By Serge Preston
ISBN-10: 3319283219
ISBN-13: 9783319283210
ISBN-10: 3319283235
ISBN-13: 9783319283234
This textual content provides and reports the tactic of so –called noncommuting diversifications in Variational Calculus. this technique was once pioneered by means of Vito Volterra who spotted that the traditional Euler-Lagrange (EL-) equations aren't appropriate in Non-Holonomic Mechanics and recommended to change the fundamental rule utilized in Variational Calculus. This ebook offers a survey of Variational Calculus with non-commutative adaptations and indicates that almost all uncomplicated houses of traditional Euler-Lagrange Equations are, with a few ameliorations, preserved for EL-equations with K-twisted (defined by means of K)-variations.
Most of the booklet could be understood by means of readers with out robust mathematical instruction (some wisdom of Differential Geometry is necessary). in an effort to make the textual content extra available the definitions and a number of other worthy ends up in Geometry are provided individually in Appendices I and II additionally in Appendix III a brief presentation of the Noether Theorem describing the relation among the symmetries of the differential equations with dissipation and corresponding s stability legislation is presented.
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Sample 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.