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Standard and non-standard Lagrangians for dissipative dynamical systems with variable coefficients. (English) Zbl 1136.37044
The method of the previous article [Z. E. Musielak, D. Roy and I. D. Swift, Chaos Solitons Fractals 38, No. 3, 894–902 (2008)] is used to determine forms of equations of motion which have standard Lagrangians. Another method is suggested to identify equations of motion which can be derived from nonstandard Lagrangians. It is shown that there are two general classes of equations of motion with standard Lagrangians and one special class ones with nonstandard Lagrangians. Each general class has also a subset of equation of motion with nonstandard Lagrangians. Conditions required for the existence of standard and nonstandard Lagrangians are derived and a relationship between these two types of Lagrangians is introduced. By obtaining Lagrangians for several dynamical systems and some basic equations of mathematical physics it is demonstrated that the presented methods can be applied to a wide range of physical problems.

MSC:
37L99Infinite-dimensional dissipative dynamical systems
70H03Lagrange’s equations