The Hamilton-Jacobi problem in nonholonomic mechanics Manuel de Le´ on Instituto de Ciencias Matem´ aticas (ICMAT) Nonholonomic mechanics and optimal control Institut Henri Poincar´ e Institute, Paris, November 25th 28th, 2014 1 / 66
A short introduction to Lagrangian mechanics Let L = L ( q i , ˙ q i ) be a lagrangian function, where ( q i ) are coordinates in q i ) are the generalized velocities. a configuration n -manifold Q , and (˙ The Hamilton ’s principle produces the Euler-Lagrange equations dt ( ∂ L d q i ) − ∂ L ∂ q i = 0 , 1 ≤ i ≤ n . (1) ∂ ˙ A geometric version of Eq. (1) can be obtained as follows. → R . Consider the (1,1)-tensor field S and the Liouville vector L : TQ − field ∆ defined on the tangent bundle TQ of Q : ∂ q i ∂ q i ⊗ dq i , ∆ = ˙ S = q i . ∂ ˙ ∂ ˙ We construct the Poincar´ e-Cartan 1 and 2-forms α L = S ∗ ( dL ) , ω L = − d α L , where S ∗ denotes the adjoint operator of S . The energy is given by E L = ∆( L ) − L , so that we recover the classical expressions ω L = dq i ∧ dp i , E L = ˙ q i p i − L , 2 / 66
We say that L is regular if the 2-form ω L is symplectic, which in coordinates turns to be equivalent to the regularity of the Hessian matrix of L with respect to the velocities � � ∂ 2 L W ij = ∂ ˙ q i ∂ ˙ q j In this case, the equation i X ω L = dE L (2) has a unique solution, X = ξ L , called the Euler-Lagrange vector field; ξ L is a second order differential equation (SODE) that means that its integral curves are tangent lifts of their projections on Q (these projections are called the solutions of ξ L ). The solutions of ξ L are just the ones of Eqs (1). → T ∗ TQ is the musical isomorphism, ♭ L ( v ) = i v ω L , then If ♭ L : TTQ − we have ♭ L ( ξ L ) = dE L . 3 / 66
Legendre transformation → T ∗ Q is a fibred mapping The Legendre transformation FL : TQ − → Q and π Q : T ∗ Q − (that is, π Q ◦ FL = τ Q , where τ Q : TQ − → Q denote the canonical projections of the tangent and cotangent bundle of Q , respectively) defined by FL ( q i , ˙ q i ) = ( q i , p i ) , L is regular if and only if FL is a local diffeomorphism. We will assume that FL is in fact a global diffeomorphism (in other words, L is hyperregular) which is the case when L is a lagrangian of mechanical type, say L = T − V where - T is the kinetic energy defined by a Riemannian metric on Q , → R is a potential energy. - V : Q − 4 / 66
Hamiltonian description The hamiltonian counterpart is developed in the cotangent bundle T ∗ Q of Q . Denote by ω Q = dq i ∧ dp i the canonical symplectic form, where ( q i , p i ) are the canonical coordinates on T ∗ Q . The Hamiltonian energy is just h = E L ◦ FL − 1 and the Hamiltonian vector field is the solution of the symplectic equation i X h ω Q = dh . The integral curves ( q i ( t ) , p i ( t )) of X h satisfies the Hamilton equations. Since FL ∗ ω Q = ω L we deduce that ξ L and X h are FL -related, and consequently FL transforms the Euler-Lagrange equations into the Hamilton equations. 5 / 66
Non-holonomic mechanics Consider a disc rolling without slipping in a rough plane. Let ( x , y ) be the coordinates of the point of contact of the disc with the ground, ψ the angle between a point fixed in the circle and the point of contact (the angle of rotation), φ the angle between the tangent to the disc at the point of contact and the axis x , and θ the angle of inclination of the disc. 2 × S 1 × S 1 × S 1 . The lagrangian is The configuration space is Q = R L = T − V where 1 x 2 + ˙ y 2 + R 2 ˙ θ 2 + R 2 ˙ φ 2 sin 2 θ ) − mR ( ˙ x sin φ − ˙ T = 2 m (˙ θ cos φ (˙ y cos φ ) y sin φ )) + 1 φ 2 cos 2 θ ) + 1 θ 2 ˙ ˙ 2 I 1 ( ˙ 2 I 2 ( ˙ ψ + ˙ φ sin θ ) 2 + φ sin θ (˙ x cos φ + ˙ and V = mgR cos θ m is the mass of the disc, R is the radius, I 1 and I 2 are the principle moments of inertia. 6 / 66
The non-slipping condition implies the following constraints. Φ 1 = ˙ ψ = 0 , Φ 2 = ˙ x − ( R cos φ ) ˙ y − ( R sin φ ) ˙ ψ = 0 . All the configurations are possible, but not all the velocities. 7 / 66
