By A. A. Mironov; P. S. Selin
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Extra info for Method of Partition of Networks with Fixed Degrees of Nodes and Network Flows
Example text
81. Consider a case where vector field ξ = ξk = ∂xk is lifted to the space Y by a connection Γ in the bundle π : Y → X: ∂xk → ξˆk = ∂xk + Γμk ∂μ . Introduce the characteristic Q of the vector field ξˆk - the quantity playing the principal role in the prolongation of vector fields and in studying symmetries and conservation laws associated with the differential equations (see [106] and sections 72,75 of Appendix. Characteristic of the vector field ξˆk has the components Qμ = Γμk −ykμ and we define the Energy-Momentum tensor as Tki = Lδki − (Γμk − ykμ )L,yiμ .
NOTES ON THE NONCOMMUTING VARIATIONS. 19 Example 2. In a case of Mechanics (n=1), there is one independent variable t and m dynamical variables y μ . A NC-tensor Kβα defines K-twisted variations of variables y μ and their derivatives: δy μ = ξ μ (t, y), δ y˙ μ = dt ξ μ + Kβμ ξ β . 6) The full prolongation formula contains the second (horizontal) component K ν (see bellow, Section 16 for the general prolongation procedure). As a result, the K-twisted 1-prolongation of a vector field v = ξ∂t + ξ μ ∂yμ is 1 P rK v = (ξ∂t + ξ μ ∂μ ) + [(dt (ξ μ − y˙ μ ξ) + (Kνμ ξ ν − K μ ξ)]∂y˙ μ .
6) Such prolongations for all vector fields ξ (normal to the surface s∗ (D) ⊂ Y ) and for small t, say for t , scan the neighborhood of 1-jet section j 1 s∗ . Using the Sup-norm for sections of 1-jet bundle π1 : J 1 (π) → X we obtain the (equivalent of) C 1 -norm for the sections s∗ (x) of the bundle π. For a chosen > 0, there exists δ > 0 such that if t δ, the distance between φt s∗ and s∗ is less then in the C 1 -norm. Taking small enough we prove that to check the condition of weak minimum for section s∗ , it is sufficient to use weak neighborhoods obtained this way - by using flows of variational vector fields.