Non-standard fractional Lagrangians

Two mathematical physics’ approaches have recently gained increasing importance both in mathematical and in physical theories: (i) the fractional action-like variational approach which founds its significance in dissipative and non-conservative systems and (ii) the theory of non-standard Lagrangians which exist in some group of dissipative dynamical systems and are entitled “non-natural” by Arnold. Both approaches are discussed independently in the literature; nevertheless, we believe that the combination of both theories will help identifying more hidden solutions in certain classes of dynamical systems. Accordingly, we generalize the fractional action-like variational approach for the case of non-standard power-law Lagrangians of the form L1+γ$(\gamma\in\mathbb{R})$ recently introduced by the author (Qual. Theory Dyn. Syst. doi:10.1007/s12346-012-0074-0, 2012). Many interesting features are discussed in some details.