# What is Lagrange: Definition and 537 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. ### Generalized coordinates and the Lagrangian

So I think the mass can only move in two "coordinates" the axis of which the mass is connected to ##k_1## and the axis connecting it to ##k_2##. Therefore, the D.O.F is 2. I don't understand what it the meaning of "variables of integration" What does it mean? Apart from that, I attempted to...
2. ### I Requirement of Holonomic Constraints for Deriving Lagrange Equations

While deriving the Lagrange equations from d'Alembert's principle, we get from $$\displaystyle\sum_i(m\ddot x_i-F_i)\delta x_i=0\tag{1}$$ to $$\displaystyle\sum_k (\frac {\partial\mathcal L}{\partial\ q_k}-(\frac d {dt}\frac {\partial\mathcal L}{\partial\dot q_k}))\delta q_k=0\tag{2}$$ However...

16. ### A Euler Lagrange and the Calculus of Variations

Good Morning all Yesterday, as I was trying to formulate my confusion properly, I had a series of posts as I circled around the issue. I can now state it clearly: something is wrong :-) and I am so confused :-( Here is the issue: I formulate the Lagrangian for a simple mechanical system...

33. ### Finding Specific Extrema when grad(F)=constant & Lagrange Gives y=-z/2

I found that f= x -2yz. To maximize f, I can first inspect the solutions to grad(F)=0. z=y=0 pops out, but I'm not sure what to do with the x-component equaling 1. Do we just include (x,0,0) as a solution? I think the problem wants specifics though, based on what I've seen previously from...
34. ### Lagrange method to find extremes

ƒ(x,y) = 3x + y x² + 2y² ≤ 1 It is easy to find the maximum, the really problem is find the minimum, here is the system: (3,1) = λ(2x,4y) x² + 2y² ≤ 1 how to deal with the inequality?
35. ### I Lagrangian and the Euler Lagrange equation

I am new to Lagrangian mechanics and I am unable to comprehend why the Euler Lagrange equation works, and also what really is the significance of the lagrangian.
36. ### Small deviations from equilibrium and Lagrange multipliers

According to the book "Principles of Statistical Mechanics" by Amnon Katz, page 123, ##\alpha## must be such that ##\exp ( -\alpha N ) ## can be expanded in powers of ##\alpha## with only the first order term kept. Is this the necessary and sufficient condition for small deviations from...
37. ### Euler Lagrange equation and a varying Lagrangian

Hello, I have been working on the three-dimensional topological massive gravity (I'm new to this field) and I already faced the first problem concerning the mathematics, after deriving the lagrangian from the action I had a problem in variating it Here is the Lagrangian The first variation...
38. ### A The tautological 1-form: Lagrange vs. Hamilton formalism

Classical mechanics is based on conservation laws which represent the symmetries of spacetime. The lagrangian function L is a function of position and velocity while the hamiltonian is a function of position and momentum. The velocity and momentum descriptions are related by a legendre...
39. ### Lagrange equations of the first kind

We cannot make it anyhow
40. ### How to model a function of a box's volume using Lagrange multiplier methods

I started to understand how to apply Lagrange multiplier methods. But, for problem like this, I have difficulty to build the function to describe the volume that will be maximized. For the second question, I know from the example (in ML Boas) that ##V=8xyz## becase (x,y,z) is in the 1st octant...
41. ### Lagrange equations: Two blocks and a string

I've problems understanding why the kinetic energy of the string is only $$T_{string}=\frac{1}{2}m\dot{y}$$ Why the contribution of the string in the horizontal line isn't considered?
42. ### I Question about Lagrange multipliers

I'm having some trouble understanding the following proof (##a_{ik}## and ##b_{ik}## are constants) Shouldn't it be ##a_{ik}q_iq_k - \frac 1 {\lambda} (b_{ik}q_iq_k-1)## ? (Summation convention is used) Thanks Ric
43. ### Lagrange Multipliers Problem

Hi there! Kindly help me to solve the problem below. A company is using frustum of a cone containers for their products. What are the dimensions of the least expensive container that can hold 300 cubic cm? Use Lagrange Multipliers to solve the problem. Thanks.
44. ### D'Alembert's principle and the work done by constraint forces

From what I understand, constraint forces do no work because they are perpendicular to the allowed virtual displacements of the system. However, if you consider an unbalanced Atwood machine, in which both masses are accelerating in opposite directions, you'll find that the tension force of the...
45. ### Lagrange Mechanic Dynamics

A homogen box with the mass M rolls without sliding on two round wheels. The wheels with mass mass m are also homogen and roll without sliding, on top of the banked Surface. We use Gravitation g. Find the accelration xM of the box I don't know which solution is correct. i got 0.67 m for xM...
46. ### Deriving the Equation of Motion out of the Action

Exercise statement: Given the action (note ##G_{ab}## is a symmetric matrix, i.e. ##G_{ba} = G_{ab}##): $$S = \int dt \Big( \sum_{ab} G_{ab} \dot q^a\dot q^b-V(q)\Big)$$ Show (using Euler Lagrange's equation) that the following equation holds: \ddot q^d +...
47. ### Horizontal Circular Motion With Lagrange

In the situation described in the problem, the mass is moving on a horizontal circular path with constant velocity. Wouldn’t this make L and U both constant? Then the Lagrange equation would give 0 = 0, which isn’t what I’m looking for. Any help would be appreciated.
48. ### Lagrange equation, of a hoop?

I couldn't even get the position vector. Help!
49. ### Lagrange Equations of Motion for a particle in a vessel

The final answer should have a negative b^2⋅r(dot)^2⋅r term but I have no idea how that term would become negative. Also I know for a fact that my Lagrangian is correct.
50. ### Lagrange Equations of Motion for a particle in a vessel

I start out by substituting rcos(Θ) and rsin(Θ) for x and y respectively. This gives me z=(b/2)r^2. The Lagrangian of this system is (1/2)m(rdot^2+r^2⋅Θdot^2+zdot^2)-mgz. (rdot and such is the time derivative of said variable). I then find the time derivative of z, giving me zdot=br⋅rdot and...