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. C

    Max and Min of function via Lagrange multipliers

    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}\,\...
  2. S

    Need help understanding Lagrange multipliers at a more fundamental level.

    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...
  3. S

    Use Lagrange multipliers to find the max & min

    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) = <...
  4. M

    Understanding Lagrange multipliers in the Lagrangian

    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...
  5. jfy4

    Bead and hoop in 2D using Lagrange

    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...
  6. A

    Using Lagrange Multipliers to Maximize a Quantity Under Constraint

    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...
  7. C

    Proof of Lagrange multipliers method in calculus of variations.

    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...
  8. R

    Liearization of Lagrange EOMs of an Inverted Pendulum

    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...
  9. M

    MATLAB: How do you find the lagrange multipliers

    When I try to solve a linear program using matlab,after using linprog(f,A,b,...) I can find the Lagrange multiplier associated with the inequality constraints and the lower bound constraints by using: lambda.ineqlin ; lambda.lower But if I want to solve a quadratic program (using...
  10. U

    Bonus (Unexpected) solution to lagrange equation?

    Homework Statement The lagrange equations are obtained as in the picture. I am only showing the final part of the solution, where they consider the final case of x≠y≠z. Homework Equations The equation at the second paragraph is obtained by subtracting: (5.34 - 5.35). The final equations are...
  11. U

    Lagrange undetermined multipliers

    Homework Statement This section describes the "Lagrange undetermined multipliers" method to find a maxima/minima point, which i have several problems at the end. The Attempt at a Solution Why are they adding the respective contributions d(f + λg), instead of equating df = λdg ? Imagine...
  12. C

    What is the width of the Lagrange points?

    Hello everyone! I just finishexd reading Death By Black Hole and I was interested in the Lagrange points. Neil talks about how if you placed objects inside of them you could use the points as place holders for objects while building in space. I couldn't seem to find anything about the width of...
  13. C

    Lagrange Multipliers to Find Extreme Values of a Multi-Variable Function

    Homework Statement I need to find the extrema of f(x,y) = 3x^{2} + y^{2} given the constraint x^{2} + y^{2} = 1 Homework Equations I'm not sure what goes here. I've been trying to solve it with this: ∇f(x,y) = λ∇g(x,y) The Attempt at a Solution f(x,y) = 3x^{2} + y^{2} g(x,y)...
  14. W

    Help with Derivation of Euler Lagrange Equation

    Hello all, I am having some frustration understanding one derivation of the Euler Lagrange Equation. I think it most efficient if I provide a link to the derivation I am following (in wikipedia) and then highlight the portion that is giving me trouble. The link is here If you scroll...
  15. M

    Max & Min Values of f(x,y) = x^2 + y^2 in Constraint 3x^2+4xy+6y^2=140

    Homework Statement Show that , the maximum value of function f(x,y) = x^2 + y^2 is 70 and minimum value is 20 in constraint below. Homework Equations Constraint : 3x^2 + 4xy + 6y^2 = 140 The Attempt at a Solution Book's solution simply states the Lagrange rule as ...
  16. T

    Finding the minimum and maximum distances using Lagrange Multipliers

    Homework Statement What I don't understand is why you can maximize the distances squared - d2. Isn't d2 different from d? I don't see how they can get you the same value.
  17. P

    Topological sigma model, Euler Lagrange equations

    Homework Statement My question refers to the paper "Topological Sigma Models" by Edward Witten, which is available on the web after a quick google search. I am not allowed to include links in my posts, yet. I want to know how to get from equation (2.14) to (2.15). We consider a theory of maps...
  18. H

    Using lagrange mutilpliers when lamda has two values

    Consider $${f (x, y) = x^2 + 2 y^2}$$ subject to the constraint $${x^2 + y^2 = 1}$$. What would be the minimun and the maximum values of the f. Trouble is when I tried solving the problem lamda comes out to have two values 1 and 2 respectively. How do I proceed in order to get the answers?
  19. L

    Fluid mechanics Lagrange & Euler formalism

    Lagrange & Euler formalism How we get relation (\frac{\partial T^{(L)}}{\partial t})_{r_0}=(\frac{\partial T^{(E)}}{\partial t})_{r}+\frac{\partial T^{(E)}}{\partial x}(\frac{\partial x}{\partial t})_{r_0}+\frac{\partial T^{(E)}}{\partial y}(\frac{\partial y}{\partial...
  20. Y

    The method of Lagrange multipliers

    Homework Statement The problem of minimizing f(x1, x2) = x1^3 subject to (x1 + 1)^3 = (x2 − 2)^2 is known to have a unique global solution. Use the method of Lagrange multipliers to find it. You should deal with the issue of whether a constraint qualification holds. Homework Equations...
  21. B

    Euler Lagrange equation as Einstein Field Equation

    I want to prove that Euler Lagrange equation and Einstein Field equation (and Geodesic equation) are the same thing so I made this calculation. First, I modified Energy-momentum Tensor (talking about 2 dimension; space+time) : T_{\mu\nu}=\begin{pmatrix} \nabla E& \dot{E}\\ \nabla p &...
  22. R

    Is Lagrange's Theorem the Key to Solving This Vector Equation?

