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. O

    A Lagrange vs Hamilton: Clarifying the Distinction

    Hello, When doing a little internet search today on generalized coordinates I stumbled on this document: http://people.duke.edu/~hpgavin/cee541/LagrangesEqns.pdf If you are willing, would you be so kind as to open it up and look at the top of (numbered) page 6? OK, so the very existence of...
  2. O

    A Generalized Coordinates and Porn

    Yes, that is a serious title for the thread. Could someone please define GENERALIZED COORDINATES? In other words (and with a thread title like that, I damn well better be sure there are other words ) I understand variational methods, Lagrange, Hamilton, (and all that). I understand the...
  3. defaultusername

    Lagrange Multipliers / Height of a Rocket

    Homework Statement I am going to paste the problem word for word, so you can have all the exact information that I have: You’re part of a team that’s designing a rocket for a specific mission. The thrust (force) produced by the rocket’s engine will give it an acceleration of a feet per second...
  4. C

    Finding the geodesic equation from a given line element

    Homework Statement We've got a line element ds^2 = f(x) du^2 + dx^2 From that we should find the geodesic equation Homework Equations Line Element: ds^2 = dq^j g_{jk} dq^k Geodesic Equation: \ddot{q}^j = -\Gamma_{km}^j \dot{q}^k \dot{q}^m Christoffel Symbol: \Gamma_{km}^j = \frac{g^{jl}}{2}...
  5. M

    I Making Eulers eqs. comply with Lagrange eqs.

    Lately when doing a simulation for a quadrocopters most reports I've come across regarding modeling use Eulers equation of motion. That makes sense, as the quadrocopter is a body rotating in 3 dimensions. Then I tried to model the system using Lagrange equations instead but I don't get the...
  6. JulienB

    Barbell in gravitational field (Lagrange)

    Homework Statement Hi everybody! Here is a new Lagrange problem I am trying to solve, and I would like to have your opinion about my solution so far! A barbell composed of two masses ##m_1## and ##m_2##, idealised as particles and separated by a distance ##a## from each other, moves in the...
  7. C

    Discrete Lagrangian Homework: Minimize S, Find EoM's & Discrete Trajectory

    Homework Statement In this exercise, we are given a discrete Lagrangian which looks like this: http://imgur.com/TL0P61r. We have to minimize the discrete S with fixed point r_i and r_f and find the the discrete equations of motions. In the second part we should derive a discrete trajectory for...
  8. D

    Lagrangian of a centrifugal regulator

    Homework Statement Homework Equations L = T-V The Attempt at a Solution I got a forumla for the lagrangian as
  9. N

    I Solve Lagrange Interpolation Problem with Pen Position Detection

    Hi all I am facing a problem and I hope that you can give me a hand. Here I describe the situation I am working on a digitizer that can detect the pen position by measuring the antennae energy that are placed in a grid fashion. To get the x coordinate of the pen I measure the energy of three...
  10. S

    I Lagrange Multiplier where constraint is a rectangle

    Hello, How can I use Lagrange Multipliers to get the Extrema of a curve f(x,y) = x2+4y2-2x2y+4 over a rectangular region -1<=x<=1 and -1<=y<=1 ? Thanks
  11. 1

    1-D Lagrange and Hamilton equation gives different results.

    Homework Statement This was supposed to be an easy question. I have a question here that wants you to describe a yoyo's acceleration (in one dimension) using Lagrangian mechanics. I did and got the right answer. Now I want to use Hamilton's equations of motion but I get a wrong number. Here is...
  12. 1

    How do I set up this Legendre Transform for Hamiltonian

    Homework Statement Im trying to understand the Legendre transform from Lagrange to Hamiltonian but I don't get it. This pdf was good but when compared to wolfram alphas example they're slightly different even when accounting for variables. I think one of them is wrong. I trust wolfram over the...
  13. a255c

    Lagrange optimization: cylinder and plane intersects,

    Homework Statement The cylinder x^2 + y^2 = 1 intersects the plane x + z = 1 in an ellipse. Find the point on the ellipse furthest from the origin. Homework Equations $f(x) = x^2 + y^2 + z^2$ $h(x) = x^2 + y^2 = 1$ $g(x) = x + z = 1$ The Attempt at a Solution $\langle 2x, 2y, 2z \rangle...
  14. H

    I Lagrange multiplier in Hamilton's and D'Alembert's principles

    Why do displaced paths need to satisfy the equations of constraint when using the method of Lagrange multiplier? I thought that with the multiplier, all the coordinates are free and hence should not be required to satisfy the equations of constraint. Source...
  15. evinda

    MHB Prove Lagrange Property w/ Algebra - A Hint for You!

