A new version of energy for involute of slant helix with bending energy in the Lie groups

: In this paper, we study energy of involute curves for slant helix in the Lie group. With this new representation, we illistrate some ﬁgures of energy by elastica. Finally, we have an original and satisfactorily connection between energy of the curve on Lie groups


Introduction
e innovation that Lie group brings to mathematics is that it has three different structures of mathematical form that enable us set a connection between these different forms.Primarily, it has structure of group.Further, the elements belonging to this group form a topological space such that it can be defined as being a certain case of a topological group.Lastly, the elements also form an analytic manifold (Catmull & Clark, 1978;Crouch & Silva, 1995;Capovilla, Chryssomalakos, & Guven, 2002;).
Lie groups play a key role not only in physical systems also in mathematical studies such as loop groups, gauge groups, and Fourier integral's groups operators that occur as symmetry groups and phase spaces (Esprito, Fornari, Frensel, & Ripoll, 2003).Lie groups are also useful in mechanics (Milnor, 1976;).Since incomprehensible inviscid fluid motion and rigid body motion correspond to geodesic flow of le (or right) invariant metric defined on a Lie group (Arnold, 1966;Kolev, 2004).
e study of computing an energy of given vector field depending on the structure of the geometrical spaces has earned such attention in the last couple years.It has been shown that these types of computations have numerous applications in various fields and thus multidisciplinary subjects have been evolved.For instance, Wood (Wood, 1997) studied on the unit vector field's energy firstly.Gil-Medrano (Gil, 2001) worked on relation between energy and volume of vector fields.(Chacon, Naveira, & Weston, 2001;Chacon & Naveira, 2004) investigated on the energy of distrubutions and corrected energy of distrubutions on Riemannian manifolds.Altin computed energy of a Frenet vector fields for a given nonlightlike curves (Altin, 2011).Körpınar discussed timelike biharmonic particle's energy in Heisenberg spacetime (Körpınar, 2014).Also, curves and its flows are researched simply by various experts (Asil, 2007;Turhan & Altay, 2014;Turhan & Altay, 2015;Yeneroğlu, 2016).
e corresponding theory for the energy of curvature-based energy is considered to be at its early stages of evolution.Some of the prolific fields and pioneering studies for this theory can be found in mathematical physics, membrane chemistry, computer aided geometric design and geometric modeling, shell engineering, biology and thin plate (Kirchhoff, 1850;Roberts, Schleif, & Dlugosz, 2007;Weber, 1961).One of the well-known functional and related work is 'bending energy functional', which appeared firstly Bernoulli-Euler elastica formulation for energy (Euler, 1744;Einstein, 1920;Guven, Valencia, & Vazquez, 2014).
In this study we compute the energy of the particle lying on the 3-dimensional Lie group to investigate its connection between mass-energy and motion-energy concept which is a topic of special relativity (Einstein, 1905a;1905b).Furthermore we gain a different perspective by calculating curve's energy on Lie groups to see the relation between energy of particle and curvature-based bending energy functional.e method we use for computing the energy of vector fields is that considering a vector field as a map from manifold M to the Riemannian manifold (TM, p s ) where TM is tangent bundle of a Riemannian manifold and p s is a Sasaki metric induced from TM naturally.
We organize the manuscript by starting to state fundamental definitions and proposition for a Lie group.en we recall interpretation of geometrical meaning of the energy for unit vector fields.Based on these relations we compute the energy of curves defined on the Lie group.Finally, we give examples about particle's energy for different cases by computing their value and drawing thier graph in Figure 1-6.

Material and methods
For a Lie group R with a bi-invariant metric and Levi-Civita connection of Lie group R if h shows the Lie algebra of R then we get isomorphism between h and TjR where j is natural element of R. Let be a biinvariant metric on R then it is obtained, Equation 1: (1) and Equation 2(2) for all For an arc-lenghted curve and orthonormal basis [K 1 , K 2 , ... K n ] of h it is written that for any two vector fields P and Q along the curve , where and are smooth fucntions.Let P and Q be any two vector fields then Lie bracket is written in the following form Equation 3: (3) and for a vector field P the covariant derivative along the curve is stated by Equation 4: (4) where: and (Cici, 2009).Frenet apparatus of the curve can be represented by elemets for a 3-dimensional Lie group R.
Definition 2.1: For a parametrized curve and Frenet apparatus we have Equation 5: (5) or equivalently Equation 6: (6) For a parametrized arc-lenghted curve in 3-dimensional Lie group.If are linearly dependent for all then it is said that is Frenet curve of osculating order three.It can be constructed to an orthonormal Frenet frame for each Frenet curve of order three in the following way Equation 7: (7) where: Lie group R has the Levi-Civita connection .Proposition 2.2: For a 3-dimensional Lie group Rinduced with a bi-invariant metric we have following statements that can be obtained for different Lie groups (Cici, 2009), Equation 8:

