What is Lagrange's equation: Definition and 26 Discussions

In the calculus of variations and classical mechanics, the Euler-Lagrange equations is a system of second-order ordinary differential equations whose solutions are stationary points of the given action functional. The equations were discovered in the 1750s by Swiss mathematician Leonhard Euler and Italian mathematician Joseph-Louis Lagrange.
Because a differentiable functional is stationary at its local extrema, the Euler–Lagrange equation is useful for solving optimization problems in which, given some functional, one seeks the function minimizing or maximizing it. This is analogous to Fermat's theorem in calculus, stating that at any point where a differentiable function attains a local extremum its derivative is zero.
In Lagrangian mechanics, according to Hamilton's principle of stationary action, the evolution of a physical system is described by the solutions to the Euler equation for the action of the system. In this context Euler equations are usually called Lagrange equations. In classical mechanics, it is equivalent to Newton's laws of motion, but it has the advantage that it takes the same form in any system of generalized coordinates, and it is better suited to generalizations. In classical field theory there is an analogous equation to calculate the dynamics of a field.

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  1. deuteron

    Solving two body central force motion using Lagrangian

    For the central force ##F=-\nabla U(r_r)## where ##\vec r_r=\vec r_1-\vec r_2##, and ##\vec r_1## and ##\vec r_2## denote the positions of the masses, we get the following kinetic energy using the definition of center of mass ##\vec r_{cm}= \frac{m_1\vec r_1+m_2\vec r_2}{m_1+m_2}##: $$T= \frac...
  2. Pironman

    I How to find the equation of motion using Lagrange's equation?

    Good morning, I'm not a student but I'm curious about physics. I would like to calculate the equation of motion of a system using the Lagrangian mechanics. Suppose a particle subjected to some external forces. From Wikipedia, I found two method: 1. using kinetic energy and generalized forces...
  3. gromit

    Using Lagrangian to show a particle has a circular orbit

    Hi :) This is a problem from David Tong's Classical Dynamics course, found here: http://www.damtp.cam.ac.uk/user/tong/dynamics.html. In particular it is problem 6ii in the first problem sheet. Firstly we can see the lagrangian is ##L = \frac{1}{2}m(\dot{r}^2+r^2\dot{\theta}^2+\dot{z}^2) -...
  4. K

    A Two equations of generalized forces

    Wikipedia article under generalized forces says Also we know that the generalized forces are defined as How can I derive the first equation from the second for a monogenic system ?
  5. K

    I Independent coordinates are dependent

    (This is not about independence of ##q##, ##\dot q##) A system has some holonomic constraints. Using them we can have a set of coordinates ##{q_i}##. Since any values for these coordinates is possible we say that these are independent coordinates. However the system will trace a path in the...
  6. juardilag

    Help with a modified Kepler potential

    I have tried to solve the problem through the use of a rotating reference frame, since I should have as a solution an orbit given by the Kepler potential, but I haven't come up with anything. Any ideas ?
  7. K

    Virtual work in Atwood's machine

    The first chapter in Goldstein's Classical Mechanics ends with 3 examples about how to apply Lagrange's eqs. to simple problems. The second example is about the Atwood's machine. The book says that the tension of the rope can be ignored, but I don't understand why. The two masses can move...
  8. Amitayas Banerjee

    What is the Lagrangian, equations of motion for this system?

    <<Moderator's note: Moved from a technical forum, no template.>> Description of the system: The masses m1 and m2 lie on a smooth surface. The masses are attached with a spring of non stretched length l0 and spring constant k. A constant force F is being applied to m2. My coordinates: Left of...
  9. Phylosopher

    Conservation laws from Lagrange's equation

    My question is related to the book: Classical Mechanics by Taylor. Section 7.8 So, In the book Taylor is trying to derive the conservation of momentum and energy from Lagrange's equation. I understood everything, but I am struggling with the concept and the following equation...
  10. Amitayas Banerjee

    Problem in getting correct coefficients of frictional forces

    I am getting correct equations on using the Lagrangian method in Systems with no non conservative forces, but when I use it in Systems with friction, sometimes I get correct equations, and sometimes I do not. Most of the equations have some problem with the coefficients of the frictional forces...
  11. G

    Relativistic Harmonic Oscillator Lagrangian and Four Force

    Homework Statement Consider an inertial laboratory frame S with coordinates (##\lambda##; ##x##). The Lagrangian for the relativistic harmonic oscillator in that frame is given by ##L =-mc\sqrt{\dot x^{\mu} \dot x_{\mu}} -\frac {1}{2} k(\Delta x)^2 \frac{\dot x^{0}}{c}## where ##x^0...
  12. P

    Is the Lagrange Equation Valid for All Holonomic Systems?

