We consider the inverse problem of the calculus of variations for any system by writing its differential equations of motion in first-order form. We provide a way of constructing infinitely many Lagrangians for such a system in terms of its constants of motion using a covariant geometrical approach. We present examples of first-order Lagrangians for systems for which no second-order Lagrangians exist. The Hamiltonian theory for first-order (degenerate) Lagrangians is constructed using Dirac’s method for singular Lagrangians.
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Hojman et al. (1981) studied this question.
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