Results and discussion
Energy on the frenet vector field Definition 3.1.:Let be an arc length parametrized curve.en is called a slant helix if its principal normal vector makes a constant angle with a le-invariant vector field X which is unit length (Okuyucu, Gök, Yaylı, & Ekmekçi, 2013).Definition 3.2.: Let be an arc length parametrized curve with the Frenet apparatus .en the harmonic curvature function of the curve is defined by Equation 9: (9) where: Proposition 3.3.:If the curve a is a slant helix in R, then the axis of is Equation 10: (10) where: is a constant angle.Definition 3.4.:For two Riemannian manifolds (M, ) and the energy of a differentiable map can be defined as Equation 11: (11) where: {e a } is a local basis of the tangent space and v is the canonical volume form in M (Altin, 2011).
Let T 1 M be the unit tangent bundle endowed with the restriction of the Sasaki metric on TM. en the energy of a unit vector field X is defined to be the section's energy of .For the bundle projection , vertical/horizontal splitting induced by the Levi-Civita connection can be stated as Further, we write where F shows the line bundle generated by X and G is the orthogonal complement.Proposition 3.5.: Let be the connection map.en following two conditions hold: i) and where is the tangent bundle projection; ii)for and a section ; we have Equation 12: (12) where: is the Levi-Civita covariant derivative (Chacon & Naveira, 2004).Definition 3.6.:For we define Equation 13: is yields a Riemannian metric on TM: As we know p s is called the Sasaki metric that also makes the projection a Riemannian submersion.
Definition 3.7.: Angle is known as the angle between arbitrary Frenet vectors given any curve K.For an initial point the angle between Frenet vectors can be stated with the help of the curvature function of the curve K as the following Equation 14: (14) where: V i represents Frenet vector.

Energy of spherical images in lie groups and bending energy functional
In the theory of relativity, all the energy moving through an object contributes to the body's total mass that measures how much it can resist to acceleration.Each kinetic and potential energy makes a highly proportional contribution to the mass (Carmelli, 1965).In this study not only we compute the energy of surface curves but we also investigate its close correlation with bending energy of elastica which is a variational problem proposed firstly by Daniel Bernoulli to Leonard Euler in 1744.Euler elastica bending energy formula for a space curve in the 3-dimensional Frenet curvature along the curve is known as Equation 15: (15) Definition 4.1.:Let be an arc-lengthed regular curve in R. en the curve is called the involute of the curve if the tangent vector field of the curve is perpendicular to the tangent vector field of the curve .at is where T and T u are the tangent vector fields of the curves and , respectively.eorem 4.2.: Let be an arc-lengthed regular curve in R and be an involute of .en is a slant helix in a three dimensional Lie group if and only if is a general helix.

Final considerations
Now, we consider following results for involute of slant helix in Lie group.i) Let R be an Abelian group.us we have and we get following graph respectively for the energy of Frenet fields.
ii) Let R be SU 2 .us we have and we get following graph respectively for the energy of Frenet fields.iii) Let R be SO 3 .us we have and we get following graph respectively for the energy of Frenet fields.Energy in Abelian group.

Energy in SO3.
Corollary 4.5.: T u , N u , B u have not constant energy in the Lie group R with a bi-invariant metric.eorem 4.6.: Angle of Frenet vectors can be respectively given by using Def.3.4 as Equation 24 a t 28 : Now, we consider following results for angle in Lie group.i) Let R be an Abelian group.us we have and we get following graph respectively for the angle of Frenet fields.
ii) Let R be SU2.us we have and we get following graph respectively for theangle of Frenet fields.iii) Let R be SO3.us we have and we get following graph respectively for the angle of Frenet fields.Angle in SO 3 .

Conclusion
Lie groups and energy play an significant role in geometric design and theorical physics.
In this work, we study energy of involute curves for slant helix in the Lie group.With this new representation, we illistrate some figures of energy by elastica.Also, we have an original and satisfactorily connection between energy of the curve on Lie groups.
In the light of these results, we will study energy of magnetic curves in the Lie groups.We also aimed to obtain other useful and original results about the relationship between the spherical indicatrices of magnetic curves and its energy.
eorem 4.3.: Energy of Frenet vectors by using Sasaki metric is stated by Equation 16 a t 20 .