Integrability of the constrained rigid body

The integrability theory for the differential equations, which describe the motion of an unconstrained rigid body around a fixed point is well known. When there are constraints the theory of integrability is incomplete. The main objective of this paper is to analyze the integrability of the equations of motion of a constrained rigid body around a fixed point in a force field with potential U(γ ) = U(γ1, γ2, γ3). This motion subject to the constraint 〈ν,ω〉 = 0 with ν is a constant vector is known as the Suslov problem, and when ν = γ is the known Veselova problem, here ω= (ω1,ω2,ω3) is the angular velocity and 〈 , 〉 is the inner product of R3. We provide the following new integrable cases. J. Llibre ( ) Departament de Matemàtiques, Universitat Autònoma de Barcelona, Bellaterra, 08193 Barcelona, Catalonia, Spain e-mail: [email protected] R. Ramírez Departament d’Enginyeria Informàtica i Matemàtiques, Universitat Rovira i Virgili, Avinguda dels Països Catalans 26, 43007 Tarragona, Catalonia, Spain e-mail: [email protected] N. Sadovskaia Departament de Matemàtica Aplicada II, Universitat Politècnica de Catalunya, C. Pau Gargallo 5, 08028 Barcelona, Catalonia, Spain e-mail: [email protected] (i) The Suslov’s problem is integrable under the assumption that ν is an eigenvector of the inertial tensor I and the potential is such that U =− 1 2I1I2 ( I1μ 2 1 + I2μ2 ) , where I1, I2, and I3 are the principal moments of inertia of the body, μ1 and μ2 are solutions of the firstorder partial differential equation γ3 ( ∂μ1 ∂γ2 − ∂μ2 ∂γ1 ) − γ2 ∂μ1 ∂γ3 + γ1 ∂μ2 ∂γ3 = 0. (ii) The Veselova problem is integrable for the potential U =− Ψ 2 1 +Ψ 2 2 2(I1γ 2 2 + I2γ 2 1 ) , where Ψ1 and Ψ2 are the solutions of the first-order partial differential equation (I2 − I1)γ1γ2 〈 γ, ∂Ψ2 ∂γ 〉 + I1γ2 ∂Ψ2 ∂γ1 − I2γ1 ∂Ψ2 ∂γ2 − p ( γ3 〈 γ, ∂Ψ1 ∂γ 〉 − ∂Ψ1 ∂γ3 ) = 0, where p = √ I1I2I3( γ 2 1 I1 + γ 2 2 I2 + γ 2 3 I3 ). 2274 J. Llibre et al. Also it is integrable when the potential U is a solution of the second-order partial differential equation 2 ∂U ∂τ3 + I1I2I3 ∂ 2U ∂τ 2 2 + (τ2 − I1 − I2 − I3) ∂ 2U ∂τ3∂τ2 + τ3 ∂ 2U ∂τ 2 3 = 0, where τ2 = I1γ 2 1 + I2γ 2 2 + I3γ 2 3 and τ3 = γ 2 1 I1 + γ 2 2 I2 + γ 2 3 I3 . Moreover, we show that these integrable cases contain as a particular case the previous known results.