Spray (mathematics)

In differential geometry, a spray is a vector field H on the tangent bundle TM that encodes a quasilinear second order system of ordinary differential equations on the base manifold M. Usually a spray is required to be homogeneous in the sense that its integral curves t→Φ_H^t(ξ)∈TM obey the rule Φ_H^t(λξ)=Φ_H^λt(ξ) in positive re-parameterizations. If this requirement is dropped, H is called a semi-spray.

Sprays arise naturally in Riemannian and Finsler geometry as the geodesic sprays whose integral curves are precisely the tangent curves of locally length minimizing curves. Semisprays arise naturally as the extremal curves of action integrals in Lagrangian mechanics. Generalizing all these examples, any (possibly nonlinear) connection on M induces a semispray H, and conversely, any semispray H induces a torsion-free nonlinear connection on M. If the original connection is torsion-free it coincides with the connection induced by H, and homogeneous torsion-free connections are in one-to-one correspondence with full sprays.

Let M be a differentiable manifold and (TM,π_TM,M) its tangent bundle. Then a vector field H on TM (that is, a section of the double tangent bundle TTM) is a semi-spray on M, if any of the three following equivalent conditions holds:

A semispray H on M is a (full) spray if any of the following equivalent conditions hold:

Let ${\displaystyle (x^{i},\xi ^{i})}$ be the local coordinates on ${\displaystyle TM}$ associated with the local coordinates ${\displaystyle (x^{i}}$) on ${\displaystyle M}$ using the coordinate basis on each tangent space. Then ${\displaystyle H}$ is a semi-spray on ${\displaystyle M}$ if it has a local representation of the form

on each associated coordinate system on TM. The semispray H is a (full) spray, if and only if the spray coefficients Gⁱ satisfy

A physical system is modeled in Lagrangian mechanics by a Lagrangian function L:TM→R on the tangent bundle of some configuration space M. The dynamical law is obtained from the Hamiltonian principle, which states that the time evolution γ:[a,b]→M of the state of the system is stationary for the action integral

In the associated coordinates on TM the first variation of the action integral reads as