The dynamical rigid body with memory

Transcription

1 The dynamical rigid body with memory Ion Doru Albu, Mihaela Neamţu and Dumitru Opriş Abstract. In the present paper we describe the dynamics of the revised rigid body, the dynamics of the rigid body with distributed delays and the dynamics of the fractional rigid body. We analyze the stationary states for given values of the rigid body s parameters. M.S.C. 2: 26A33, 53C63,58A5, 58A4. Key words: rigid body, revised rigid body, distributed delay, fractional derivative. 1 Introduction In mechanical problems, the dynamics of the rigid body has an important role. In many papers M. Puta analyzed the dynamics of the rigid body with control and he obtained important results. Recently, the dynamics of revised rigid body, the dynamics of the rigid body with distributed delay and the dynamics of the rigid body with fractional derivative have been studied. The last two aspects represent the dynamics of the rigid body with memory. Our paper studies the dynamics of the revised rigid body obtained by a metriplectic structure which is canonically associated. We define the dynamics of the rigid body with distributed delays. For the Euler-Poincare dynamics of the rigid body we analyze the linearized system in an equilibrium point. We obtain the the existence conditions for the Hopf bifurcation with respect to the parameter of the repartition density which defines the distributed delay. Also, we define the fractional rigid body dynamics using the Caputo fractional derivative. 2 The revised differential equations for the rigid body The differential equations for the rigid body in R 3 are described by a 2-antisymmetric tensor field P and the Hamiltonian function h given by: BSG Proceedings 15. The International Conference Differential Geometry - Dynamical Systems DGDS-27, October 5-7, 27, Bucharest-Romania, pp c Balkan Society of Geometers, Geometry Balkan Press 28.

2 2 Ion Doru Albu, Mihaela Neamţu and Dumitru Opriş (2.1) P (x) = (P ij (x)) = x 3 x 2 x 3 x 1 x 2 x 1, h = 1 2 (a 1(x 1 ) 2 + a 2 (x 2 ) 2 + a 3 (x 3 ) 2 ), where (x 1, x 2, x 3 ) T R 3, a i R +, i = 1, 2, 3, a 1 > a 2 > a 3. These differential equations are: (2.2) ẋ(t) = P (x(t)) x h(x(t)), where ẋ(t) = (ẋ 1 (t), ẋ 2 (t), ẋ 3 (t)) T and x h is the gradient of h with respect to the canonical metric on R 3. The differential equations (2.2) have been studied by M. Puta in [8]. Let R 3 be the space interpreted as the space of body angular velocities Ω equipped with the cross product as the Lie bracket. On this space, we consider the standard Lagrangian kinetic energy L(Ω) = 1 2 I Ω, where I = diag(i 1, I 2, I 3 ) is the moment of inertial tensor, so that the general Euler-Poincare equations become the standard rigid body equations for a freely spinning rigid body: (2.3) I Ω = (I Ω) Ω. If M = I Ω is the angular momentum, then, (2.3) is: (2.4) Ṁ = M Ω. If M = (I 1 x(t), I 2 y(t), I 3 z(t)) T, Ω = (x(t), y(t), z(t)) T from (2.4) results: (2.5) ẋ(t) = I 2 I 3 y(t)z(t), ẏ(t) = I 3 I 1 x(t)z(t), ż(t) = I 1 I 2 x(t)z(t), I 1 I 2 with I 1 > I 2 > I 3. The differential equations (2.5) have been studied by M. Puta in [8]. The revised differential equations for the rigid body given by (2.2) have been studied in [5]. They are described by P, h and the tensor fields g = (g ij (x)), where P and h are given by (2.1) and g is defined by g(x) = (g ij (x)), g ij (x) = h(x) x i h(x) x j, i j, g ij (x) = 3 k=1,k i and the Casimir function of the Poisson structure P is: c(x) = 1 2 ((x1 ) 2 + (x 2 ) 2 + (x 3 ) 3 ). I 3 ( h(x) x k )2, i = 1, 2, 3 The structure (R 3, P, g, h, c) is called a metriplectic manifold of second kind. The revised differential system associated to (2.2) is given by: (2.6) ẋ(t) = P (x) x h(x) + g(x) x c(x).

