Symplectic geometry arises as the natural geometry of phase-space in the equations of classical mechanics. In this study, we obtain new characterizations of regular symplectic curves with respect to the Frenet frame in four-dimensional symplectic space. We also give the characterizations of the symplectic circular helices as the third- and fourth-order differential equations involving the symplectic curvatures.
It is well known that there exist characterizations for curve in Euclidean space. Also, a lot of authors extended this characterizations for Minkowski space and obtained very different results. In this paper, we introduce the geometric properties of Symplectic Curve in 4-Dimensional Symplectic Space which given by [1, 2]. Later we obtained the conditions for Symplectic Curve to lie on some subspaces of 4-Dimensional Symplectic Space and we give some characterizations and theorems for these curves.
In the present work, we have dealt with the properties of associated curves of a Frenet curve in R14. In addition to this, we define principal direction curve, B_1 -direction curve, B_2- direction curve of a given Frenet curve by using integral curves of 4-dimensional Minkowski space. Then we introduce some characterizations for general helix and slant helix. Finally, some new associated curves and theorems obtained for space-like curves and time- like curves in R14. Also, an example is given.
In this study, we have expressed the notion of $k$-type slant helix in $4$-symplectic space. Also, we have generated some differential equations for $k$-type slant helix of symplectic regular curves.
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