Nonholonomic algebroids, Finsler geometry, and Lagrange-Hamilton spaces. (English) Zbl 1264.70039
Summary: We elaborate a unified geometric approach to classical mechanics, Riemann-Finsler spaces and gravity theories on Lie algebroids provided with nonlinear connection (\(N\)-connection) structure. There are investigated conditions when the fundamental geometric objects (anchor, metric and linear connection, almost symplectic, and related almost complex structures) may be canonically defined by an \(N\)-connection induced from a regular Lagrangian (or Hamiltonian), in mechanical models, or by generic off-diagonal metric terms and nonholonomic frames, in gravity theories. Such geometric constructions are modelled on nonholonomic manifolds provided with nonintegrable distributions and related chains of exact sequences of submanifolds defining \(N\)-connections. We investigate the main properties of the Lagrange, Hamilton, Finsler-Riemann and Einstein-Cartan algebroids, construct and analyze exact solutions describing such objects.
MSC:
| 70G45 | Differential geometric methods (tensors, connections, symplectic, Poisson, contact, Riemannian, nonholonomic, etc.) for problems in mechanics |
| 70H03 | Lagrange’s equations |
| 70H05 | Hamilton’s equations |
| 53D17 | Poisson manifolds; Poisson groupoids and algebroids |
| 53B40 | Local differential geometry of Finsler spaces and generalizations (areal metrics) |
| 83C99 | General relativity |
Keywords:
Lie algebroids; Lagrange; Hamilton and Riemann-Finsler spaces; nonlinear connection; nonholonomic manifold; geometric mechanics and gravity theoriesReferences:
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