3 The dynamical rigid body with memory 3 Proposition 2.1. (i) The differential equations (2.2) are given by: (2.7) ẋ 1 (t)=(a 2 a 3 )x 2 (t)x 3 (t), ẋ 2 (t)=(a 3 a 1 )x 1 (t)x 3 (t), ẋ 3 (t)=(a 1 a 2 )x 1 (t)x 2 (t); (ii) The differential equations (2.6) are given by: (2.8) ẋ 1 (t)=(a 2 a 3 )x 2 (t)x 3 (t)+a 2 (a 1 a 2 )x 1 (t)(x 2 (t)) 2 +a 3 (a 1 a 3 )x 1 (t)(x 3 (t)) 2 ẋ 2 (t)=(a 3 a 1 )x 1 (t)x 3 (t)+a 3 (a 2 a 3 )x 2 (t)(x 3 (t)) 2 +a 1 (a 2 a 1 )x 2 (t)(x 1 (t)) 2 ẋ 3 (t)=(a 1 a 2 )x 1 (t)x 2 (t)+a 1 (a 3 a 1 )x 3 (t)(x 1 (t)) 2 +a 2 (a 3 a 2 )x 3 (t)(x 2 (t)) 2. The equilibrium points of the system (2.7) are studied in [8] and the equilibrium points of the system (2.8) in [5]. 3 The differential system with distributed delay for the rigid body We consider the space R 3, the product R 3 R 3 = {( x, x), x R 3, x R 3 } and the canonical projections π i : R 3 R 3 R 3, i = 1, 2. A vector field X X(R 3 R 3 ) satisfying the condition X(π 1f) =, for any f C (R 3 ) is given by: X( x, x) = 3 i=1 X i ( x, x) x i. The differential system associated to X is given by: ẋ i (t) = X i ( x(t), x(t)), i = 1, 2, 3. A differential system with distributed delay is a differential system associates to a vector field X X(R 3 R 3 ) for which X(π 1f) =, for any f C (R 3 ) and it is given by (1), where x(t) is: x(t) = k k(s)x(t s)ds, where k(s) is a density of repartition. In what follows, we will consider the case of the following densities of repartition: (i) the uniform density with:, s a 1 k τ (s) = τ, a s a + τ,, s > a + τ where a >, τ > are given numbers; (ii) the exponential density, with k α (s) = αe αs, α > ; (iii) the Erlang density, with k α (s) = α 2 se αs, α > ; (iv) the Dirac density, with k τ (s) = δ(s τ), τ >.

