Conjugate loci of Riemannian metrics on 2D manifolds related to the - PowerPoint PPT Presentation

Conjugate loci of Riemannian metrics on 2D manifolds related to the Euler top problem Nataliya Shcherbakova Laboratoire de Génie Chimique, ENSIACET - INP Toulouse L2S seminar, April 4th, 2013 Nataliya Shcherbakova (LGC-ENSIACET-INPT) April 4, 2013 1 / 45

Collaborators Bernard Bonnard, Institut de Mathématiques de Bourgogne; Olivier Cots, INRIA Sophia-Antipolis, McTao; Jean-Baptiste Pomet, INRIA Sophia-Antipolis, McTao Nataliya Shcherbakova (LGC-ENSIACET-INPT) April 4, 2013 2 / 45

Related Publications B. Bonnard, O. Cots, J.-B. Pomet, N. Shcherbakova. Riemannian metrics on 2D manifolds related to the Euler-Poinsot rigid body problem (2012, preprint); B. Bonnard, O. Cots, N. Shcherbakova. The Serret-Andoer Riemannian metric and Euler-Poinsot rigid body motion (2012, to appear in Math. Control and Relat. Fields); H. Yuan, R. Zeier, N. Khaneja, S. Lloyd. Elliptic functions and efficient control of Ising spin chains with unequal couplings. Ph. Rev. A 77, 032340, 2008; Nataliya Shcherbakova (LGC-ENSIACET-INPT) April 4, 2013 3 / 45

Plan Euler’s top problem: classical formulation, optimal control model and Euler’s equations. Integrability via the Serret-Andoyer trasformation. The Seret-Andoyer metric on a 2D surface : geodesics, normal form and conjugate locus. Optimal control of a linear chain of three spin- 1 2 particles as a sub-Riemannian version of the Euler top problem. First results on the conjugate locus of the tree spins problem. Perspectives. Nataliya Shcherbakova (LGC-ENSIACET-INPT) April 4, 2013 4 / 45

The Euler top problem Classical Mechanics viewpoint: a rigid body, modeled by its inertia ellipsoid with principal momenta of inertia I 1 , I 2 and I 3 , freely rotates about its center of mass, which is fixed. Optimal control formulation: the Euler’s top rotations are the extremals of the optimal control problem on SO ( 3 ) :   0 − u 3 u 2 3 ˙  = � ( P ) R ( t ) = R ( t ) u 3 0 − u 1 u i A i ( R ( t )) ,  − u 2 u 1 0 i = 1 T 3 � 1 � u 2 i I i → min u ( · ) , T − fixed , 2 i = 1 0 where R ( t ) ∈ SO ( 3 ) and A i ∈ so ( 3 ) are the elements of the orthonormal basis       0 0 0 0 0 1 0 − 1 0  ,  , A 1 = 0 0 − 1 A 2 = 0 0 0 A 3 = 1 0 0     − 1 0 1 0 0 0 0 0 0 verifying [ A 1 , A 2 ] = A 3 , [ A 2 , A 3 ] = A 1 , [ A 3 , A 1 ] = A 2 . Nataliya Shcherbakova (LGC-ENSIACET-INPT) April 4, 2013 5 / 45

Euler’s equations and integrability Assume R ( 0 ) and R ( T ) are fixed. Denote H i = p ( A i ( R )) and 3 3 � � I i u 2 H u = u i H i − ν i , ν ∈ { 0 , 1 } . i = 1 i = 1 ⇒ ν = 1, u i = H i I − 1 Pontryagin’s maximal principle = and optimal trajectories are i projections of the extremals, i.e., solutions of the Hamiltonian system associated to � H 2 I 1 + H 2 I 2 + H 2 � H n = 1 1 2 3 . 2 I 3 Since ˙ H i = { H n , H i } , we get Euler’s equations : � 1 � 1 � 1 � � � dH 1 I 3 − 1 dH 2 I 1 − 1 dH 3 I 2 − 1 = H 2 H 3 , = H 1 H 3 , = H 1 H 2 . dt I 2 dt I 3 dt I 1 Nataliya Shcherbakova (LGC-ENSIACET-INPT) April 4, 2013 6 / 45

