Integral Characterization of a System of Differential Equations and Applications

under the Creative Commons Attribution License, which permits unrestricted use, distribution , and reproduction in any medium, provided the original work is properly cited. Abstract In this study, we obtain the solution sets of the Frenet-Like system of differential equations in the form of integral relations, then show the main applications of results to the normal systems which occur in differential geometry, physics, and kinematics.


Introduction
The system of differential equations in the normal form usually appears in the concept of differential geometry.For instance, a system of differential equations characterize E 4 spherical curves can be given as where s is arc parameter, ρ(s) = 1/κ(s) is curvature diameter; κ, τ and μ are curvatures and, f (s) and g(s) are in the class of C 2 [3,5].
The system characterize curves of constant breadth are also the same type and can be given as where θ(s) = s 0 κ(s) ds and, λ(θ), μ(θ) and δ(θ) are the coefficients of the curve [4,6].
Also, the well known Serret-Frenet Equations , where κ 1 and κ 2 are Euclidean curvatures, lead us to the following system of differential equations [2]: The solution of this system also gives as a criterion for periodicity of a space curve.
Since the normal systems (1), ( 2) and ( 4) are all the same type, it is possible to form them as where a,b,x,y and z are assumed to be continuous on a closed interval.This kind of systems are called Frenet-Like system of differential equations.For further reading on the differential geometry of the curves, we refer readers [2].
In this study, we show that the system (5-7) is equivalent to several integral relations and also third order differential equations, then conclude that solution set can be obtained by using this relations.To achieve this goal, we also use the methods presented in [1].Furthermore, we show the feasibility of the results to the systems (1), (2), the focal curve of the light in 3 dimensional Euclidean space, and the curve of unit speed line of sight which plays an important role to investigate proportional navigation in kinematics; and suitability to define and construct these kind of curves.

Constructing Differential Equations and Integral Relations
Let us consider the system (5-7).Since we can obtain the solution as where R is arbitrary constant.It is possible to observe that the solution set {x(t), y(t), z(t)} of the system (5-7) lies on the sphere (9), i.e. the first curvature of the curve obtained from the solution set of the system (5-7) never vanishes.Now, let us consider the equations ( 5) and ( 6), and obtain the solution set {x(t), y(t), z(t)} by choosing arbitrary z.In this case, the differential equation where A and B are arbitrary constants.By following the same procedure, we may also obtain a relation where A and B are arbitrary constants.
If we substitute relations (10) and (11) to the equation ( 9), we obtain the integral relation which only depends on z.
If variable z is chosen as to satisfy equation (12), then (10), ( 11) and (12) construct the solution set of the system (5-7).On the other hand, if we eliminate the variables x and y in the system (5-7), then we obtain the following third order linear differential equation: or where Since the equation ( 14) is obtained by the elimination of the arbitrary constants A, B and R of the integral relation ( 12), the integral relation ( 12) is the solution of the equation ( 14) in closed form.Furthermore, equations ( 10), (11) and (12) satisfy the system (5-7) and therefore equivalent to this system.Now, let us consider the equations ( 6) and (7), then by the following the same procedure above we may obtain where C, D and R are arbitrary constants.
By the elimination of the variables y and z of the system (5-7), we obtain the following third order linear differential equation: where On the other hand, if we eliminate arbitrary constants C, D and R in equation ( 17) we obtain the equation ( 19), and therefore we may conclude that equation ( 17) is the solution of equation ( 19) in closed form.Hence, equations ( 15), ( 16) and ( 17) are also equivalent to the system (5-7).Now, let us consider the point (0, y, 0) as a special case.By the equations (10), ( 11) and ( 12) we obtain Thus, at the point (0, y, 0) relations ( 28) or (29) holds.