    Hello! (Wave) With use of algebra I want to prove the Lagrange property:For any real numbers $x_1, \dots, x_n$ and $y_1, \dots, y_n$, $$\left( \sum_{i=1}^n x_i y_i\right)^2=\left(\sum_{i=1}^n x_i^2 \right)\left(\sum_{i=1}^n y_i^2 \right)- \sum_{i<j} (x_i y_j-x_j y_i)^2$$ Could you give me a...
  16. S

    Equations of motion for 4 dof

    Hi all, I'm working on a project to control the angles of a beam(purple) with a quadcopter(orange),see figure below. The angles for both the ground-beam and beam-quadcopter will be measured with joysticks, so only roll and pitch angles will be measured and the yaw rotation is fixed. To obtain...
  17. Alain De Vos

    Line element and derivation of lagrange equation

    With coordinates q en basis e ,textbooks use as line element : ds=∑ ei*dqi But ei is a function of place, as one can see in deriving formulas for covariant derivative. Why don't they use as line element the correct: ds=∑ (ei*dqi+dei*qi) Same question in deriving covariant derivative,
  18. bananabandana

    Euler Lagrange Derivation (Taylor Series)

    Mod note: Moved from Homework section 1. Homework Statement Understand most of the derivation of the E-L just fine, but am confused about the fact that we can somehow Taylor expand ##L## in this way: $$ L\bigg[ y+\alpha\eta(x),y'+\alpha \eta^{'}(x),x\bigg] = L \bigg[ y, y',x\bigg] +...
  19. H

    Help with contour plots of effective potential in R3BP.

    Hello everyone! I'm currently trying to plot the effective potential for Sun-Jupiter system, to show the lagrangian points in this system. I've converted to a system of units where G=1, m_sun+m_jupiter=1 and R=1, whereby I get the following equation describing the effective potential of a third...
  20. M

    Lagrange multipliers method?

    regarding question number 10, we have h = f + λg where g is the constraint (the ellipsoid) and f is the function we need to maximize or minimize (the rectangular parallelpiped volume), now my question : is it right that f is 8xyz ? i mean if we take f to be xyz not 8xyz and solved till we got...
  21. F

    Proving Euler Lagrange equations

    Hi everybody; I am looking for the deduction of the euler lagrange equations (d/dt)(∂L/∂v)-(∂L/∂x) from the invariance of the action δ∫Ldt=0. Can someone please tell me where can I find It? Thanks for reading.
  22. H

    MHB Lagrange nodal basis question.

    $\text{Let } L_{n,i}, i = 0,...,n, \text{be the Lagrange nodal basis at} x_0 < x_1<...<x_n$. Show that, for any polynomial $q \in P_n$ $$\sum_{i=0}^nq(x_i)L_{n,i}(x)= q(x)$$ I don't know how to begin this proof. I know what a lagrange polynomial is, but I am not sure how to begin. If someone...
  23. M

    Grassmann Integral into Lagrange for scalar superfields

    I have a more philosophical question about the interpretation of a mathematical process. We have a chiral superscalarfield shown as partiell Grassmann Integral and transform it into a lagrange. where S and P are real components of a complex scalarfield and D and G are real componentfields of...
  24. ognik

    MHB Learning Lagrange Mesh Method for 1D Schrödinger Eqtn.

