Variational calculus with conformable fractional derivatives

Invariant conditions for conformable fractional problems of the calculus of variations under the presence of external forces in the dynamics are studied. Depending on the type of transformations considered, different necessary conditions of invariance are obtained. As particular cases, we prove fractional versions of Noether U+02BC s symmetry theorem. Invariant conditions for fractional optimal control problems, using the Hamiltonian formalism, are also investigated. As an example of potential application in Physics, we show that with conformable derivatives it is possible to formulate an Action Principle for particles under frictional forces that is far simpler than the one obtained with classical fractional derivatives.

[1]  Delfim F. M. Torres,et al.  Fractional Noether's theorem in the Riesz-Caputo sense , 2010, Appl. Math. Comput..

[2]  Thabet Abdeljawad,et al.  On conformable fractional calculus , 2015, J. Comput. Appl. Math..

[3]  Delfim F. M. Torres,et al.  The DuBois–Reymond Fundamental Lemma of the Fractional Calculus of Variations and an Euler–Lagrange Equation Involving Only Derivatives of Caputo , 2012, J. Optim. Theory Appl..

[4]  Agnieszka B. Malinowska,et al.  Advanced Methods in the Fractional Calculus of Variations , 2015 .

[5]  Delfim F. M. Torres,et al.  Fractional Optimal Control in the Sense of Caputo and the Fractional Noether's Theorem , 2007, 0712.1844.

[6]  Delfim F. M. Torres,et al.  Necessary and sufficient conditions for the fractional calculus of variations with Caputo derivatives , 2010, 1007.2937.

[7]  J. Logan.  Invariant Variational Principles , 1977 .

[8]  O. Agrawal,et al.  Fractional hamilton formalism within caputo’s derivative , 2006, math-ph/0612025.

[9]  Delfim F. M. Torres,et al.  A Conformable Fractional Calculus on Arbitrary Time Scales , 2015, 1505.03134.

[10]  M. J. Lazo,et al.  The action principle for dissipative systems , 2014, 1412.5109.

[11]  Delfim F. M. Torres,et al.  A formulation of Noether's theorem for fractional problems of the calculus of variations , 2007 .

[12]  T. Kaczorek,et al.  Fractional Differential Equations , 2015 .

[13]  Agnieszka B. Malinowska,et al.  Introduction to the Fractional Calculus of Variations , 2012 .

[14]  Delfim F. M. Torres,et al.  Non-conservative Noether's theorem for fractional action-like variational problems with intrinsic and observer times , 2007, 0711.0645.

[15]  Agnieszka B. Malinowska,et al.  Fractional Calculus of Variations in Terms of a Generalized Fractional Integral with Applications to Physics , 2012, 1203.1961.

[16]  H. Srivastava,et al.  Theory and Applications of Fractional Differential Equations, Volume 204 (North-Holland Mathematics Studies) , 2006 .

[17]  Delfim F. M. Torres.  On the Noether theorem for optimal control , 2001, 2001 European Control Conference (ECC).

[18]  Nonconservative Noether's Theorem in Optimal Control , 2005, math/0512468.

[19]  K. Miller,et al.  An Introduction to the Fractional Calculus and Fractional Differential Equations , 1993 .

[20]  Delfim F. M. Torres.  Conservation Laws in Optimal Control , 2002 .

[21]  Quasi-Invariant Optimal Control Problems , 2003, math/0302264.

[22]  Delfim F. M. Torres,et al.  Nonsymmetric and symmetric fractional calculi on arbitrary nonempty closed sets , 2015, 1502.07277.

[23]  Juan J. Nieto,et al.  Three-Point Boundary Value Problems for Conformable Fractional Differential Equations , 2015 .

[24]  D. Djukić,et al.  Noether's theorem for optimum control systems , 1973 .

[25]  Jacky Cresson,et al.  Fractional embedding of differential operators and Lagrangian systems , 2006, math/0605752.

[26]  Delfim F. M. Torres,et al.  A fractional calculus on arbitrary time scales: Fractional differentiation and fractional integration , 2014, Signal Process..

[27]  F. Mainardi,et al.  Recent history of fractional calculus , 2011 .

[28]  M. Sababheh,et al.  A new definition of fractional derivative , 2014, J. Comput. Appl. Math..

[29]  Delfim F. M. Torres,et al.  Fractional order optimal control problems with free terminal time , 2013, 1302.1717.

[30]  Frans Cantrijn,et al.  GENERALIZATIONS OF NOETHER'S THEOREM IN CLASSICAL MECHANICS* , 1981 .

[31]  Delfim F. M. Torres,et al.  Fractional conservation laws in optimal control theory , 2007, 0711.0609.

[32]  Om P. Agrawal,et al.  Formulation of Euler–Lagrange equations for fractional variational problems , 2002 .

[33]  Douglas R. Anderson,et al.  FRACTIONAL-ORDER BOUNDARY VALUE PROBLEM WITH STURM-LIOUVILLE BOUNDARY CONDITIONS , 2014, 1411.5622.

[34]  Delfim F. M. Torres,et al.  Existence of solution to a local fractional nonlinear differential equation , 2016, J. Comput. Appl. Math..

[35]  Agnieszka B. Malinowska,et al.  Fractional Variational Calculus with Classical and Combined Caputo Derivatives , 2011, 1101.2932.

[36]  S. Pooseh.  Computational Methods in the Fractional Calculus of Variations , 2013, 1312.4064.

[37]  H. Srivastava,et al.  Theory and Applications of Fractional Differential Equations , 2006 .

[38]  P. S. Bauer.  Dissipative Dynamical Systems: I. , 1931, Proceedings of the National Academy of Sciences of the United States of America.

[39]  Delfim F. M. Torres.  Proper extensions of Noether's symmetry theorem for nonsmooth extremals of the calculus of variations , 2004 .

[40]  Frederick E. Riewe,et al.  Mechanics with fractional derivatives , 1997 .

[41]  M. Shapiro,et al.  Towards a physics on fractals: Differential vector calculus in three-dimensional continuum with fractal metric , 2016 .