Take a fibred manifold . Choose a vector field over that satisfies for the Jacobian map over all coordinate vectors . A spray field G over is a globally defined smooth vector field associated to the first jet bundle of that is given in local coordinates as
The spray coefficients are second degree homogeneous functions which correspond up to nonlinear connections on . Thus by the integral curves of must be of second order, and so given the constraints of the spray coefficients, satisfy . Subsequently, the pair is called a spray space.
Example 1: Choose a system of second order quasilinear ordinary differential equations that satisfy
for a family of parameterized curves , and let the system induce its corresponding spray. Then when is also a Finsler geodesic in with constant speed so that the covariant derivative gives along a vector field , the corresponding autoparallels of the spray coefficients completely characterize a path space for .
|Date of creation||2013-05-03 16:21:46|
|Last modified on||2013-05-03 16:21:46|
|Last modified by||jacou (1000048)|