Lagrange Definition and 541 Threads

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. A

    How Are Libration Points Computed in Orbital Mechanics?

    How does one go about computing the locations of the 5 largrange points? I know where they are but do not know how to derive the equilibrium equations. I know you will get a quintic polynomial which can be solved numerically depending on the masses of the two large bodies, but using forces, how...
  2. L

    C/C++ Lagrange Interpolation: Investigating Interpolation Points with C++/Fortran

    In my comp physics class, we've been introduced to both c++ and fortran languages. For instance, for our first assignment, I am not sure how to go about investigating the quality of interpolation points for i.e f(x)=sin(x^2) by using n-point langrange interpolation, where n is an input...
  3. X

    Trying to prove inequality with Lagrange multipliers

    Show that if we have N positive numbers \left[ p_{i}\right]_{i=1}^{N} such that \sum_{i} p_{i} =1 then for any N numbers \left\{x_{i}\right\}_{i=1}^{N} we have the inequality \prod_{i=1}^{N} x_{i}^{2 p_{i}} \leq \sum_{i=1}^{N} p_{i}x_{i}^{2} So I am thinking...
  4. B

    Lagrange multiplers - contraints and matrices

    Hi, I'm stuck on the following questions and would like some help. 1. A percel delivery service requires that the dimension of a rectangular box be such that the length plus twice the width plus twice the height be no more than 108 centimetres. What is the volume of the largest box that the...
  5. M

    Help Me Solve a Physics Homework Problem - Lagrange Points

    Hey a friend asked me for help on his physics homework, and I found this place and was wondering if you guys could help me out. 2: In 1772, the famed Italian-French mathematician Joseph Louis Lagrange was working on the infamous three-body problem when he discovered an intersting quirk in the...
  6. M

    Calculating Escape Speed, Lagrange Points, & Stress/Strain of Rope

    1: Determine the escape speed of a rocket on the far side of Ganymede, the largest of Jupiter's moons. The radius of Ganymede is 2.64 X 10^6m, and its mass is 1.495 X 10^23 kg. The mass of Jupiter is 1.90 x 10^27 kg, and the distance between Jupiter and Ganymede is 1.071 X 10^9m. Be sure to...
  7. S

    Lagrange Multipliers: A Long First Post

    A long first post, but not too hard! dont worry about this i already solved it thanks anyway! The lagrangian of a particle of mass m moving under constant gravity is \mathcal{L} = \frac{1}{2} m (\dot{x}^2 + \dot{z}^2 - mgz = \frac{1}{2}m (\dot{\rho}^2 + \rho^2 \dot{\phi}^2) - mg \rho...
  8. B

    Lagrange Multipliers (and finding extrema of a function with two restraints)

    I need to find the extrema of f(x,y,z)=x+y+z subject to the restraints of x^2 - y^2 = 1 and 2x+z = 1. So the gradient of f equals (1,1,1) = lambda1(2x,-2y,0) + lambda2(2,0,1). Solving for the lambdas I found that lambda1 = -1/(2x) = -1/(2y), or x=y. But this isn't possible if x^2 - y^2 = 1...
  9. E

    Finding Extrema with Lagrange Multipliers

    I'm stuck on the following question "Find the maximum and minimum values of f(x,y,z) = x^2y^2-y^2z^2 + z^2x^2 subject to the constraint of x^2 + y^2 + z^2 = 1 by using the method of lagrange multipliers. Write the 4 points where the minimum value is achieved and the 8 points where the...
  10. L

    Can Lagrange Multipliers solve optimization problems with multiple constraints?

    Hi all, I was wondering how to go about solving an optimization problem for a function f(x,y,z) where the two constraint equations are given by: a is less than or equal to g(x,y,z) is less than or equal to b (a and b are two distinct numbers) h(x,y,z) is less than or equal to c (c is...
  11. L

    Studying for test : Lagrange Multipliers

    Help Please! Studying for test : Lagrange Multipliers! Good morning all. I am having trouble with the next step to the following problem: Q.Find all realtive extrema of x^2y^2 subject to the constraint 4x^2 + y^2 = 8. g(x)= x^2y^2 f(x) = 4x^2 + y^2 = 8. the gradiant of f = <8x,2y>...
  12. B

    Solving Maximum & Minimum of f(x,y) with Lagrange Multipliers

    Hi, I'm having trouble with the following question. Q. Find the maximum and minimum of the function f(x,y) = x^2 + xy + y^2 on the circle x^2 + y^2 = 1. I started off by writing: Let g(x,y) = x^2 + y^2 then \nabla f = \lambda \nabla g,g\left( {x,y} \right) = 1 \Rightarrow 2x + y...
  13. D

    Why Does the Disk Have Two Moments of Inertia?

