Abstract
We derive optimality conditions for variational problems of Herglotz type whose Lagrangian depends on fractional derivatives of both real and complex order, and resolve the case of subdomain when the lower bounds of variational integral and fractional derivatives differ. Moreover, we consider a problem of the Herglotz type that corresponds to the case when the Lagrangian depends on the fractional derivative of the action and give an example of the problem that corresponds to the oscillator with a memory. Since our assumptions on the Lagrangian are weaker than in the classical theory, we analyze generalized Euler–Lagrange equations by the use of weak derivatives and the appropriate technics of distribution theory. Such an example is discussed in detail.
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Notes
We note that in the case when different boundary conditions on u are prescribed, the boundary conditions on \(\eta \) will be different. This case can be treated similarly.
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Acknowledgements
This work is supported by Projects 174005 and 174024 of the Serbian Ministry of Education, Science, and Technological Development and Project 142-451-2384 of the Provincial Secretariat for Higher Education and Scientific Research.
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Atanacković, T.M., Konjik, S. & Pilipović, S. Variational problems of Herglotz type with complex order fractional derivatives and less regular Lagrangian. Acta Mech 230, 4357–4365 (2019). https://doi.org/10.1007/s00707-019-02521-9
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DOI: https://doi.org/10.1007/s00707-019-02521-9