Statement Find the absolute maximum and minimum values for f(x,y) = sin x + cos y on the rectangle R defined by 0<=x<=2pi and 0<=y<=2pi using the method of Lagrange Multipliers. The Attempt at a Solution I don't know where to start in getting the constraint into something I...

Why when doing a Lagrange Multipler with two constraints, why do you add the gradients of the two constriant funcions and set it parallel to the function to be maximized...

Let F and f be functions of the same n variables where F describes a mechanical system and f defines a constraint. When considering the variation of these functions why does eliminating the nth term (for example using the Lagrange multiplier method) result in a free variation problem where it...

Homework Statement Consider the problem of finding the points on the surface xy+yz+zx=3 that are closest to the origin. 1) Use the identity (x+y+z)^2=x^2+y^2+z^2+2(xy+yz+zx) to prove that x+y+z is not equal to 0 for any point on the given surface. 2) Use the method of Lagrange...

Homework Statement Use Lagrange Multipliers to find the Maximum and Minimum values of f(x,y) = x2-y. Subject to the restraint g(x,y) = x2+y2=25 Homework Equations gradient f(x,y)= gradient g(x,y) The Attempt at a Solution I have found the gradients of f and g to be f(x,y) =...

Homework Statement Let f be a function whose seventh derivative is f7(x) = 10,000cos x. If x = 1 is in the interval of convergence of the power series for this function, then the Taylor polynomial of degree six centered at x = 0 will approximate f(1) with an error of not more than a.)...

1. Homework Statement [/b] f\left(x,y\right) = x^2 +y^2 g\left(x,y\right) = x^4+y^4 = 2 Find the maximum and minimum using Lagrange multiplier Homework Equations The Attempt at a Solution grad f = 2xi +2yj grad g= 4x^3i + 4y^3j grad f= λ grad g 2x=4x^3λ and 2y=...

Hi, I am supposed to find the point on the cone z^2=x^2+y^2 which is closest to the point(9,12,5). here is my work: http://img27.imageshack.us/my.php?image=lagrange001.jpg Is it correct so far? If it is: I get stuck when trying to solve the equations z^2=x^2+y^2, x=9/12*y, and...

Homework Statement A uniform disk of mass M and radius a can roll along a rough horizontal rail. A particle of mass m is suspended from the center C of the disk by a light inextensible string b. The whole system moves in a vertical plane through a rail. Take as generalized coordinates x...

Hello, I am aware that magnetic forces can do no work. I am also aware that, in a conservative system, equations of motion that minimize the "action" (which are the true equations of motion) can be found with the euler-lagrange equation. The only information the euler-lagrange equation needs...

Could somebody explain to me how lagrange multipliers works in finding extrema of constrained functions? also, what is calculus of variations and lagrangian mechanics, and can somebody explain to me what the lagrangian function is and the euler-lagrange equation. And, i read something about...

Lagrange Polynomals are defined by: lj(t)= (t-a0) ...(t-aj-1)(t-aj+1)...(t-an) / (aj-a0)...(aj-aj-1)(aj-aj+1)...(aj-an) A) compute the lagrange polynomials associated with a0=1, a1=2, a2=3. Evaluate lj(ai). B) prove that (l0, l1, ... ln) form a basis for R[t] less than or equal to n...

help me out on this proble i am confuse a sport center is to be constructed.it consists of a rectangular region with a semicircle ach end .if the perimater of the room is to be a 500 meter running truck find the dimetion that will make the area as large as possible. i can find if the...

Hi people, here's my problem: A uniform, flexible rope of length D, mass M, hangs off a frictionless table-top of height greater than D. The length of the section of rope hanging off is A. Gravity accelerates the part of the rope that is hanging off, so the length of the hanging part increases...

Homework Statement A disk moves on an inclined plane, with the constraint that it's velocity is always at the same direction as it's plane (similar to an ice skate, maybe). In other words: If \hat{n} is a vector normal to the disk's plane, we have at all times: \hat{n} \cdot \vec{v} = 0. Also...

I'm looking for a derivation of the method of Lagrange multipliers as used in the calculus of variations for extremizing a functional subject to constraints. More specifically, I'm trying to understand the relationship between the "method of Lagrange multipliers" from standard calculus and the...

Homework Statement suppose that H and K are subgroups of a group G such that K is a proper subgroup of H which is a proper subgroup of G and suppose (H : K) and (G : H) are both finite. Then (G : K) is finite, and (G : K) = (G : H)(H : K). **that is to say that the proof must hold for...

Homework Statement Find the maximum and minimum values of f = (x-1)^2 + (y-1)^2 on the boundary of the circle g = x^2 + y^2 = 45. Homework Equations f=(x-1)^2 + (y-1)^2 g=x^2+y^2=45 gradf(x,y)=lambda*gradg(x,y) The Attempt at a Solution gradf(x,y)=<2x-2,2y-4>...

Use Lagrange Multipliers to find the maximum and minimum values of f(x,y)=x^{2}y subject to the constraint g(x,y)=x^{2}+y^{2}=1. \nablaf=\lambda\nablag \nablaf=<2xy,x^{2}> \nablag=<2x,2y> 1: 2xy=2x\lambda ends up being y=\lambda 2: x^{2}=2y\lambda ends up being(1...

Homework Statement derive from the Lagrange function and with the use of the midpoint rule, the Schrodinger equation for a particle with charge e coupled to a vector potential? Homework Equations L=(1/2)m Xt2+ e Xt . A(Xt,t) I mean by X== X dot, but A is a fun. of X and t The...

Homework Statement Point P(x,y,z) lies on the part of the ellipsoid 2x^2 + 10y^2 + 5z^2 = 80 that is in the first octant of space. It is also a vertex of a rectangular parallelpiped each of whose sides are parallel to a coordinate plane. Use Method of LaGrange Multipliers to determine the...