Douglas' Theorem and The Helmholtz Conditions
To simplify the notation, let
and define a collection of n2 functions Φji by
Theorem. (Douglas 1941) There exists a Lagrangian L : × TM → R such that the equations (E) are its Euler–Lagrange equations if and only if there exists a non-singular symmetric matrix g with entries gij depending on both u and v satisfying the following three Helmholtz conditions:
(The Einstein summation convention is in use for the repeated indices.)
Read more about this topic: Inverse Problem For Lagrangian Mechanics
Famous quotes containing the words theorem and/or conditions:
“To insure the adoration of a theorem for any length of time, faith is not enough, a police force is needed as well.”
—Albert Camus (1913–1960)
“If there is a species which is more maltreated than children, then it must be their toys, which they handle in an incredibly off-hand manner.... Toys are thus the end point in that long chain in which all the conditions of despotic high-handedness are in play which enchain beings one to another, from one species to another—cruel divinities to their sacrificial victims, from masters to slaves, from adults to children, and from children to their objects.”
—Jean Baudrillard (b. 1929)