Starting with an extended complex backwardforward derivative operator in differential geometry which is motivated from non-local-in-time Lagrangian dynamics, higher-order geodesic equations are obtained within classical differential geometrical settings. We limit our analysis up to the 2^{nd}-order derivative where some applications are discussed and a number of features are revealed accordingly.
[1] F. Ben Adda and J. Cresson, Quantum derivatives and the Schrödinger equation, Chaos Solitons Fractals, 9, (2004), 1323-1334
[2] Y. Ciann-Dong, On the existence of complex spacetime in relativistic quantum mechanics, Chaos Solitons Fractals, (2008), 316-331
[3] R. Colistete Jr., C. Leygnac, and R. Kerner, Higher-order geodesic deviations applied to the Kerr metric, Class. Quantum Grav, (2002), 4573-4590
[4] M. Davidson, A study of the Lorentz-Dirac equation in complex spacetime for clues to emergent spacetime, J. Phys.: Conf. Series, Conf.1, (2012), (11 pages).
[5] R. A. El-Nabulsi, Non-standard non-local-in-time Lagrangians in classical mechanics, Qual. Theor. Dyn. Sys., (2014), 149-160
[6] G. Esposito, From spinor geometry to complex general relativity, Int. J. Geom. Meth. Mod. Phys, (2005), 675-731
[7] D. Eberly, Computing geodesics on a Riemannian manifold, Geometric Tools, LLC, 2015
[8] R. S. Herman, Derivation of the geodesic equation and defining the Christoffel symbols, a lecture given at the University of North Carolina Wilmington, March 13, 2008
[9] E. L. Hill, On the kinematics of uniformly accelerated motions and classical magnetic theory, Phys. Rev, (1947), 143-149
[10] W. P. Joyce, Gauge freedom of Dirac theory in complexified spacetime algebra, J. Phys. A: Math. Gen, (2002), 4737-4747
[11] R. Kerner, J. W. von Holten, and R. Colistete Jr, Relativistic epicycles: another approach to geodesic deviations, Class. Quantum Grav, (2001), 4725-4742
[12] N. Koike, The complexifications of pseudo-Riemannian manifolds and anti-Kaehler geometry, SUT J. Math, (2014), 271-295
[13] Z.-Y. Li, J.-L. Fu, and L.-Q. Chen, Euler–Lagrange equation from nonlocal-intime kinetic energy of nonconservative system, Phys.Lett, (2009), 106-109
[14] C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation, W. H. Freeman and Co, San Francisco, 1973
[15] R. Ya. Matsyuk, The variational principle for the uniform acceleration and quasispin in two-dimensional space-time, SIGMA, 4, (2008), 016-027
[16] R. Ya. Matsyuk, Lagrangian analysis of invariant third-order equations of motion in relativistic classical particle mechanics, English transl.: Soviet Phys. Dokl, 30, (1985), 458-460
[17] E. Nelson, Derivation of the Schrödinger Equation from Newtonian Mechanics, Phys. Rev, (1966), 1079-1085
[18] J. Saucedo and V. M. Villanueva J. A. Nieto, Relativistic top deviation equation and gravitational waves, Phys. Lett, (2003), 175-186
[19] L. Nottale, Fractal Space-Time and Microphysics: Towards a Theory of Scale Relativity, World Scientific, 1993
[20] L. Nottale, The theory of scale-relativity: Non-differentiable geometry and fractal space-time, Computing Anticipatory Systems, CASYS’03-Sixth International Conference (Liege, Belgium, 11-16 August 2003), Daniel M. Dubois Editor, American Institute of Physics Conference Proceedings, (2004), 68-95
[21] G. O. Okengo, On the complexification of Minkowski spacetime, Africa J. Phys. Sci, (2015), 73-76
[22] R. Penrose, On the twistor description of massless fields, In Proceedings Complex Manifold Techniques In Theoretical Physics, ed. by Lawrence, (1978), 55-91
[23] R. Penrose and M.A. MacCallum, Twistor theory: an approach to the quantization of fields and space-time, Phys. Rep, (1972), 241-316
[24] T. Popiel, Higher-order geodesics in Lie groups, Math.Contr. Sign. Syst, No. 3, (2007), 235-253
[25] T. Roubicek, Calculus of variations, Chap. 17 in Mathematical Tools for Physicists, (Ed. M. Grinfeld), J. Wiley, Weinheim, 2014
[26] S. L. Shapiro and S. A. Teukolsky, Black Holes, White Dwarfs and Neutron Stars, Wile-Interscience, New York, 1983
[27] B. G. Sidharth, Complexified spacetime, Found. Phys. Letts, No. 1, (2003), 91-97
[28] H. Stephani, General Relativity-An Introduction to the Theory of the Gravitation Field, Cambridge University Press, 1982
[29] R. Szoke, Complex structures on tangent bundles of Riemannian manifolds, Math. Ann, (1991), 409-428
[30] J. A. K. Suykens, Extending Newton’s law from nonlocal-in-time kinetic energy, Phys. Lett, (2009), 1201-1211
[31] S. Taniguchi, On almost complex structures on abstract Wiener spaces, Osaka J. Math, (1996), 189-206
[32] J. Vines, Geodesic deviation at higher order via covariant bitensors, Gen. Rel. Grav, (2015), 49-65
[33] K. Yano, Concircular geometry I. Concircular transformations, Proc. Imp. Acad. Jap, (1940), 195-200