Kinematics and dynamic differential equations of rotation of a rigid body for the «non-classical» conjugate three- dimensional new vectors of rotation are considered in this article. The vectors modules contain tangent and cotangent of one-fourth of the rotation angle. This paper also delivers applications of the conjugate equations in the dynamic and of orientation problems of a rigid body. Different new (polar) dynamical differential equations of RB rotation come out on the basis of kinematics equations.

In classical Euler occasion, for example, problem of decision of the system of six dynamical differential equations of Euler-Poisson reduces to integration of the system of just three dynamical equations with two independent classical first integrals of energy and square. Three new "tangent" vector coordinates-parameters definitely determine RB orientation in contrast to three direction cosines determined orientation of just one unit vector of supporting basis I in Poisson equations.

New vectors and conjugated polar equations define a (essentially) new line of fundamental research in solid mechanics.

1. Three-dimensional vectors of rotation with minimal possible quantity of generalized coordinate lining with three freedoms of rigid body (RB)[1-4] with fixed point are of the most interest in the problem of dimensional orientation such as a task of orientation order parameter identification and directing of rotation (spherical) motion of RB.

Classical Rodrigues vector [1,4] has a module with tangent (tg(φ/2)) of half-angle φ of finite rotation of RB. This vector can not be used in the condition of φ=π(180º) in the problems of RB dimensional orientation. Considered in the works [2-4] tangent, cotangent conjugate rotation vectors τ=τk, ρ=ρk, where τ=k_τtg(φ/4), ρ=k_ρctg(φ/4) (k_τ, k_ρ - arbitrary constant coefficients), k - unit vector of Euler axis of RB finite rotation are of the most interest (by 0< φ<2π) because of its conjugate features [2,4,5].

2. The article deals with conjugative polar non-linear vector kinematics differential equations of RB rotation in the transformation of lineal vector ω(t) of RB angular velocity [3,4]

τ* = τ 'Θ _τ ω, ρ*= ρ 'Θ^T_ρ ω , (1)

where τ*= dτ/dt, ρ*= dρ/dt - local derivative of vectors in the time t (derivative towards some connected with RB coordinate base); τ '=ðτ/ðφ, ρ'=ðρ/ðφ - particular derived modules τ, ρ of vectors of angle φ, defined as function

τ'=(k _τ+ τ ²/ k _τ )/4 , ρ'= - (k_ρ + ρ ²/ k_ρ )/4 ; Θ_τ, Θ^T_ρ

- orthogonal operators semi-rotation of RB (angle φ/2) [3]:

Θ_τ=E+2(k _τT+T²)/(k²_τ+τ²),

Θ ^T_ρ =E+2(-k_ρR+R²)/( k_ρ²+ρ ²),

«T» - transposing.

Therein T, R - skew-symmetric operators of vector multiplication [2, 4], satisfies the identities Tτ= τ × τ = 0 , R ρ = ρ × ρ = 0 - vector products, 0 - zero vector, E - unit operator. Operators Θ_τ, Θ_ρ satisfy the identity (Θ_τΘ_ρ)²=E, determined conjugacy of vectors τ, ρ and equations (1) [4] as features of "duality" and isomorphic correspondence.

3. Equations (1) have the first («trigonometric») integral in the form of scalar product (τ · ρ) = τ ρ=k_τ k_ρ = C (arbitrary constant) and admit simple and graphic kinematics interpretation - polar precession-nutation model [4] with arbitrary vector ω _τ and for any RB. In the first equation vector ω is transformed (at every moment of the traveling time of RB) by operator Θ_τ into vector ω _τ as a result of precession ω _τ on the angle ψ _τ = φ/2 (rotation vector ω with module ω (t) about the surface of some circular cone of «velocity» precession cone angle 2υ_ω) around Euler axis with unit vector k . Angle υ_ω - nutation angle (vector deviation ω from the unit vector k). And then a vector ω _τ is multiplied by scalar operator τ'E. At the second equation vector ω precesses (rotates n the surface of that) at the angle ψ_Θρ = (π+φ/2) and is multiplied by scalar operator ρ'E. Angle υ_ω of nutation is determined from scalar product (ω · τ), (τ* · τ) or (ω · ρ), (ρ*· ρ).

On the basis of such interpretation equation - models (1) come out, for example, kinematics anholonomic equality:

(a· b)² +((a × b) · (a × b)) = (a·a)(b · b),

where a = ω × τ , b = (Θ_τ ω) × τ.

Equations (1) have general decisions in the form of Cauchy [4, 5]. General decision of the first equation in (1) determines the form shape of the new vector representations of three-dimensional rotation (a Lie group) [6]. In this group inverse element τ ^-1 is equal to adverse vector, i. e. τ ^-1 = - τ , but unit element is equal to zero vector 0 .

4. Different new (polar) dynamical differential equations of RB rotation come out on the basis of kinematics equations (1).

In classical Euler occasion, for example, problem of decision of the system of six dynamical differential equations of Euler-Poisson [7] reduces to integration of the system of just three dynamical equations with two independent classical first integrals (of energy and square [7]). These two integrals are enough to consider the system of three equations to be integrated [7].

At this case change is introduced in the equations (1) ω=S^-1g , where g - constant vector of kinetic moment (constant module and in the line of supporting basis I [7]); S^-1 - inverse operator S of RB inertia [3,4,7] (constant in connected with RB basis J).

Contained of (1), for example, polar matrix differential equation with vector coordinates-parameters τ has the form (see also [3]):

τ*_{J =} τ '(Θ_{τ J} S _J ^-1 Θ_{τ J} ^T) Θ_{τ J} ^T g _I , (2)

where τ * _J =[ τ_j^.₁ τ_j^.₂ τ_j^.₃ ] ^T - column matrix with derivative coordinates τ* in basis J; Θ_{τ J} - matrix (3x3) of operator Θ_τ in basis J; S _J - diagonal matrix (3x3) with three constant the main moments of RB inertia; S_J^-1 - inverse matrix, g_I = [g_i1 g_i2 g_i3]^T - constant column matrix with vector coordinates g in basis I.

Three coordinates of the new vector-parameter τ is uniquely determined RB orientation (as opposed to the three direction cosines of only one unit ortho-vertical supporting basis I in the Poisson equations [7]). In the problems of the dynamics of RB and synthesis problems of attitude control laws (definition of control moments) with the use of Lyapunov functions (see, eg, [8]), the scalar equations (1), (2) are used in conjunction with the classical dynamic Euler equations [1, 7, 8].

5. Conjugate vectors τ, ρ are considering as parts of vectors with non-traditional non-normalized (non-Hamiltonian) rotation new quaternions [8]. Such non-normalized quaternions can be effectively used instead of classical normalized quaternions (with parameters of Rodrigues-Hamilton, Euler- Rodrigues [4]) in different «particular» tasks of inertial orientation [1, 4, 9, 10], including RB orientation control problem [8].

Sets of coordinates-parameters of conjugate vectors τ, ρ, corresponded to them quaternions sufficiently broaden the circle of new RB orientation parameters. Such parameters can be of interest in the problem of global parameterization [9] of three-dimensional rotation group and new representation Lorenz group in quantum mechanics [11].

New vectors τ_, ρ and conjugated polar equations (1), (2) define the (essentially) the new direction of fundamental research in solid mechanics.

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