A method for constructing general null Lagrangians and their higher harmonics is presented for dynamical systems with one degree of freedom. It is shown that these Lagrangians can be used to obtain non-standard Lagrangians, which give equations of motion for the law of inertia and some dissipative dynamical systems. The necessary condition for deriving equations of motion by using null Lagrangians is presented, and it is demonstrated that this condition plays the same role for null Lagrangians as the Euler-Lagrange equation plays for standard and non-standard Lagrangians. The obtained results and their applications establish a novel role of null Lagrangians in classical dynamics.

The space of null Lagrangians is the least investigated territory in dynamics as these Lagrangians are identically sent to zero by their Euler-Lagrange operator, and thereby they are having no effects on equations of motion. A procedure that significantly generalizes the previous work, which appeared in Physica Scripta 97, 125213 (2022), is developed and used to construct null Lagrangians and then the corresponding non-standard Lagrangians, which represent a range of interesting dynamical systems. By using the generalized procedure, derivation of equations of motion for a harmonic oscillator as well as for the Bateman and Duffing oscillators is presented. The obtained results demonstrate a new role played by the null Lagrangians and their corresponding non-standard Lagrangians in describing linear and nonlinear, and dissipative and non-dissipative dynamical systems.