Novel gauge functions are introduced to non-relativistic classical mechanics and used to define forces. The obtained results show that the gauge functions directly affect the energy function and allow for converting an undriven physical system into a driven one. This is a novel phenomenon in dynamics that resembles the role of gauges in quantum field theories.

The Lagrange formalism is developed for Bateman oscillators, which includes both damped and amplified systems, and a novel method to derive the Caldirola-Kanai and null Lagrangians is presented. For the null Lagrangians, the corresponding gauge functions are obtained. It is shown that the gauge functions can be used to convert the undriven Bateman oscillators into the driven ones. Applications of the obtained results to quantizatation of the Bateman oscillators are briefly discussed.

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A method to derive general standard and null Lagrangians for second-order differential equations whose solutions are special function of mathematical physics is presented. The general null Lagrangians are used to find the corresponding general gauge functions. All derived Lagrangians are new and in special cases they reduce to those published in the literature. The obtained results are applied to the Bessel, Hermite and Legendre equations, which have many applications in physics, applied mathematics and engineering.