A comprehensive study is reported herein for the evaluation of Lagrangian functions for linear systems possessing symmetric or nonsymmetric coefficient matrices. Contrary to popular beliefs, it is shown that many coupled linear systems do not admit Lagrangian functions. In addition, Lagrangian functions generally cannot be determined by system decoupling unless further restriction such as classical damping is assumed. However, a scalar function that plays the role of a Lagrangian function can be determined for any linear system by decoupling. This generalized Lagrangian function produces the equations of motion and it contains information on system properties, yet it satisfies a modified version of the Euler–Lagrange equations. Subject to this interpretation, a solution to the inverse problem of linear Lagrangian dynamics is provided.

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