    Hi, I have been trying to find some articles that would cover the Lagrange mesh method applied to the 1-D Schrödinger eqtn. - using the Laguerre mesh. I want to develop some fortran programs, for example building Lagrange functions, the kinetic energy matrix elements ... LMM is totally new to...
  25. G

    Lagrange equation of motion for tensegrity

    Hi, I have read this paper “Dynamic equations of motion for a 3-bar tensegrity based mobile robot” (1) and this one “Dynamic Simulation of Six-strut Tensegrity Robot Rolling”. 1) http://digital.csic.es/bitstream/10261/30336/1/Dynamic%20equations.pdf I am trying to implement a tensegrity...
  26. S

    Virtual Work & Quadcopter Torques: Exploring Rotational Dynamics

    Here is what we know from virtual work: $$ \delta W=\sum_{i=1}^N{\vec F_i\cdot\delta\vec r_{i}} $$ Where ##N## is the number of bodies in the system. I am considering a quadcopter, modeled as a rigid body so it is just one body and we have: $$ \delta W=\vec F\cdot\delta\vec r $$ My question...
  27. hideelo

    Deriving Commutation of Variation & Derivative Operators in EL Equation

    I am trying to do go over the derivations for the principle of least action, and there seems to be an implicit assumption that I can't seem to justify. For the simple case of particles it is the following equality δ(dq/dt) = d(δq)/dt Where q is some coordinate, and δf is the first variation in...
  28. P

    What would it be like to be at a Lagrange Point?

    According to this NASA factsheet (http://www.nasa.gov/pdf/664158main_sls_fs_master.pdf ) on the Space Launch System (SLS), NASA identifies missions to a Lagrange point as a possibility. From what I understand, a Lagrange point is simply a point where the gravitational fields of two massive...
  29. BiGyElLoWhAt

    Excellent video series raises good question:

    www.youtube.com/watch?v=oW4jM0smS_E That's the video I'm referencing in particular, but 1 and 3 are necessary prereqs if you're new to the matter (as I am). He goes through and derives the product rule and power rule for polynomials using algebra. My question is this: why don't we teach...
  30. Amat3r

    How Do You Model a Torsion Spring in a Pendulum's Lagrangian?

    Hello! I'd like to ask for help with one problem :) thank you in advance. 1. Write the equations for kinetic and potential energy for the pendulum with rectangular prism of size a*b*c (width, length, depth). With the Lagrange's equation get the equation of motion. The block is homogeneous...
  31. Y

    MHB Solve Lagrange Multipliers Problem with x-4y=1 Constraint

    Hello all I have this problem: Use Lagrange Multipliers to find the min and max of: \[f(x,y)=xy^{2}\] under the constraint: \[x-4y=1\] \[-1\leqslant x\leq 2\] My problem is: I know how to solve if \[-1\leqslant x\leq 2\] wasn't given. I calculate the Lagrangian function, find it's...
  32. B

    Euler Lagrange equation of motion

    Homework Statement Find the equations of motion for both r and \theta of Homework Equations My problem is taking the derivative wrt time of and \dfrac{\partial\mathcal{L}}{\partial\dot{r}}=m \dot{r} \left( 1 + \left( \dfrac{\partial H}{\partial r}\right)^2 \right) The Attempt at a...
  33. M

    Classical Mechanics Notes needed:

    Hello Seniors, I have done BSc in Physics but couldn't take lectures of Classical Mechanics. I am Almost blind in this subject. Since it's a core course in Physics, so i need your help to understand the basics in this course. If anyone of you have any helping material/notes/slides etc which...
  34. amjad-sh

    Lagrange and Hamiltonian formulations

    Hello. Do I need to grasp well the Lagrangian and Hamiltonian formulations in classical mechanics to go through quantum mechanics without struggles?
  35. Dethrone

    MHB Lagrange Multiplier Ellipsoid

    Use Lagrange multipliers to find $a,b,c$ so that the volume $V=\frac{4\pi}{3}abc$ of an ellipsoid $\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$, passing through the point $(1,2,1)$ is as small as possible. I just need to make sure my setup is correct. $\triangledown...
  36. B

    Constant solutions to the Lagrange equations of motion

    Homework Statement A particle moves on the surface of an inverted cone. The Lagrangian is given by Show that there is a solution of the equations of motion where and take constant values and respectively Homework Equations The equations of motion and are (1) (2) So can be...
  37. A

    The pendulum using Lagrange in cartisian

    I am newly learning Lagrange formalism and I learned how to get the equation of motion for a simple pendulum using Lagrange in the spherical coordinate system. But I am unable to derive the same using the Cartesian system. If someone can please tell me what is wrong with the following...
  38. B

    Lagrange equation particle on an inverted cone

    Homework Statement Derive the equations of motion and show that the equation of motion for \phi implies that r^2\dot{\phi}=K where K is a constant Homework Equations Using cylindrical coordinates and z=\alpha r The kinetic and potential energies are...
  39. T

