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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 amount of time, independent of the starting point.
These equations for solution of a first-order partial differential equation are identical to the Euler–Lagrange equations if we make the identification = ˙ ˙. We conclude that the function ψ {\displaystyle \psi } is the value of the minimizing integral A {\displaystyle A} as a function of the upper end point.
In mathematics, the inverse problem for Lagrangian mechanics is the problem of determining whether a given system of ordinary differential equations can arise as the Euler–Lagrange equations for some Lagrangian function. There has been a great deal of activity in the study of this problem since the early 20th century.
However, the Euler–Lagrange equations can only account for non-conservative forces if a potential can be found as shown. This may not always be possible for non-conservative forces, and Lagrange's equations do not involve any potential, only generalized forces; therefore they are more general than the Euler–Lagrange equations.
Hilbert calls this a "regular variational problem". [9] Property means that these are minimum problems. Property is the ellipticity condition on the Euler–Lagrange equations associated to the given functional, while property is a simple regularity assumption about the function F. [10]
This is similar to solving the Euler–Lagrange equation with Dirichlet boundary conditions. Additionally there are settings in which there are minimizers in , (,) but not in , (,). The idea of solving minimization problems while restricting the values on the boundary can be further generalized by looking on function spaces where the trace is ...
Building on the early work of Pierre Louis Maupertuis, Leonhard Euler, and Joseph-Louis Lagrange defining versions of principle of least action, [34]: 580 William Rowan Hamilton and in tandem Carl Gustav Jacob Jacobi developed a variational form for classical mechanics known as the Hamilton–Jacobi equation. [35]: 201
Minimal surface theory originates with Lagrange who in 1762 considered the variational problem of finding the surface = (,) of least area stretched across a given closed contour. He derived the Euler–Lagrange equation for the solution