On the Rheonomic Finslerian Mechanical Systems
نویسنده
چکیده
In this paper it will be studied the dynamical system of a rheonomic Finslerian mechanical system, whose evolution curves are given, on the phase space TM×R, by Lagrange equations. Then one can associate to the considered mechanical system a vector field S on TM ×R, which is called the canonical semispray. All geometric objects of the rheonomic Finslerian mechanical system one can be derived from S. So we have the fundamental notion as the nonlinear connection N , the metrical N -linear connection, etc. 1. The geometry of phases space (TM ×R, π, M) Let be M a smooth C∞ manifold of finite dimension n, called the space of configurations and (TM, π,M) be its tangent bundle.The 2n-dimensional manifold TM is called the phases space of M . We denote by (x), i = 1, 2, . . . , n, the local coordinates on M and by (x, y) the canonical local coordinates on TM . We consider the manifold TM×R and we shall use the differentiable structure on TM ×R as product of the manifold TM fibered over M with R. The manifold E = TM × R is a 2n + 1−dimensional, real manifold. In a domain of a local chart U × (a, b), the point u = (x, y, t) ∈ E have the local coordinates (x, y, t). A change of local coordinates on E has the following form: (1.1) x̃ = x̃(x, x, . . . , x); ỹ = ∂x̃ ∂xj y ; t̃ = φ(t) with rank ( ∂x̃ ∂xj ) = n and φ′ := dφ dt 6= 0. 2000 Mathematics Subject Classification. 53C60, 53C80.
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