Complex Lagrangian dynamics

In this article one introduces a formalism of classical mechanics where complex Lagrangian functions are admitted. The results include complex versions of the Lagrangian function, of the Euler-Lagrange equation, of the Hamilton principle, a geometric formulation, and the relation to a previous complex Hamiltonian formalism. The framework is particularly suitable for non-stationary motion, and various pathways can be followed in future investigation.

💡 Research Summary

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The paper proposes a systematic extension of classical mechanics by allowing the Lagrangian to be a complex‑valued function. Starting from the standard real Lagrangian (L(q,\dot q,t)) and its Euler‑Lagrange equations, the author introduces a complex coordinate
(w=\frac{1}{\sqrt{2}}(\dot q+i\omega_{0}q))
and the associated derivative (\partial/\partial w). The complex Lagrangian is defined as
(\mathcal L = L + iM)
where both (L) and (M) are real functions of the usual variables. The fundamental dynamical equation becomes
(u = i\omega_{0},\partial\mathcal L/\partial w)
with (u = \frac{1}{\sqrt{2}}(\dot p + i\omega_{0}p)). Splitting into real and imaginary parts yields a pair of modified Euler‑Lagrange equations:

\