4 4 Ion Doru Albu, Mihaela Neamţu and Dumitru Opriş The initial condition is: x(s) = ϕ(s), s (, ], with ϕ : (, ] R 3 a smooth map. Some systems of differential equations with distributed delay in R 3 were studied in [1], [2]. For such a system, we consider relevant the geometric properties of the vector field which defines the system, for example first integrals (constants of the motion), Morse functions, almost metriplectic structure, etc. The differential equations with distributed delay for rigid body are generated by a 2-antisymmetric tensor field P on R 3 R 3 that satisfies the following relations: P (π 1f 1, π 2f 2 ) =, P (π 2f 1, π 2f 2 ) =, for all f 1, f 2 C (R 3 ) and h C (R 3 R 3 ). The differential equation is given by: (3.1) ẋ = P ( x, x) x h( x, x). Let P ( x, x) be the tensor field with the components given by: x3 x 2 (3.2) (P ij (x, x)) = x 3 x 1 x 2 x 1 and (3.3) h ( x, x) = a 1 x 1 x 1 + a 2 x 2 x 2 + a 3 x 3 x 3. The differential equations (3.1) are: The differential system: ẋ 1 (t) = a 2 x 2 (t)x 3 (t) a 3 x 2 (t) x 3 (t), ẋ 2 (t) = a 3 x 1 (t) x 3 (t) a 1 x 1 (t)x 3 (t), ẋ 3 (t) = a 1 x 1 (t) x 2 (t) a 2 x 1 (t) x 2 (t). (3.4) ẋ = P ( x, x) x h( x, x) + g( x, x) x c( x, x), where the components of g( x, x) are: (3.5) g ij (x) = h(x, x) x i c(x, x) x j, i j, g ii (x) = 3 k=1,k i h(x, x) c(x, x) x k x k, i = 1, 2, 3 is called the revised differential system with distributed delay associated to the differential system (3.1). From (3.2), (3.3), (3.5) and c( x, x) = 1 2 (x1 ) 2 + x 2 x (x3 ) 2 results: a 2 2x 2 x 2 a 3 x 3 x 3 a 1 a 2 x 1 x 2 a 1 a 3 x 1 x 3 (g ij )= a 1 a 2 x 1 x 2 a 2 1x 1 x 1 a 3 x 3 x 3 a 2 a 3 x 2 x 3. a 1 a 3 x 1 x 2 a 2 a 3 x 2 x 3 a 2 1x 1 x 1 a 2 2x 2 x 2 The differential system (3.4) is given by: (3.6) ẋ 1 (t) = a 2 x 2 (t)x 3 (t) a 3 x 2 (t) x 3 (t) + a 1 a 2 x 1 (t)(x 2 (t)) 2, ẋ 2 (t) = a 3 x 1 (t) x 3 (t) a 1 x 1 (t)x 3 (t) a 2 1x 1 (t) x 1 (t)x 2 (t) a 2 3x 2 (t)x 3 (t) x 3 (t), ẋ 3 (t) = a 1 x 1 (t)x 2 (t) a 2 x 1 (t) x 2 (t) + a 2 a 3 x 2 (t)x 2 x 3.

5 The dynamical rigid body with memory 5 For the Dirac distribution, system (3.6) was analyzed in [6]. The other types of densities will be analyzed in our future papers. The Euler-Poincare equation for the free rigid body with distributed delay is defined by: Ṁ = M Ω + αm ( M Ω) where M = (I 1 x(t), I 2 y(t), I 3 z(t)) T, Ω = (x(t), y(t), z(t)) T, Ω = ( x(t), ỹ(t), z(t)) T, M = I Ω, I 1 >, I 2 >, I 3 > and α R. The equilibrium points of our system are Ω 1 = ( m I 1,, ) T, Ω 2 = (, m I 2, ) T, Ω 3 = (,, m I 3 ) T, m R. Proposition 3.1. The equilibrium point Ω 1 has the following behavior: (i) The corresponding linear system is given by: U(t) = AU(t) + αbũ(t) where U(t) = (u 1 (t), u 2 (t), u 3 (t)) T and A = I 3 I 1 I 1 I 2 m, B = I 1 I 2 I 1 I 3 m I 2 I 1 I 1 I 2 m 2 I 3 I 1 I 1 I 3 m 2 ; (ii) The characteristic equation is: λ[λ 2 αm2 I 1 ( I 2 I 1 I 2 + I 3 I 1 I 3 )λk (1) (λ) + α2 m 4 I 2 1 I 2I 3 (I 2 I 1 )(I 3 I 1 )k (1) (λ) 2 (I 1 I 2 )(I 3 I 1 ) I 2 1 I 2I 3 m 2 ] = ; (iii) On the tangent space at Ω 1 to the sphere of radius m 2 the linear operator given by the linearized vector field has the characteristic equation: λ 2 αm2 I 1 ( I 2 I 1 I 2 + I 3 I 1 I 3 )λk (1) (λ) + α2 m 4 I 2 1 I 2I 3 (I 2 I 1 )(I 3 I 1 )k (1) (λ) 2 (I 1 I 2 )(I 3 I 1 ) I 2 1 I 2I 3 m 2 ] = ; (iv) If I 1 > I 2, I 1 > I 3 and k (1) (λ) = e τλ, τ >, for τ < τ c, where τ c = I 1(I 3 (I 1 I 2 ) + I 2 (I 1 I 3 )) 3 α m 2, (I 1 I 2 )(I 1 I 3 ) then the equilibrium point Ω 1 is asymptotically stable. The analysis of the equilibrium point Ω 1 for the Dirac density is given in [1].