    Understanding the Euler Lagrange Equation and Its Boundary Condition

    I am trying to derive it but I am stuck at the boundary condition. What is this boundary comdition thing such that the value must be zero?
  40. M

    Calculus of Variations & Lagrange Multiplier in n-dimensions

    extremize $$S = \int \mathcal{L}(\mathbf{y}, \mathbf{y}', t) dt $$ subject to constraint $$g(\mathbf{y}, t) = 0 $$ We move away from the solution by $$y_i(t) = y_{i,0}(t) + \alpha n_i(t) $$ $$\delta S = \int \sum_i \left(\frac{\partial\mathcal{L} }{\partial y_i} - \frac{d}{dt} \frac{\partial...
  41. kostoglotov

    Discontinuity of a constraint in Lagrange Method

    Homework Statement My question is quite specific, but I will include the entire question as laid out in the text Consider the problem of minimizing the function f(x,y) = x on the curve y^2 + x^4 -x^3 = 0 (a piriform). (a) Try using Lagrange Multipliers to solve the problem (b) Show that the...
  42. throneoo

    Constrained Extrema and Lagrange Multipliers

    Suppose I have a function f(x,y) I would like to optimize, subject to constraint g(x,y)=0. Let H=f+λg, The extrema occurs at (x,y) which satisfy Hy=0 Hx=0 g(x,y)=0 Suppose the solutions are (a,b) and (c,d). If f(a,b)=f(c,d) , how do I determine whether they are maxima or minima?
  43. Coffee_

    A Lagrange multiplier approach to the catenary problem

    In general, when dealing with mechanics problems using a function ##f(q1,q2,...)=0## that represent constraints one is minimizing the action ##S## while adding a term to the Lagrangian of the not-independent coordinates ##L + \lambda f ##. One can show that this addition doesn't change the...
  44. Coffee_

    How can I know when the Lagrange multiplier is a constant?

    Consider a holonomic system where I have ##n## not independent variables and one constraint ##f(q1,q2,...,qN,t)=0##. One can rewrite the minimal action principle as: ##\frac{\partial L}{\partial q_i} - \frac{d}{dt} \frac{\partial L}{\partial q'_i} - \lambda \frac{\partial f}{\partial q_i} = 0...
  45. S

    Lagrange EOM for 2 masses on a string

    Homework Statement Derive the equation of motion for the system in figure 6.4 using Lagrange's equations [/B] Homework Equations m1=.5m m2=m strings are massless and in constant tension Lagrange=T-V The Attempt at a Solution I currently have the kinetic energy as .5m1y'12 + .5m2y'22 I am...
  46. S

    Lagrange for Rod/nail swinging from horizontal plane

    Hi! I need to figure out the Lagrange Equation for a rod or nail swinging from a horizontal plane. The thing is, that while it is swinging back and forth, the while nail is moving along the X axis as well. I was thinking to use 1/2mv^2+(1/2)Iø^2 . Any help would be appreciated! Thanks.
  47. I

    Lagrange multipliers, guidance needed

    Homework Statement f(x,y) is function who's mixed 2nd order PDE's are equal. consider k_f: determine the points on the graph of the parabloid f(x,y) = x^2 + y^2 above the ellipse 3x^2 + 2y^2 = 1 at which k_f is maximised and minimised. The Attempt at a Solution is this the langrange...
  48. V

    Violation in diffraction? Lagrange (Optical) invariant

    It says you can not change with lenses the value L - radiance. Below I have an example where it proves that you can or where am I wrong? (I made L for 2D case, in 3D case everything the same - L2>L1)
  49. J

    Can Euler-Lagrange Equations Explain Mirages?

    Homework Statement On very hot days there sometimes can be a mirage seen hovering as you drive. Very close to the ground there is a temperature gradient which makes the refraction index rises with the height. Can we explain the mirage with it? Which unit do you need to extremalise? Writer the...
  50. C

    Lagrange Multipliers: Deriving EOM & Conditions for Contact Loss

    Homework Statement An object of mass m, and constrained to the x-y plane, travels frictionlessly along a curve f(x), while experiencing a gravitational force, m*g. Starting with the Lagrangian for the system and using the method of Lagrange multipliers, derive the equations of motion for the...
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