Euler’s equations Observations: In Classical Mechanics H = ( H 1 , H 2 , H 3 ) describes the vector of the angular momentum of the body written in the moving frame related to the principal axes of inertia of the body, while u = ( u 1 , u 2 , u 3 ) is the angular velocity vector. Euler’s equations are integrable by quadratures using two first integrals: the Hamiltonian H n = h and � H � 2 = H 2 1 + H 2 2 + H 2 3 . Solutions to Euler’s equations can be seen as the curves on the energy ellipsoid formed by its intersection with the sphere of the constant angular momentum called polhodes in Classical Mechanics. If I 3 > I 2 > I 1 , the physically � � � H � 2 2 I 1 , � H � 2 realizable motions take place iff h ∈ . 2 I 3 The full system on SO ( 3 ) is integrable by quadratures. Nataliya Shcherbakova (LGC-ENSIACET-INPT) April 4, 2013 7 / 45

Polhodes on the energy ellipsoid Nataliya Shcherbakova (LGC-ENSIACET-INPT) April 4, 2013 8 / 45

Full system Observations: Remaining equations of motion can be obtained using some suitable coordinates on SO ( 3 ) , for instance, the Euler angles Φ i , i = 1 , 2 , 3 so that R = exp (Φ 1 A 3 ) ◦ exp (Φ 2 A 2 ) ◦ exp (Φ 3 A 3 ) . A complete picture of motion can be seen as the rotation of the energy ellipsoid on a fixed plane (Poinsot model), where the point of contact on the ellipsoid moves along a polhode, while its trace on the plane is called a herpolhode : Nataliya Shcherbakova (LGC-ENSIACET-INPT) April 4, 2013 9 / 45

Serret-Andoer’s transformation The Serret-Andoyer coordinates is a set of symplectic coordinates x , y , z , p x , p y , p y , defined by � � H 1 = p 2 x − p 2 y sin y , H 2 = p 2 x − p 2 y cos y , H 3 = p y , Then � sin 2 y + cos 2 y y ) + p 2 � H n = H a = 1 ( p 2 x − p 2 y 2 I 3 . 2 I 1 I 2 x and z are cyclic variables, p x and p z are first integrals with p x = � H � , and moreover, z = const . The dynamics on the ( x , y ) plane is described by solutions to the system dx dp x dt = p x ( A sin 2 y + B cos 2 y ) , dt = 0 , dy dp y dt = p y ( C − A sin 2 y − B cos 2 y ) , dt = ( B − A )( p 2 x − p 2 y ) sin y cos y . where A = 1 B = 1 C = 1 I 1 , I 2 , I 3 . Nataliya Shcherbakova (LGC-ENSIACET-INPT) April 4, 2013 10 / 45

Serret-Andoer’s transformation Observations : The Serret-Andoer transformation is not fiber-preserving. The Serret-Andoyer variables are not the action-angle variables. A further transformation using the Hamilton - Jacobi method leads to the standard action-angle representation (Yu. Sadov 1970, H. Konoshita 1972). If A < B < C , then H a defines a Riemannian metric on a 2D surface: 2 2 w ( y ) dx 2 + 2 C − w ( y ) dy 2 , g a = where w ( y ) = 2 ( A sin 2 y + B cos 2 y ) . Nataliya Shcherbakova (LGC-ENSIACET-INPT) April 4, 2013 11 / 45

Extremals of Serret-Andoer metric dx dp x dt = p x ( A sin 2 y + B cos 2 y ) , ( ∗ ) dt = 0 , dy dp y dt = p y ( C − A sin 2 y − B cos 2 y ) , dt = ( B − A )( p 2 x − p 2 y ) sin y cos y . Nataliya Shcherbakova (LGC-ENSIACET-INPT) April 4, 2013 12 / 45