    Proof using lagrange! Homework Statement (A x B) . (C x D) = (A . B) (C . D) - (A . D) (B . C) Homework Equations This is all that's given..I am sort of lost on how to proof this. Spent 4hrs + The Attempt at a Solution Completely lost and don't know where to start
  23. O

    Very important, Lagrange multiplier

    Guys, i would be really greatfull if someone help me with this because i really don't know how to deal with this math problem: Find the maximum and minimum values of f = x^(1/4) + y^(1/3) on the boundary of g = 4*x+ 6*y = 720. Please help me someone, i am desperate from this :(
  24. J

    Problems with Lagrange Multipliers

    Does anyone have any tips for solving the system of equations formed while trying to find Lagrange Multipliers? I have searched for videos online (patrickjmt and the MIT lecture on Lagrange Multipliers) but I still find it a bit confusing.
  25. H

    Optimizing Multivariate Function with Lagrange Multiplier Method

    Homework Statement Find the stationary value of $$ f(u,v,w) = \left( \frac{c}{u} \right)^m + \left( \frac{d}{v} \right)^m + \left( \frac{e}{w} \right)^m $$ Constraint: $$ u^2 + v^2 + w^2 = t^2 $$ Note: $$ u, v, w > 0 $$. $$ c,d, e, t > 0 $$. $$ m > 0 $$ and is a positive integer.Homework...
  26. I

    Lagrange Multipliers. Maximizing Area.

    Homework Statement An open gutter with cross section in the form of a trapezoid with equal base angles is to be made by bending up equal strips along both sides of a long piece of metal 12 inches wide. Find the base angles and the length of the sides for maximum carrying capacity. For more...
  27. S

    Lagrange interpolation formula

    Homework Statement (a) If x_{1},\ldots, x_{n} are distinct numbers, find a polynomial function f_{i} of degree n - 1 which is 1 at x_{i} and 0 at x_{j} for j \ne i. Hint: the product of all (x - x_{j}) for j \ne i is 0 at x_{j} if j \ne i. This product is usually denoted by \prod_{\substack{j...
  28. T

    How to Solve Lagrange Multiplier Problems for Function Extremes?

    Homework Statement Find the product of the maximal and the minimal values of the function z = x - 2y + 2xy in the region (x -1)2+(y + 1/2)2≤2 Homework Equations The Attempt at a Solution I have taken the partial derivatives and set-up the problem, but I am having difficulty...
  29. S

    Lagrange Multiplier -> Find the maximum.

    Lagrange Multiplier --> Find the maximum. Homework Statement Find the maximum value, M, of the function f(x,y) = x^4 y^9 (7 - x - y)^4 on the region x >= 0, y >= 0, x + y <= 7. Homework Equations Lagrange multiplier method and the associated equations. The Attempt at a Solution...
  30. A

    Lagrange multiplier problem - function of two variables with one constraint

    Homework Statement Find the maximum and minimum values of f(x,y) = 2x^2+4y^2 - 4xy -4x on the circle defined by x^2+y^2 = 16. Homework Equations Lagrange's method, where f_x = lambda*g_x, f_y= lambda*g_y (where f is the given function and g(x,y) is the circle on which we are looking...
  31. ElijahRockers

    Quadratic Forms & Lagrange Multipliers

    Homework Statement I'm having trouble grasping http://www.math.tamu.edu/~vargo/courses/251/HW6.pdf. Our teacher has decided to combine elements from Linear Algebra, and understanding Quadratic forms with our section on lagrange multipliers. I am barely able to follow his lectures. If I look...
  32. S

    Total production function using Lagrange Multipliers

    Homework Statement Attached as Question.jpg. Homework Equations Partial differentiation. Lagrange multiplier equation. The Attempt at a Solution Attached as MyWork.jpg. Is my work correct? I'm still not confident with myself for these problems and it would be great if someone...
  33. M

    Lagrange Equation: Solution for J(q_1,...,q_n)

    For functional J(q_1,...,q_n)=\int^{t}_{t_0}L(q_1,...,q_n;\dot{q}_1,...,\dot{q}_n;t) Why isn't J(q_1,...,q_n;\dot{q}_1,...,\dot{q}_n;t)?
  34. ElijahRockers