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Nonholonomic mechanical systems A nonholonomic mechanical system is given by a lagrangian function → R subject to contraints determined by a linear distribution D L : TQ − on the configuration manifold Q . We will denote by D the total space of the corresponding vector sub-bundle ( τ Q ) |D : D − → Q defined by D , where ( τ Q ) |D is the restriction of the canonical projection τ Q : TQ − → Q . We will assume that the lagrangian L is defined by a Riemannian metric g on Q and a potential energy V ∈ C ∞ ( Q ), so that L ( v q ) = 1 2 g ( v q , v q ) − V ( q ) or, in bundle coodinates ( q i , ˙ q i ) q i ) = 1 q i ˙ q j − V ( q i ) L ( q i , ˙ 2 g ij ˙ 9 / 66
If { µ a } , 1 ≤ a ≤ k is a local basis of the annihilator D o of D , then the constraints are locally given by q i = 0 , µ a i ( q ) ˙ where µ a = µ a i ( q ) dq i . The nonholonomic equations can be written as � ∂ L � d − ∂ L ∂ q i = λ a µ a i ( q ) dt ∂ ˙ q i q i = 0 , µ a ( q ) ˙ for some Lagrange multipliers λ a to be determined. If we modify (11) as follows: i X ω L − dE L ∈ S ∗ (( T D ) o ) (3) X ∈ T D (4) the unique solution X nh is again a SODE whose solutions are just the ones of the nonholonomic equations. 10 / 66
Let → T ∗ Q FL : TQ − be the Legendre transformation given by q i ) = ( q i , p i = ∂ L FL ( q i , ˙ q j ) q i = g ij ˙ ∂ ˙ FL is a global diffeomorphism which permits to reinterpret the nonholonomical mechanical system in the hamiltonian side. Indeed, we denote by h = E L ◦ FL − 1 the hamiltonian function and by M = FL ( D ) the constraint submanifold of T ∗ Q . The nonholonomic equations are then given by dq i dt = ∂ h ∂ p i dp i dt = − ∂ h ∂ q i + ¯ λ a µ a i , where ¯ λ a are Lagrange multipliers to be determined. 11 / 66
As above, the symplectic equation i X h ω Q = dh which gives the hamiltonian vector field X h should be modified as follows: i X ω Q − dh ∈ F o (5) X ∈ TM (6) where F is a distribution along M whose annihilator F o is obtained from S ∗ (( T D ) o ) throught FL . Equations (14) and (15) have a unique solution, the nonholonomic vector field X nh . 12 / 66
The Hamilton-Jacobi theory for Hamiltonian systems The standard formulation of the Hamilton-Jacobi problem is to find a function S ( t , q i ) (called the principal function ) such that ∂ S ∂ t + h ( q i , ∂ S ∂ q i ) = 0 , (7) where h = h ( q i , p i ) is the hamiltonian function of the system. If we put S ( t , q i ) = W ( q i ) − tE , where E is a constant, then W satisfies h ( q i , ∂ W ∂ q i ) = E ; (8) W is called the characteristic function . Equations (7) and (8) are indistinctly referred as the Hamilton-Jacobi equation . 13 / 66
The Hamilton-Jacobi equation helps to solve the Hamilton equations for the hamiltonian h dq i dt = ∂ h , dp i dt = − ∂ h (9) ∂ p i ∂ q i Indeed, if we find a solution W of the Hamilton-Jacobi equation (8) then a solution ( q i ( t )) of the first set of equations (9) gives a solution of the Hamilton equations by taking p i ( t ) = ∂ W ∂ q i . R. Abraham, J.E. Marsden: Foundations of Mechanics (2nd edition). Benjamin-Cumming, Reading, 1978. 14 / 66
Let λ be a closed 1-form on Q , say d λ = 0; (then, locally λ = dW ) Hamilton-Jacobi Theorem The following conditions are equivalent: (i) If σ : I → Q satisfies the equation dq i dt = ∂ h ∂ p i then λ ◦ σ is a solution of the Hamilton equations; (ii) d ( h ◦ λ ) = 0 15 / 66
� � � Define a vector field on Q : X λ h = T π Q ◦ X h ◦ λ Xh T ∗ Q � T ( T ∗ Q ) π Q T π Q λ X λ h � TQ Q The following conditions are equivalent: (i) If σ : I → Q satisfies the equation dq i dt = ∂ h ∂ p i then λ ◦ σ is a solution of the Hamilton equations; (i)’ If σ : I → Q is an integral curve of X λ h , then λ ◦ σ is an integral curve of X h ; (i)” X h and X λ h are λ -related, i.e. T λ ( X λ h ) = X h ◦ λ 16 / 66
Hamilton-Jacobi Theorem Let λ be a closed 1-form on Q . Then the following conditions are equivalent: (i) X λ h and X h are λ -related; (ii) d ( h ◦ λ ) = 0 If λ = λ i ( q ) dq i then the Hamilton-Jacobi equation becomes h ( q i , λ i ( q j )) = const . and we recover the classical formulation when λ i = ∂ W ∂ q i 17 / 66
PROBLEM: How to extend the classical Hamilton-Jacobi theory for nonholonomic mechanical systems? 18 / 66
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