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217, 393 (2005). G. Misiolek, J. Geom. Phys. 24, 203 (1998). E. Reyes, Lett. Math. Phys. 59, 117 (2002). 41 42 FERMI-WALKER PARALLEL TRANSPORT, TIME EVOLUTION OF A SPACE CURVE AND THE ¨ SCHRODINGER EQUATION AS A MOVING CURVE R. fr Based on Fermi-Walker parallel transport along a space curve, we discuss the possible geometric phases that can occur. We give a general condition for the time evolution of a space curve. We identify two types of local geometric phases associated with the evolution of a space curve, the “Fermi-Walker” and “incompatibility” phases, and derive a relationship between them.
Let us take the following explicit parametrization for the angular velocity: Camassa-Holm equation as a geodesic ﬂow 0 −ω3 ω2 ωL = ω3 0 −ω1 ∈ g −ω2 ω1 0 35 ↔ ω1 ω L ≡ ω 2 ∈ R3 ω3 (2) The quantities related to the Euler top are schematically presented at Fig. 1, (the dot denotes the time derivative) [1,18,21]. The identiﬁcation between the algebra g and its dual is given by the inertia operator , see Fig. 2: mL = J(ωL ) ≡ AωL + ωL A, (3) where A = diag(a1 , a2 , a3 ) is a constant symmetric matrix.
Ivanov, Phys. Lett. SI/0507005. SI/0601066. 16. 17. R. Johnson, J. Fluid. Mech. 457, 63 (2002). 18. B. Khesin and G. Misiolek, Adv. Math. 176, 116 (2003). 19. A. Kirillov, Funct. Anal. Appl. 15, 135 (1981). , Contemp. Math. 145, 33 (1993). 20. Camassa-Holm equation as a geodesic ﬂow 21. 22. 23. 24. B. Kolev, J. Nonlinear Math. Phys. 11, 480 (2004). J. Lenells, J. Diﬀ. Eq. 217, 393 (2005). G. Misiolek, J. Geom. Phys. 24, 203 (1998). E. Reyes, Lett. Math. Phys. 59, 117 (2002). 41 42 FERMI-WALKER PARALLEL TRANSPORT, TIME EVOLUTION OF A SPACE CURVE AND THE ¨ SCHRODINGER EQUATION AS A MOVING CURVE R.