Application
First, let us apply the results to the system (1) which characterize E 4 spherical curves.This system is a special case of the system (5-7) when x = ρ, y = f , z = g, a = τ , b = μ and s = t.Thus, by the equations ( 9), ( 15), ( 16), ( 17) and ( 19) we may obtain As a result; equation (34) is the differential equation that characterize E 4 spherical curves which lie on sphere (30) and satisfy the integral relation (33).
In geometrical optics, a curve Γ in Euclidean 3-space can be considered as a source of light.The envelope of all light rays normal to Γ is the focal surface or caustic of Γ [7].The Frenet vectors of the curve C Γ : θ → C Γ (θ) consisting of the centers of the osculating spheres of a regular curve Γ satisfy the following equation: where { t * , n * , b * } is the Frenet frame, and c 1 , c 2 are focal curvatures of the curve C Γ .If we consider the special point (0, y, 0), then by the equations ( 28) and (29); we may conclude that in the case of A = C and B = D and in the case of A = D and B = C or Therefore; through the second Euclidean axis, the focal curvatures of the light satisfy the integral relations (41) or (43).Finally, the kinematic equations of unit speed "line of sight" can be obtained as follow: [8] where e r , e θ , e ω are principle vectors of line of sight rotation coordinates; and ω, V , and Ω are , angular velocity of a coordinate system, relative velocity vector, and angular velocity of a plane; respectively.Therefore, by the equations ( 9), (10), ( 11), ( 12) and ( 14) we may obtain and ; and the third order linear differential equation Therefore; if the third axis of the rotation coordinates is chosen as satisfying the equation (48), then the integral characterization of the line of sight can be obtained as the equation (47).By following same procedure, it is also possible to characterize line of sight by the first axis of the rotation; i.e., if first axis is chosen by satisfying the third order linear differential equation then the relation is the integral characterization of the line of sight.

Conclusions and Results
In Section 2, the system (5-7) is studied by the equation duos (5,6) and (6,7), since these equations are enough to obtain the set of independent solutions.However, it is also possible to use the equation duo (5,7) for some special conditions, by following same procedure.Explicit solutions of the equations ( 14) and (19) haven't been obtained yet.Nevertheless, we obtained integral relations (12) and (17) which are the closed solutions to the equations, respectively.If the functions a(t) and b(t) in these relations are constant or the fraction a(t) b(t) is constant, then it is possible to find explicit solutions by changing the independent variable.
Methods presented in this study are restricted to some special cases.Therefore, they are useful to study some special curves.For instance, beside the spherical curves and curves of constant breadth presented in Section 3, the relations (21-29) can be useful for Bertrand curves which are special case of curves of constant breadth.Especially, when a initial condition is given one of the functions x,y, and z, complexity of calculation will be reduced significantly.Also in Section 3, we may obtain integral relation bz obtained by the elimination of y, can be reduced to the differential equation with constant coefficients by the change of variable ξ = a dt [1].Then solution of this equation is x = A − bz sin( a dt) dt cos( a dt) + B + bz cos( a dt) dt sin( a dt) (10) equations (10), (11) and (12) lead us D cos( b dt) − C sin( b dt) = 0 (24) C cos( b dt) + D sin( b dt) = y(= R) (25) C 2 + D 2 = R 2 this point A = C and B = D or A = D and B = C.If A = C and B = D, then by equations (23) and (27) tan( a dt) tan( b dt) = −1 or a dt − b dt = kπ + π 2 , k ∈ Z .(28) If A = D and B = C, then tan( a dt) + tan( b dt) = 0 or a dt + b dt = π + kπ, k ∈ Z .(29)

ρ 2 + C − ρτ ds 2 = R 2 −
D 2 = K 2 by setting g = 0 and μ = 0 in the equations (31-33), where K is a constant.The differential equation of this relation is 1 τ ρ + τρ = 0 which characterize the spherical curves [3].Therefore, the obtained integral relation is an integral property for spherical curves.At the same time, well known explicit solution ρ = A cos τ ds + B sin τ ds can be obtained easily by the equation (10).Finally, it is also possible to study the system (5-7) by considering it as a characteristic band of a first order partial differential equation, since it is possible to rewrite this system as dx ay = dy −ax + bz = dz −by = dt .