What is Lagrange: Definition and 538 Discussions

Joseph-Louis Lagrange (born Giuseppe Luigi Lagrangia or Giuseppe Ludovico De la Grange Tournier; 25 January 1736 – 10 April 1813), also reported as Giuseppe Luigi Lagrange or Lagrangia, was an Italian mathematician and astronomer, later naturalized French. He made significant contributions to the fields of analysis, number theory, and both classical and celestial mechanics.
In 1766, on the recommendation of Swiss Leonhard Euler and French d'Alembert, Lagrange succeeded Euler as the director of mathematics at the Prussian Academy of Sciences in Berlin, Prussia, where he stayed for over twenty years, producing volumes of work and winning several prizes of the French Academy of Sciences. Lagrange's treatise on analytical mechanics (Mécanique analytique, 4. ed., 2 vols. Paris: Gauthier-Villars et fils, 1788–89), written in Berlin and first published in 1788, offered the most comprehensive treatment of classical mechanics since Newton and formed a basis for the development of mathematical physics in the nineteenth century.
In 1787, at age 51, he moved from Berlin to Paris and became a member of the French Academy of Sciences. He remained in France until the end of his life. He was instrumental in the decimalisation in Revolutionary France, became the first professor of analysis at the École Polytechnique upon its opening in 1794, was a founding member of the Bureau des Longitudes, and became Senator in 1799.

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  1. MarkFL

    MHB Solve Box Cost Minimization w/ Lagrange Multipliers

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

    Variational calculus Euler lagrange Equation

    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...
  3. D

    Maximizing Distance from Point on Sphere (1,1,-1): Lagrange Multipliers Method

    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=...
  4. R

    Lagrange Mechanics Homework: 2 DoF, Eqns of Motion, Constants

    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...
  5. M

    Maximizing volume of a box without lagrange multipliers

    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...
  6. M

    In proving that the Lagrange equations hold in any coordinate system

    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...
  7. D

    MHB Min of an Integral lagrange multipliers for E-L

    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*}...
  8. D

    MHB Coupled ODEs from Euler Lagrange eq

    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] &=...
  9. D

    MHB What is the integral of f(x) and q(x) in the Euler Lagrange equations?

    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*}
  10. L

    Optimisation - Using the lagrange method

    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)...
  11. E

    Euler - Lagrange Equation(changing variable)

    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...
  12. M

    Euler Lagrange Equations for 1 particle in 3-dimensions

    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...
  13. A

    Parking at Lagrange point at Phobos & Mars

    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...
  14. K

    Three masses two strings system: lagrange and eigenvalues

    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...
  15. G

    MHB Lagrange Multipliers: Find Extrema of f(x,y)=x^2y

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

    MHB GWR309's question at Yahoo Answers regarding Lagrange multipliers

    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.
  17. G

    Lagrange multipliers in Lagrangian Mechanics

    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...
  18. G

    Generality of the Euler Lagrange equations

    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...
  19. O

    Lagrange Multiplier-to find out the dimensions when metal used min.

    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...
  20. T

    Lagrange equations of a spinning parabola

    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...
  21. C

    Maximizing C_t with Lagrangian: First Order Condition Explained

    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...
  22. icystrike

    Lagrange Remainder: Clarifying MVT Statement

    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...
  23. T

    Max/Min f subject to g: Lagrange Multipliers

    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 -...
  24. H

    Euler Lagrange Equation Question

    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...
  25. MarkFL

    MHB JOHN's question at Yahoo Answers involving Lagrange multipliers

    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.
  26. B

    Derivation of Lagrange Family of Interpolation functions

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

    Lagrangian and lagrange equations of a system of two masses

    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...
  28. skate_nerd

    MHB Lagrange multipliers with a summation function and constraint

    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...
  29. skate_nerd

    MHB Lagrange multipliers for extreme values

    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...
  30. B

    LaGrange Multiplier Problem

    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...
  31. G

    Equilibrium state proof with lagrange

    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...
  32. B

    Solve Lagrange Multiplier Problem | f(X,Y,Z) = 2XY + 6YZ + 8XZ

    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 = <...
  33. T

    Optimization program using Lagrange multipliers.

    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...
  34. D

    MHB Lagrange Coefficients for Two-Body Equation of Motion

    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} =...
  35. S

    Dual vector space - Lagrange Interpolating Polynomial

    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...
  36. E

    Closest approach of a parabola to a point, using lagrange multipliers

    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...
  37. L

    Lagrange multipliers for finding geodesics on a sphere

    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...
  38. D

    MHB Maximizing Gamma with Lagrange Multipliers

    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?
  39. C

    Optimizing Multivariate Function with Constraint: Lagrange Multiplier Troubles?

    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...
  40. P

    Lagrange Remainder for Taylor Expansion of ln(4/5) ≤ 1/1000?

    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...
  41. F

    Exercise with lagrange and derivatives

    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...
  42. L

    Advanced Lagrange Maxima Minima Problems

    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...
  43. Y

    MHB Using Lagrange's Theorem to Show 1.71<\sqrt{3}<1.75

    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 !
  44. Y

    Lagrange on an Ellipse to find Max/Min Distance

    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} +...
  45. 5

    Use Lagrange multipliers to find the eigenvalues and eigenvectors of a matrix

    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...
  46. M

    Lagrange multipliers and combinations of points

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

    Find the max and min using lagrange

    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...
  48. F

    Lagrange Equations: Rectilinear Rod Rotating with Constant Angular Velocity

    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...
  49. V

    The error caused by the lagrange inversion

    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...
  50. STEMucator

    Finding max/min using Lagrange Multipliers

    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...
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