| dbo:abstract |
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→ΦHt(ξ)∈TM obey the rule ΦHt(λξ)=ΦHλt(ξ) in positive reparameterizations. If this requirement is dropped, H is called a semispray. 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. (en) |
| dbo:wikiPageExternalLink |
https://arxiv.org/abs/1011.5799 https://math.stackexchange.com/q/166955 |
| dbo:wikiPageID |
1425185 (xsd:integer) |
| dbo:wikiPageLength |
12549 (xsd:nonNegativeInteger) |
| dbo:wikiPageRevisionID |
1089901972 (xsd:integer) |
| dbo:wikiPageWikiLink |
dbr:Vector_field dbr:Double_tangent_bundle dbr:Integral_curve dbc:Differential_geometry dbr:Tangent_bundle dbr:Quasiconvex_function dbr:Geodesics dbc:Finsler_geometry dbr:Damodar_Dharmananda_Kosambi dbr:Finsler_manifold dbr:Riemannian_geometry dbr:Riemannian_manifold dbr:Hamiltonian_vector_field dbr:Lagrangian_mechanics dbr:Differentiable_manifold dbr:Differential_geometry dbr:Section_(fiber_bundle) dbr:Finsler_geometry dbr:Symplectic_form dbr:Geodesic_spray dbr:Dynamical_covariant_derivative dbr:Kosambi_biderivative_operator |
| dbp:wikiPageUsesTemplate |
dbt:Anchor dbt:Citation dbt:Expand_section dbt:Further dbt:Main dbt:Short_description |
| dcterms:subject |
dbc:Differential_geometry dbc:Finsler_geometry |
| gold:hypernym |
dbr:H |
| rdf:type |
dbo:ChemicalCompound |
| rdfs:comment |
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→ΦHt(ξ)∈TM obey the rule ΦHt(λξ)=ΦHλt(ξ) in positive reparameterizations. If this requirement is dropped, H is called a semispray. (en) |
| rdfs:label |
Spray (mathematics) (en) |
| owl:sameAs |
freebase:Spray (mathematics) wikidata:Spray (mathematics) https://global.dbpedia.org/id/4vgXK |
| prov:wasDerivedFrom |
wikipedia-en:Spray_(mathematics)?oldid=1089901972&ns=0 |
| foaf:isPrimaryTopicOf |
wikipedia-en:Spray_(mathematics) |
| is dbo:wikiPageDisambiguates of |
dbr:Spray |
| is dbo:wikiPageWikiLink of |
dbr:Double_tangent_bundle dbr:Geodesic dbr:Spray dbr:Linear_connection dbr:Finsler_manifold dbr:Jet_bundle |
| is foaf:primaryTopic of |
wikipedia-en:Spray_(mathematics) |