    Lagrange Problem redux -- super urgent... See the attachment to help you visualize this. A rod of length L and mass m is povoted at the origin and swings in the vertical plane. The other end of the rod is attached/pivoted to the center of a thin disk of mass m and radius r. OK, I know that...
  14. D

    Solve LaGrange Problem: Rod of Length L & Disk of Radius R

    We have a rod of length L and mass M pivoted at a point at the origin. This rod can swing in the vertical plane. The other end of the rod is pivoted to the center of a thin disk of mass M and radius R. Derive the equations of motion for the system. I have attached a drawing :) If you...
  15. W

    How Do You Use Lagrange Multipliers to Find the Closest Points to the Origin?

    LaGrange Multipliers! Help! Use the Lagrange multiplier method for 3 variables to find the points on the surface 3xy-z^2=1 that are closest to the origin. I tried using the gradient= lamda(granient) and ended up getting (-3/2,0,-1). but i think i did it way wrong. Can someone please help...
  16. E

    Solving Lagrange Multiplier Question: Find Nearest Point to Origin

    Hi, I would appreciate if anyone can help me out with the following question. I've been asked to find the point on the surface z = xy + 1 nearest to the origin by using the Lagrange Multiplier method. But all the examples I've been given in class and for coursework gave you the constraint...
  17. T

    Solving Max of x^2+y^2 w/ Lagrange Multipliers

    Find the shortest and longest distance from the origin to the curve x^2 + xy + y^2=16 and give a geometric interpretation...the hint given is to find the maximum of x^2+y^2 i am not sure what to do for this problem thanks
  18. Y

    Lagrange and Hamiltonian question

    I can't understand what the question is asking~ hope somebody can help me~ A particle of mass m moves in a plane under the influence of Newtonian gravitation force, described by the potential V(r) = - GmM/r (symbol in conventional meaning) Now introduce a new variable u(theta) = 1/r(theta)...
  19. Cyrus

    Can Lagrange Multipliers Avoid Meaningless Critical Points in Optimization?

    For the proof of lagrange multipliers, it is based on the assumption that the function you are optimizing, f(x,y,z), takes on an extreme value at the point (x0,y0,z0), and that any curve that passes through this point has the tangent vector perpendicular to the gradient vector. That seems fair...
  20. C

    How Can Lagrange Multipliers Be Used to Solve Optimization Problems in Calculus?

    This problem was given in my calc class during the semester, "Find the lowest point on the intersection of the sphere x^2+y^2 +z^2 = 30 and the cone 2*x^2 +y^2 = c^2". I don't know how to solve this problem with lagrange multipliers. How is it done? Thanks! Callisto
  21. A

    Lagrange Multipliers: Exploring G, f, and g

    I'm not entirely sure what the english terms are for some of the things I'm about to say but i hope it's clear what I mean exactly. I'n my handbook the theorom is said to be: Say G is a part (wich is open) of R^n, f and g are functions from G to R (f:G->R, g:G->R) and both are differentiable...
  22. Pyrrhus

    Intermediate Dynamics Books for Lagrange, Hamilton & Canonical Transformations

    Hello, I'm looking for a good Dynamics Book. I got Engineering Mechanics: Dynamics by Andrew Pytel and Jaan Kiusalaas, but it's fairly introductional, i also got Classical Mechanics by Goldstein, which is advanced. I am looking for intermediate level. I am looking mainly to learn the Lagrange...
  23. C

    Resonance pde wave equation u(\phi,t) involving lagrange polynomials

    1/sin(phi) * d/d\phi(sin(phi) * du/d\phi) - d^2u/dt^2 = -sin 2t for 0<\phi < pi, 0<t<\inf Init. conditions: u(\phi,0) = 0 du(\phi,0)/dt = 0 for 0<\phi<pi How do I solve this problem and show if it exhibits resonance? the natural frequencies are w = w_n = sqrt(/\_n) =2...
  24. J

    Understanding Lagrange Error Analysis

    In my BC calc class, we just finished working through most of series and sequences, and as we were reviewing years past free response questions on the topic, and in 2004, they dropped a lagrange error analysis. I've been looking at different explanations, but I'm not getting the concept. It...
  25. P

    Maximizing and Minimizing Functions with Lagrange Multipliers

    Hey guys, i need some help with this problem. It goes as follow: Find the global max and min valves of the fuction z=x^2+2y^2 on the circle x^2+y^2=1. Ok and here is what i have done. I found the derivites and have done (K is the lagrange constant) 2x=2xK K=1 4y=2yK K=2 Then i set...
  26. P

    Max/Min Values for f(x,y,z): Lagrange Multipliers

    Greetings all, Find the max and min values of f(x,y.z)=3x-y-3z subject to x+y-z=0, x^(2)+2z^(2)=1 can anybody help me get this problem started. thanks
  27. P

    Lagrange multipliers and triangles

    Use Lagrange Multipliers to prove that the triangle with the maximum area that has a given perimeter p is equilateral. [Hint: Use Heron’s formula for the area of a triangle: A = sqrt[s(s - x)(s - y)(s - z)] where s = p/2 and x, y, and z are the lengths of the sides.] I have no idea how to do...
  28. J

    How to Use LaGrange Multipliers to Find Highest and Lowest Points on an Ellipse?

