General Null Lagrangians and Their Novel Role in Classical Dynamics

Abstract

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.

New Role of Null Lagrangians in Derivation of Equations of Motion for Dynamical Systems

“…which is the equation of motion for an oscillatory system with the quadratic damping [23,9]. The above results demonstrate that the previously developed method can be used to derive the first and second law of Newtonian dynamics by using the constructed gauge functions and their corresponding null Lagrangians.…”

“…where c 1 is an integration constant. The obtained result means that the gauge function and null Lagrangian exist only for the equation of motion with β coefficient having this time dependence; for some other special cases see [23].…”

“…An interesting relationship was recently found between the non-standard and null Lagrangians [23] and it was shown that…”

“…However, it is possible to exploit these NLs for constructing non-standard Lagrangians to represent certain interesting dynamical systems. Recently, a procedure for deriving equations of motion for a given NL has been developed [23], and in the following we briefly summarize the procedure by presenting its main condition. Then, we show its limitations and a need for its generalization.…”

“…( 1) is that its form is much simpler than the E-L equation and thus it is easier to derive an equation of motion. However, its disadvantage is that the resulting equation of motion must be in a form that obeys a special relationship between the coefficients of this equation; the relationship is unique for a given dynamical system but varies for different systems [23]. Therefore, the mainpurpose of this paper is to identify limitations of the method and Eq.…”