6 6 Ion Doru Albu, Mihaela Neamţu and Dumitru Opriş 4 Fractional differential systems for the rigid body Generally speaking, the fractional derivative, Riemann-Liouville fractional derivative and Caputo s fractional derivative are mostly used. In the present paper we discuss the Caputo derivative: D α t x(t) = I m α ( d dt )m x(t), α >, where m 1 < α m, ( d dt )m = d dt... d dt, Iβ is the βth order Riemann-Liouville integral operator, which is expressed as follows: I β x(t) = 1 Γ(β) t (t s) β 1 x(s)ds, β >. In this paper, we suppose that α (, 1). Examples of the fractional differential systems are: the fractional order of Chua s system, the fractional order of Rossler s system and the fractional Duffing oscillator. The geometrical and mechanical interpretation of the fractional derivative is given in [7]. The geometry of fractional osculator bundle of higher order was made in [3] using the fractional differential forms defined in [4]. A fractional system of differential equations with distributed delay in R 3 is given by: (4.1) D α t x(t) = X(x(t), x(t)), α (, 1), where x(t) = (x 1 (t), x 2 (t), x 3 (t)) T R 3. The linearized of (4.1) in the equilibrium point x, (X(x, x ) = ) is given by the following linear fractional differential system: (4.2) D α t u(t) = Au(t) + Bũ(t), where A = ( X x ) x=x, B = ( X x ) x=x. The characteristic equation of (4.2) is: (4.3) (λ) = det(λ α I A k (1) (λ)b) =, where k (1) (λ) = k(s)e λs ds. From (4.2) we have: Proposition 4.1. ([4]) (i) If all the roots of characteristic equation (λ) = have negative real parts, then the equilibrium point x of (4.1) is asymptotically stable; (ii) If k(s) is the Dirac distribution, the characteristic equation (4.3) is given by: (λ) = det(λ α I A e λτ B) =. If τ =, α (, 1) and all the roots of the equation det(λi A B) = satisfy arg(λ) > απ 2, then the equilibrium point x is asymptotically stable; (iii) If α (.5, 1) and the equation det(λi A Be λτ ) = has no purely imaginary roots for any τ >, then the equilibrium point x is asymptotically stable.

7 The dynamical rigid body with memory 7 For the following delayed fractional equation (see [4]) (4.4) D α t x(t) = ax(t τ) where α (, 1), a R and τ > the stability condition is: If a <, ( a) 1 α 1 τ ((2k + 1)π α 2 π) and ( a) 1 α 1 τ ((2k + 1)π α 2 π), k Z, then the zero solution of (4.4) is asymptotically stable. For the following delayed fractional equation (see [4]) (4.5) D α t x(t) = y(t) k 1 x(t) D α t y(t) = (k 1 + k 2 )y(t) + x(t τ), where α (, 1), k 1, k 2 >, τ >, the stability condition is: If k 1 >, k 2 > 1 k 1 k, then the zero solution of system (4.5) is asymptotically stable. For f C (R 3 ), by Dx α f, D α 1 x f, D α 2 x f we denote the Caputo partial derivatives 3 defined by: (4.6) Dx α if(x) = 1 x i Γ(1 α) f(x 1,...x i 1, s, x i+1..., x n ) 1 x i (x i ds, i = 1, 2, 3 s) α where (x i ) are the coordinate functions on R, and ( x i ), i = 1, 2, 3 is the canonical base of the vector field on R n. From (4.6) it results: Dx α i(xi ) γ = (xi ) γ α Γ(1 + γ), Dx α Γ(1 + γ α) i(xj ) =, i j. Let X α (R 3 ) be the module of the fractional vector fields generated by the operators {D α x i, i = 1, 2, 3} and the module D(R 3 ) generated by 1-forms {d(x i ) α, i = 1, 2, 3}. The fractional exterior derivative d α : C (R 3 ) D(R 3 ) is defined by: d α (f) = d(x i ) α D α x i(f). Let α P X α (R 3 ) X α (R 3 ) be a fractional 2-skew-symmetric tensor field and d α f, d α g D(R 3 ). The bilinear map [, ] α : C (R 3 ) C (R 3 ) C (R 3 ) defined by: (4.7) [f, g] α = α P (d α f, d α g), f, g C (R 3 ) is called the fractional Leibniz bracket. If P α = P α ij Dx α D α i x then from (4.7) it follows that: j [f, g] α = α P ij D α x ifdα x j g. From the properties of the fractional Caputo, results: ( ) ( ) [fh, g] α α αp k = ij (D α k k x f)(d α i x g) j x i h k= ( ) ( ) [f, gh] α α αp k = ij (D α k x if)(dα k x g) j x i h. k=