    It has been a while and trying to brush up on LaGrange points. I want to find the highest and lowest points on the ellipse of the intersection of the cone: x^2+y^2-z^2 ;subject to the single constraint: x+2y+3z=3 (plane). I want to find the points and I am not concerned with the minimum and...
  29. J

    Find Min of f(x,y,z)=x^8+y^8+z^8 on x^4+y^4+z^4=4

    Find the points at which the function f(x,y,z)=x^8+y^8+z^8 achieves its minimum on the surface x^4+y^4+z^4=4. I know 8x^7=(lamda)4x^3 8y^7=(lamda)4y^3 8z^7=(lamda)4z^3 x^4+y^4+z^4=4 Case1: x not equal to 0, y not equal to 0, and z not equal to 0 I get 3(4th root of 4/3 to the eigth)...
  30. D

    Solve Crazy Lagrange Problem: Min Distance from Origin

    -------------------------------------------------------------------------------- Ok this is the question I had on a test today: given this constraint equation z^2-xy+1=0 find the min. distance from the origin using Lagrange method. so basically you use D^2=x^2+y^2+z^2 as the other...
  31. S

    Pendulum Using Lagrange And Hamilton

    i have been given a problem involving a pendulum, where its support point is accelerating vertically upward. The period of the pendulum is required. Anybody have any idea how to start this one? is it not just 2pi(L/g-a)^1/2?
  32. T

    How to Maximize ln x + ln y + 3 ln z on a Sphere Using Lagrange Multipliers?

    A bit of a tough one! Find the maximum of ln x + ln y + 3 ln z on part of the sphere x^2 + y^2 + z^2 = 5r^2 where x>0, y>0 and z>0. I know I need to use Lagrange multipliers but how should I go about it? Any help would be appreciated thanks!
  33. S

    Lagrange' s eequations of a suspended set of rods

    Lagrange' s equations of a suspended set of rods Three identical rods of length l and mass m are hinged together so as to form three sides of a square in a vertical plane. The two upper free ends are hinged to a rigid support. The system is free to move in its own plane. Use Lagrange' s...
  34. S

    Lagrange ' s equations for spring problem

    Ok, there are two objects of mass m on a frictionless table. The 2 masses are connected to the other by a spring of spring constant k. One mass is connected to a wall with a spring of the same constant k. Solve for the motion using Lagrange' s equations. I used generalized coordinates x...
  35. E

    Lagrange equation of motion question

    A smooth wire is bent into the form of a helix the equations of which, in cylindrical coordinates, are z=a*beta and r=b , in which a and b are constants. The origin is a center of attractive force, , which varies directly as the distance, r. By means of Lagrange’s equations find the motion...
  36. K

    Definition of Lagrange function

    I'm learning classical mechanics right now, and I have a question about the "initial definition" of the Lagrange function. In my book, it is only introduced as L=T-V, but in many cases this doesn't help a lot, since it's not obvious what T or V is. For example, how would I come to the Lagrangian...
  37. F

    Hamiltonian and Lagrangian Mechanics: Online Resource

    Does anyone know of a good online resource on Lagrangian and Hamiltonian mechanics?
  38. A

    How do I solve a Lagrange Multiplier problem with a given constraint?

    We have started to do Lagrange Multi. in my class and my book has a very short section on how to solve these. I was wondering if someone couls help. The problem is f(x,y)=x^2-y^2 with the constraint x^2+y^2=1. I have found the partial derv. but I am not sure on what else to do. Any help would...
  39. Greg Bernhardt

    How many LaGrange points does the Earth/Moon system have?

    A)How many LaGrange points does the Earth/Moon system have? B) How many are stable (no stationkeeping neccessary)? C) Roughly where are the stable points located? 1/2 point point for each answered correctly
  40. I

    Lagrange multipliers elliptic paraboloid

    Hi, I'm really stuck on this problem and I need some help?? Here's the question: The intersection of the elliptic paraboloid z=x^2+4y^2 and the right circular cylinder x^2+y^2=1. Use Lagrange multipliers to find the highest and lowest points on the curve of intersection. Your help will...
  41. D

    Understanding Lagrange Multipliers: Solving for Max and Min Values

    Find max and min value…f(x,y,z) =3x+2y+z; x2 + y2+z2 = 1 If g(x,y,z) = x2 + y2+z2 = 1 then what do I do next? I need help to further solve for this please? I am horrible at math and don't understand lagrange multipiers so can anyone better explain it to me and help me solve for difficult...
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