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
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