  An alternative class of the Lagrangian called the multiplicative form is suc-
cessfully derived for a system with one degree of freedom for both
non-relativistic and relativistic cases. This new Lagrangian can be considered
as a 1-parameter: {\lambda} extended class from the standard additive form of
the Lagrangian since both yield the same equation of motion. Remarkably, the
multiplicative form of the Lagrangian could be treated as a generating function
to produce an infinite hierar- chy of Lagrangians which give the same equation
of motion. This nontrivial set of Lagrangians confirms that indeed Lagrange
function is not unique.
