The Lagrangian description of a dynamical system from the equation of motion consists of an inverse problem in mechanics. This problem is solved for a class of exactly integrable mixed and quadratic Liénard type oscillator equations from a given first integral of motion. The dynamics of this class of equations, which contains the generalized modified Emden equation, also known as the second-order Riccati equation, and the inverted versions of the Mathews-Lakshmanan equations, is then investigated from Hamiltonian and Lagrangian points of view.
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