8 8 Ion Doru Albu, Mihaela Neamţu and Dumitru Opriş If α P is skew-symmetric we say that (R 3, [, ] α ) is a fractional almost Poisson manifold. If α 1 then we obtain the concepts from [8]. For h C (R 3 ), the fractional almost Poisson dynamic system is given by: (4.8) D α t x i (t) = [x i (t), h(t)] α, where[x i, h] α = α P ij D α x j h. Let α P be a 2-skew-symmetric fractional tensor field and α g a 2-symmetric fractional tensor field on R 3. We define the bracket [, ] α : C (R 3 ) C (R 3 ) C (R 3 ) by: [f, h] α = α P (d α f, d α g) + α g(d α f, d α h), f, h C (R 3 ). The structure (M, α P, α g, [, ]) is called fractional almost metriplectic manifold. The fractional dynamic system associated to h C (R 3 ) is (4.9) D α t x i (t) = [x i (t), h(t)] α, where[x i, h] α = α P ij D α x j h + α g ij D α x j h. If we define the bracket [, (, )] α : C (R 3 ) C (R 3 ) C (R 3 ) C (R 3 ) by: [f, (h 1, h 2 )] α = α P (d α f, d α h 1 ) + α g(d α f, d α h 2 ), f, h 1, h 2 C (R 3 ), then, the fractional vector field α Xh 1 h 2 defined by: α Xh 1 h 2 (f) = [f, (h 1, h 2 )] α, f C (R 3 ) is called the fractional almost Leibniz vector field associated to the functions h 1, h 2 C (R 3 ). The fractional almost Leibniz dynamical system is given by: D α t x i (t) = α P ij D α x j h 1 + α g ij D α x j h 2. Let P α = (P ij ), g α = (g ij ) be the fractional 2-tensor fields on R 3 and h C (R 3 ) given by: 1 h = Γ(α + 1) [a 1(x 1 ) α+1 + a 2 (x 2 ) α+1 + a 3 (x 3 ) α+1 ]. Proposition 4.2. (i) The fractional dynamic system (4.8) is: (4.1) D α t x 1 = (a 2 a 3 )x 2 x 3, D α t x 2 = (a 3 a 1 )x 1 x 3, D α t x 3 = (a 1 a 2 )x 1 x 2 ; (ii) The fractional dynamic system (4.9) is: (4.11) D α t x 1 = (a 2 a 3 )x 2 x 3 + a 2 (a 1 a 2 )x 1 (x 2 ) 2 + a 3 (a 1 a 3 )x 1 (x 3 ) 2 D α t x 2 = (a 3 a 1 )x 1 x 3 + a 3 (a 2 a 3 )x 2 (x 3 ) 2 + a 1 (a 2 a 1 )x 2 (x 1 ) 2 D α t x 3 = (a 1 a 2 )x 1 x 2 + a 1 (a 3 a 1 )x 3 (x 1 ) 2 + a 2 (a 3 a 2 )x 3 (x 2 ) 2 ; (iii) The fractional dynamic systems (4.1) and (4.11) have the equilibrium points M 1 (m,, ), M 2 (, m, ), M 3 (,, m), m R ; (iv) The characteristic equations for (4.1) are:

9 The dynamical rigid body with memory 9 in M 1 (m,, ): λ α (λ 2α + (a 1 a 3 )(a 1 a 2 )m 2 ) =, in M 2 (, m, ): λ α (λ 2α (a 1 a 2 )(a 2 a 3 )m 2 ) =, in M 3 (,, m): λ α (λ 2α +(a 1 a 3 )(a 2 a 3 )m 2 ) = ; (v) The characteristic equations in M 1 (m,, ), M 2 (, m, ), M 3 (,, m) for (4.11) are: λ α (λ 2α a 1 (a 2 + a 3 2a 1 )m 2 λ α + (a 1 a 3 )(a 1 a 2 )m 2 (a 2 1m 2 + 1)) =, λ α (λ 2α a 2 (a 1 + a 3 2a 2 )m 2 λ α (a 1 a 2 )(a 2 a 3 )m 2 (a 2 2m 2 + 1)) =, λ α (λ 2α a 3 (a 1 + a 2 2a 3 )m 2 λ α + (a 1 a 3 )(a 2 a 3 )m 2 (a 2 3m 2 + 1)) = ; The above findings allow the analysis of the equilibrium points with respect to the parameters of the characteristic equations. Equations (4.1) are called the fractional equations of the rigid body and equations (4.11) are called revised fractional equations of the rigid body. If α 1, the results from Proposition 4.2 lead to results from [8]. For a 1 := 3, a 2 := 2, a 3 := 1, α = 1 the dynamics (x 1 (t), x 2 (t), x 3 (t)) of (4.1) is given in figure Fig.1 and for α =.82 in figure Fig.2. The numerical algorithm used is Adam-Moulton-Bashford. Fig.1.alfa=1 Fig.2.alfa= x^3 1 x^ x^ x^ x^ x^1 5 Conclusions In the present paper we present the dynamics of the rigid body with memory. The memory was described by the variables with distributed delay and by the Caputo fractional derivative. References [1] I.D. Albu, M. Neamtu and D. Opris, Dissipative mechanical systems with delay, Tensor N.S. 67 (26), [2] I.D. Albu, D.Opris, Leibnitz dynamics with time delay, arxiv.math/58225.

10 1 Ion Doru Albu, Mihaela Neamţu and Dumitru Opriş [3] I.D. Albu, M. Neamtu and D. Opris, The geometry of fractional osculator bundle of higher order and applications, International Conference on Differential Geometry Lagrange and Hamilton Spaces, Iasi, September, 3-8, 27, Scintific Annals of Al. I. Cuza University, Mathematics, LIII (27):21-32, arxiv:79.2v1, math.dg, 27. [4] W. Deng, C. Li and J. Lü, Stability analysis of linear differential systems with multiple time delay, Nonlinear Syn. 48 (27), [5] D. Fish, Dissipative perturbations of 3D Hamiltonian systems, arxiv:math/5647v1, 25. [6] M. Neamtu, D. Opris and M. Popescu, Hopf Bifurcation for the Rigid Body with Time Delay, Proceedings of the 5th conference of Balkan Society of Geometers, Geometry Balkan Press, Bucharest 26, [7] I. Podlubny, Geometrical and physical interpretation of fractional integration and fractional differentiation, Fractional Calculus and Applied Analysis 5, 4 (22), [8] M. Puta, Hamiltonian systems and geometric quantisation, Mathematics and Applications, Vol. 26, Kluwer Academic Publishers 24. Authors addresses: Ion Doru Albu and Dumitru Opriş Department of Mathematics, Faculty of Mathematics, West University of Timişoara, 4 Bd. V. Parvan, 3223, Timişoara, Romania. albud@math.uvt.ro, opris@math.uvt.ro Mihaela Neamţu Department of Economic Informatics and Statistics, Faculty of Economics, West University of Timişoara, 16A Pestalozzi St., 3115, Timişoara, Romania. mihaela.neamtu@fse.uvt.ro