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On the geometry of a bundle associated with a foliated manifold and a new look at some results in mechanics. (English) Zbl 0676.53036

Semin. Mec., Univ. Timişoara 15, 18 p. (1988).
The geometrisation of the two fundamental notions from mechanics is discussed: Lagrangians and Hamiltonians led us to the concept of Lagrange spaces and Hamilton spaces. Also, in the theory of mechanical systems the Routh function is introduced. It permits the authors to construct an interesting hybrid space between Lagrange and Hamilton spaces.
If (M,D) is a foliated manifold, then the bundle \(Tr(D)\oplus T^*(D)\) is considered as base manifold of the Lagrange-Hamilton geometry. On it nonlinear connections, the canonical connection determined by the Routh function, non-degenerate vector fields and an almost tangent structure can be defined. Two particular cases \((D=\{0\}\) and \(D=(TM))\) and some applications are studied.
Reviewer: R.Miron

MSC:

53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
70G99 General models, approaches, and methods in mechanics of particles and systems
53B50 Applications of local differential geometry to the sciences