Multiplicative form of the Lagrangian

We obtain an alternative class of Lagrangians in the so-called the multiplicative form for a system with one degree of freedom in the nonrelativistic and the relativistic cases. This new form of the Lagrangian can be regarded as a one-parameter class with the parameter $\lambda$ obtained using an extension of the standard additive form of the Lagrangian because both forms yield the same equation of motion. We note that the multiplicative form of the Lagrangian can be regarded as a generating function for obtaining an infinite hierarchy of Lagrangians that yield the same equation of motion. This nontrivial set of Lagrangians confirms that the Lagrange function is in fact nonunique.