Extremals of the Serret-Andoer metric dx dp x dt = p x ( A sin 2 y + B cos 2 y ) , ( ∗ ) dt = 0 , dy dp y dt = p y ( C − A sin 2 y − B cos 2 y ) , dt = ( B − A )( p 2 x − p 2 y ) sin y cos y . H a ( y , p y ) = H a ( y + π, p y ) ; Periodicity: Symmetry: H a ( y , p y ) = H a ( y , − p y ) , H a ( y , p y ) = H a ( − y , p y ) ; The phase portrait on the ( y , p y ) -plane is of pendulum type with stable equilibrium at y = π 2 (2 h = Ap 2 x ) and unstable ones at y = 0 mod π (2 h = Bp 2 x ). Thus there are two types of periodic trajectories: oscillations ( Ap 2 x < 2 h < Bp 2 x ) ) and rotations ( Bp 2 x < 2 h ≤ Cp 2 x ) ), and separatrices (2 h = Bp 2 x ). y ( t ) describes the trajectory on the 0 energy level set of the natural mechanical system � � p 2 � � 1 C − w ( y ) x w ( y ) y 2 + 2 ˙ − 2 h = 0 . 2 2 Nataliya Shcherbakova (LGC-ENSIACET-INPT) April 4, 2013 13 / 45

Extremals of the Serret-Andoer metric dx dp x dt = p x ( A sin 2 y + B cos 2 y ) , ( ∗ ) dt = 0 , dy dp y dt = p y ( C − A sin 2 y − B cos 2 y ) , dt = ( B − A )( p 2 x − p 2 y ) sin y cos y . H a ( y , p y ) = H a ( y + π, p y ) ; Periodicity: Symmetry: H a ( y , p y ) = H a ( y , − p y ) , H a ( y , p y ) = H a ( − y , p y ) ; The phase portrait on the ( y , p y ) -plane is of pendulum type with stable equilibrium at y = π 2 (2 h = Ap 2 x ) and unstable ones at y = 0 mod π (2 h = Bp 2 x ). Thus there are two types of periodic trajectories: oscillations ( Ap 2 x < 2 h < Bp 2 x ) ) and rotations ( Bp 2 x < 2 h ≤ Cp 2 x ) ), and separatrices (2 h = Bp 2 x ). y ( t ) describes the trajectory on the 0 energy level set of the natural mechanical system � � p 2 � � 1 C − w ( y ) x w ( y ) y 2 + 2 ˙ − 2 h = 0 . 2 2 Nataliya Shcherbakova (LGC-ENSIACET-INPT) April 4, 2013 13 / 45

Extremals of the Serret-Andoer metric 2 h − Ap 2 x ( B − A ) , ξ 3 = C − A Proposition. Set ξ 1 = x B − A . Then trajectories starting at the point ( x 0 , y 0 ) can be p 2 parametrized as follows: i). oscillating trajectories: ξ 1 cn 2 ( Mt + ψ 0 | m ) cos 2 y ( t ) = 1 − ξ 1 sn 2 ( Mt + ψ 0 | m ) , Bt − ( B − A )( 1 − ξ 1 ) τ = t � � � x ( t ) − x 0 = p x Π( ξ 1 | am ( M τ + ψ 0 | m ) | m ) , � M � τ = 0 cn ( ψ 0 | m ) = ( 1 − ξ 1 ) cos 2 y 0 m = ξ 1 ( ξ 3 − 1 ) � , M = ( B − A ) p x ξ 3 − ξ 1 , ; ξ 1 sin 2 y 0 ξ 3 − ξ 1 ii). rotating trajectories: ξ 1 cn 2 ( Mt + ψ 0 | m ) cos 2 y ( t ) = ξ 1 − sn 2 ( Mt + ψ 0 | m ) , ( A + ( B − A ) ξ 1 ) t − ( B − A )( ξ 1 − 1 ) Π( 1 τ = t � � � x ( t ) − x 0 = p x | am ( M τ + ψ 0 | m ) | m ) , � M ξ 1 � τ = 0 cn ( ψ 0 | m ) = ( ξ 1 − 1 ) cos 2 y 0 ξ 3 − ξ 1 � m = ξ 1 ( ξ 3 − 1 ) , M = ( B − A ) p x ξ 1 ( ξ 3 − 1 ) , . ξ 1 − cos 2 y 0 Nataliya Shcherbakova (LGC-ENSIACET-INPT) April 4, 2013 14 / 45