The Euler–Lagrange equation was developed in the 1750s by Euler and Lagrange in connection with their studies of the tautochrone problem. This is the problem of determining a curve on which a weighted particle will fall to a fixed point in a fixed …

In physics, Lagrangian mechanics is a formulation of classical mechanics founded on the stationary-action principle (also known as the principle of least action).

The quantity \( L=T-V\) is known as the lagrangian for the system, and Lagrange’s equation can then be written \[ \dfrac{d}{dt}\dfrac{\partial L}{\partial \dot{q}_{j}}-\dfrac{\partial L}{\partial q_{j}}=0. \label{13.4.14} \] This form of the equation is seen more often in theoretical discussions than in the practical solution of problems.

This paper is intended as a minimal introduction to the application of Lagrange equations to the task of finding the equations of motion of a system of rigid bodies. A much more thorough and rigorous treatment is given in the text “Fundamentals of Applied Dynamics” by Prof. James H. Williams, Jr., published in 1996 by John Wiley and Sons.

The principle of Lagrange’s equation is based on a quantity called “Lagrangian” which states the following: For a dynamic system in which a work of all forces is accounted for in the Lagrangian, an admissible motion between specific configurations of the system at time t1 and t2

2021年6月28日 · Lagrange’s symbol δ is used to designate a virtual displacement which is called "virtual" to imply that there is no change in time t, i.e. δt = 0. This distinguishes it from an actual displacement dri of body i during a time interval dt when the forces and constraints may change.

2024年3月28日 · The Euler-Lagrange equation was given to you as: \[\begin{aligned} \frac{d}{dt}\left(\frac{\partial L}{\partial v_{x}}\right)-\frac{\partial L}{\partial x} = 0\end{aligned}\] because we are working in one dimension.