# 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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16. ### 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
17. ### 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.)
18. ### 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...
19. ### 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...
20. ### 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...
21. ### 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.
22. ### 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...
23. ### 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...
24. ### 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...
25. ### 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...
26. ### 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)...