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Since the seminal work of Emmy Noether it is well know that all conservationslaws in physics, \textrm{e.g.}, conservation of energy or conservation ofmomentum, are directly related to the invariance of the action under a familyof transformations. However, the classical Noether's theorem can not yieldsinformations about constants of motion for non-conservative systems since it isnot possible to formulate physically meaningful Lagragians for this kind ofsystems in classical calculus of variation. On the other hand, in recent yearsthe fractional calculus of variation within Lagrangians depending on fractionalderivatives has emerged as an elegant alternative to study non-conservativesystems. In the present work, we obtained a generalization of the Noether'stheorem for Lagrangians depending on mixed classical and Caputo derivativesthat can be used to obtain constants of motion for dissipative systems. Inaddition, we also obtained Noether's conditions for the fractional optimalcontrol problem.

Fractional Noether's Theorem with Classical and Caputo Derivatives: constants of motion for non-conservative systems