Noether’s theorem for variational problems of Herglotz type with real and complex order fractional derivatives

Noether’s theorem for Herglotz type variational problems utilizing complex fractional derivatives

“…In [1], the following results are formulated and proved, so we elaborate on those. We list those together with proofs in order to illustrate the concept used.…”

“…In [1], a HVP in the case when its Lagrangian involves real and complex order fractional derivatives is introduced, where the results are continuation of the work presented in [12]. The actions of a local one-parameter group of transformations related to the corresponding Euler-Lagrange equation are formulated and the infinitesimal criteria obtained, so that the results presented in [13] and [18] are generalized.…”

“…In this review article, the results obtained in [1] are elaborated. Namely, variational principles of Hamilton type are suited for conservative systems, while for systems with non-conservative forces, variational principles of Hamilton type are rare (see [2,3]).…”

“…In [1] symmetry properties of the HVP with real and complex order fractional derivatives are derived. Concerning the variational principle of Hamilton type with real fractional derivatives, important contributions are by Riewe [6,7] and recent [8][9][10][11][12][13][14][15][16].…”

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This is a review article which elaborates the results presented in [1], where the variational principle of Herglotz type with a Lagrangian that depends on fractional derivatives of both real and complex orders is formulated and the invariance of this principle under the action of a local group of symmetries is determined. The conservation law for the corresponding fractional Euler Lagrange equation is obtained and a sequence of approximations of a fractional Euler-Lagrange equation by systems of integer order equations established and analyzed.

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