Octonionic Non-Associative Geometry F; Non-Associative Geodesic Equation– Third-Order Velocity Term and Modification of Conserved Quantities

The geodesic equation is the cornerstone for describing the motion of free par
ticles in general relativity. This paper generalises the standard geodesic equation
to a non-associative geometric framework where the coordinates are taken from
the octonion algebra. On a Moufang manifold (non-associative spacetime), a left
associative action is defined and the non-associative geodesic equation is derived
via the variational principle. Unlike the standard case, the non-associativity of oc
tonion multiplication introduces a modification of the standard geodesic equation
by a term cubic in the velocity, proportional to a parameter κ. We give an explicit
expression for Cρ µνσ and show that the equation reduces to the standard geodesic
equation in the limit κ → 0. Furthermore, we analyse the modification of conserved
quantities in the presence of Killing vector fields and discuss the effect of the veloc
ity cubic term on the initial value problem — the equation remains a second-order
nonlinear ordinary differential equation, and the solution is uniquely determined
by the initial position and velocity without requiring an initial acceleration. This
work provides a theoretical foundation for predicting the trajectories of free parti
cles in non-associative geometry and offers a mathematical framework for potential
observable effects in high-energy astrophysics.