    Using Lagrange Multipliers to Solve Constrained Optimization Problems

    Homework Statement f(x,y) = y2-x2, g(x,y) = x2/4 +y2=9 Homework Equations \nabla f = \lambda \nabla g -2x = \lambda \frac{x}{2} 2y = 2\lambda y \frac{1}{4} x^2 + y^2 = 9 The Attempt at a Solution I arrived at the three equations above. So according to the first equation...
  35. S

    Maximize the volume without using Lagrange multipliers

    Homework Statement When a rectangular box is sent through the mail, the post office demands that the length of the box plus twice the sum of its height and width be no more than 250 centimeters. Find the dimensions of the box satisfying this requirement that encloses the largest possible...
  36. fluidistic

    Lagrange multipliers in a simple pendulum

    Homework Statement Not really a homework question, just want to check out if what I'm doing is right. I challenged myself to find the equation of motion and the forces in the simple pendulum system but with using the Lagrange multipliers and the constraint equation.Homework Equations In next...
  37. B

    MHB Lagrange thm: orbits as equivalence classes and cosets

    Hi all, first post, please bear with me! I am trying to understand Lagrange's Theorem by working through some exercises relating to the Orbit-Stabilizer Theorem (which I also do not fully understand.) I think essentially I'm needing to learn how to show cosets are equivalent to other things or...
  38. R

    How to Apply Lagrange Equation, Really Confused

    Homework Statement I am having trouble understanding how to apply Lagrange's equation. I will present a simplified version of one of my homework problems. Imagine an inverted pendulum, consisting of a bar attached at a hinge at point A. At point A is a torsional spring with spring...
  39. I

    Optimizing Elliptical Radius Vectors with Lagrange Multipliers

    Homework Statement The question is : Find the maximum and minimum lengths of the radius vector contained in an ellipse 5x^2 +6xy+5y^2 Homework Equations The Attempt at a Solution Hi I seem to be at a loss here because usually along with an equation a constraint is also given but in this case...
  40. P

    Constrained Optimization using Lagrange multipliers with Commerce applications

    Homework Statement Hello! I'm having some difficulty getting the objective function out of this question, any help/hints would be appreciated >.< Company A prepares to launch a new brand of tablet computers. Their strategy is to release the first batch with the initial price of p_1 dollars...
  41. S

    Minimizing function using Calc of Variations and LaGrange Equations

    My instructor likes to explain his topics at light speed and I could barely understand how to use Calculus of variations and the La Grange equations to solve this so I need some help please. This is the problem: Consider the functional for W = w(x,y) prescribed on partial(D), I(W) =...
  42. B

    How Does the Lagrange-Newton (SOLVER) Method Work for Numerical Optimization?

    I have the analytical first and second derivatives of a (multidimensional) lagrangian ( l = f - λh). X is the vector of variables of the objective function and λ is the single lagrange multiplier. where f=f(X) is the nonlinear objective function, h is the nonlinear (equality) constraint (i.e...
  43. A

    How can Lagrange Identity be used to prove a vector equation?

    Homework Statement Prove that (A x B) . (u x v) = (a.u) (b.v) - (a.v)(b.u) The Attempt at a Solution I've used lagrange indentity to proof that. but I can't go ahead Thanks
  44. C

    Finding and recognizing infeasible Lagrange multiplier points

    Maximize: 3*v*m subject to: L - m - v >= 0 V - v >= 0 m - 6 >= 0 M - m >= 0 Where L, M, and V are positive integers. Lagrangian (call it U): U = 3vm + K1(L - m - v) + K2(V - v) + K3(m - 6) + K4(M - m) Where K1-K4 are the slack variables/inequality Lagrange...
  45. A

    Procrustes Analysis and Lagrange Multipliers

    The problem: Minimize tr{RyxR} subject to RTR=I This problem is known as Procruses Analysis and can be solved using Lagrange Multipliers, so there's a tendency to write the following function: L(R) = tr{RyxR} - \Lambda(RTR-I), where \Lambda is a matrix of Lagrange Multipliers However, there...
  46. H

    Lagrange Function for a certain problem

    Homework Statement A particle of mass m is connected by a massless spring of force constant k and unstressed length r0 to a point P that is moving along a horizontal circular path of radius a at a uniform angular velocity ω. Verify the Lagrange-Function! Homework Equations Could...
  47. A

    Lagrange inversion theorem

    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...
  48. N

    Lagrange Multipliers. All variables cancel

    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...
  49. M

    Finding Functional for Euler Lagrange ODE

    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...
  50. M

    Proof involving Taylor Polynomials / Lagrange Error Bound

    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)...
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