    Homework Statement Show that for an arbitrary ideal holonomic system (n degrees of freedom) \frac{1}{2} \frac{\partial \ddot T}{\partial\ddot q_j} - \frac{3}{2} \frac{\partial T}{\partial q_j} = Q_j where T is kinetic energy and qj generalized coordinates.[/B]Homework Equations...
  13. 1

    Help finding the vibrational frequencies and normal modes

    Homework Statement Let's say that I have a potential ##U(x) = \beta (x^2-\alpha ^2)^2## with minima at ##x=\pm \alpha##. I need to find the normal modes and vibrational frequencies. How do I do this? Homework Equations ##U(x) = \beta (x^2-\alpha ^2)^2## ##F=-kx=-m\omega ^2 x## ##\omega =...
  14. T

    I Lagrangian for electromagnetic field derivation

    " We can put the Lorentz force law into this form by being clever. First, we write $$\frac{dA_j}{dt}=\frac{d}{dt}(\frac{\partial{}}{\partial{v_j}}(v.A)),$$ since the partial derivative will pick out only the jth component of the dot product. Now, since the scalar potential is independent of the...
  15. A

    Lagrangian and Slonczewski-like torque

    To include slonczewski-like torque in the Lagrangian there enter as dissipation by Rayleig function? ST= σ m x (m x mp) where m is the magnetization in free layer and mp current direction in the pinned layer (-z). The Rayleig function is: RF=(dm/dt+σ m x mp)2 Then L= ∫ (RF+E)dx Thanks
  16. Q

    A Is this constraint nonholonomic or not?

    I really want to know whether this equation is nonholonomic or not. (As far as I know, Nonholonomic constraint has a term of velocity and do non-integrable. But this formula does not dependent on a path, because it is a total differential form.)
  17. L

    Derivation of Lagrange's eqs

    Homework Statement So I'm deriving Lagrange's equations using Hamilton's principle which states that the motion of a dynamical system follows the path, consistent with any constraints, that minimise the time integral over the lagrangian L = T-U, where T is the kinetic energy and U is the...
  18. mwspice

    Lagrange's Equation with Multiple Degrees of Freedom

    Hi, I'm currently trying to learn about finding equations of motion from the Lagrange equation, and I'm a little confused about how it applies to multiple degree of freedom systems. I am using the following form of the equation with T as total kinetic energy, V as total potential energy, R as...
  19. B

    Lagrange's Equation Generalized Coordinates

    Hello, I am currently reading about the topic alluded to in the topic of this thread. In Taylor's Classical Mechanics, the author appears to be making a requirement about any arbitrary coordinate system you employ in solving some particular problem. He says, "Instead of the Cartesian...
  20. N

    Comparing Lagrange's Equation of Motion and Euler-Lagrange Equations

    Hi What is the difference between Lagrange's equation of motion and the Euler-Lagrange equations? Don't they both yield the path which minimizes the action S? Niles.
  21. O

    Block on a Cylinder, using Lagrange's Equation

    Homework Statement A hard rubber cylinder of radius r is held fixed with its axis horizontal, and a wooden cube of mass m and side 2b is balanced on top of the cylinder, with its center vertically above the cylinder's axis and four of its sides parallel to the axis. Assuming that b < r, use...
  22. N

    Pendulum motion lagrange's equation

    i have been trying to solve this past exam problem, a simple pendulum of length l and bob with mass m is attracted to a massless support moving horizontally with constant acceleration a. Determine the lagrange's equations of motion and the period of small oscillations. here's what i solved...
  23. N

    Double Pendulum Lagrange's Equation Problem

    Homework Statement A double pendulum consists of two simple pendula, with one pendulum suspended from the bob of the other. If the two pendula have equal lengths and have bobs of equal mass and if both pendula are confined to move in the same plane, find Lagrange's equations of motion for the...
  24. N

    Lagrange's Equation of Motion

    Homework Statement two blocks each of mass m are connected by an extensionless uniform string of length l. one block is placed on a smooth horizontal surface and the other block hangs over the side the string passes over a frictionless pulley. describe the motion of the system when the mass of...
  25. S

    Problem about Lagrange's equation

    A pendulum consists of a mass m suspended by a massless spring with unextended length b and. spring constant k. Find Lagrange’s equations of motion Here's how I set up my equation: x = lsin(theta) y = -lcos(theta) (x=0 at equilibrium, y=0 at the point wehre the pendulum is hung from)...