Integrability of the constrained rigid body

Jaume Llibre, Rafael Ramírez, Natalia Sadovskaia

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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,γ 2,γ 3). This motion subject to the constraint 〈ν, ωâŒ=0 with ν is a constant vector is known as the Suslov problem, and when ν=γ is the known Veselova problem, here ω=(ω 1,ω 2,ω 3) is the angular velocity and 〈, 〉 is the inner product of ℝ. We provide the following new integrable cases. (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 21 + I2μ12, where I1, I2, and I3 are the principal moments of inertia of the body, μ 1 and μ 2 are solutions of the first-order partial differential equation γ3(;μ1/ ;γ1 - ;μ2/;γ2 -γ2 ;μ1/;γ3 + γ1 ;μ2/ ;γ2 = 0. (ii) The Veselova problem is integrable for the potential U = -Ψ12 + Ψ22/2(I1γ22 + I 2γ2, where Ψ1 and Ψ 2 are the solutions of the first-order partial differential equation (l 2 - l1)γ1γ2〈γ, Ψ 2/ γ〉 + I1γ2Ψ2/ γ1 - I2γ1Ψ2/ γ2- P(γ3〈γ,Ψ1/γ〉 - Ψ1/γ3= 0, where p = √I1 I2 I3(γ12/I1+γ 22/I2 + γ32/I 3. Also it is integrable when the potential U is a solution of the second-order partial differential equation 2U/τ3 + I 1I2I32U/τ22 + (τ2 - I1 - I2 + I3) 2U/τ3τ2 + τ3 2U/ τ32 = 0, where τ2 = I1γ 12 + I2γ22 + I 3γ22 and τ = γ12/ I1 + γ22/ I2 + γ32/ I1. Moreover, we show that these integrable cases contain as a particular case the previous known results. © 2013 Springer Science+Business Media Dordrecht.
Original languageEnglish
Pages (from-to)2273-2290
JournalNonlinear Dynamics
Issue number4
Publication statusPublished - 1 Sept 2013


  • Constraint
  • Integrability
  • Invariant measure
  • Mechanical systems
  • Nonholonomic system
  • Ordinary differential equation
  • Rigid body
  • Suslov problem
  • Veselova problem


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