The inverse problem for Lagrangian systems with certain non-conservative forces

Transcription

1 The inverse problem for Lagrangian systems with certain non-conservative forces Tom Mestdag Ghent University (Belgium) tmestdag Joint work with Mike Crampin and Willy Sarlet

2 Example (1.) The harmonic oscillator: Equations of motion are m q = kq. F = kq is a conservative force (F = dv/dq, with V (q) = 1 2 kq2 ). Take L( q, q) = 1 2 m q2 1 2 kq2, then d dt ( ) L q L q = 0. If E = 1 2 m q kq2, then E = cte. (2.) The damped oscillator: Equations of motion are m q = kq b q. F 2 = b q is a non-conservative force (F 2 dv/dq). But F 2 = D/ q, with D( q, q) = 1 2 b q2, then d dt ( ) L q L q = D q. If E = 1 2 m q kq2, then de dt = 2D (dissipation). Both (1.) and (2.) are differential equations of second-order: Λ( q, q, q) = 0.

3 Goal of the talk Find necessary and sufficient conditions for a given set of functions Λ i ( q, q, q) to take the form Λ i = d dt ( ) L q i L q i D q i for functions L(q, q) (a Lagrangian) and D(q, q) (a dissipation function). The equations Λ i = 0 are then of Lagrangian form with dissipative forces Obvious condition: Λ i ( q, q, q) is of the form g ij (q, q) q j + B i (q, q). A set of necessary and sufficient conditions was given in: G. Kielau and P. Maisser, A generalization of the Helmholtz conditions for the existence of a first-order Lagrangian, Z. Angew. Math. Mech. 86 (2006)

4 However: 1. Their conditions are not independent: some are redundant. What is the smallest set? 2. In a follow-up paper a version of the conditions expressed in terms of quasi-velocities (nonholonomic velocities) is rederived from scratch. This should be unnecessary: a truly satisfactory formulation of the conditions should be tensorial and thus independent of a choice of coordinates. It s better to use coordinate independent methods = differential geometry. 3. Let D = 0. Their cond. test whether a given system Λ i = 0 is Lagrangian. In case Λ i = 0 is equivalent with q i = f i ( q, q), there is a more general approach for equivalent systems: When does a so-called multiplier g ij (a non-singular matrix) exists, such that g ij ( q j f j ) takes the Euler-Lagrange form? (when the conditions are satisfied g ij (q, q) will be the Hessian of L wrt q). Is it possible to do the same for D 0?

5 Goal of the talk (2) Starting from second-order equations in normal form q i = f i (q, q), what is the smallest set of (coordinate-independent) conditions for existence of a non-singular multiplier (g ij (q, q)) such that g ij ( q j f j ) = d dt ( ) L q i L q i D q i. for some (regular) Lagrangian L(q, q) and some D(q, q).

6 Geometry of a tangent bundle τ : T M M One can associate a SODE field Γ on T M to the equations q i = f i (q, q): Γ = v i q i + f i (q, v) v i. Two ways of lifting vfs X = X i (q) / q i on M canonically to vfs on T M: X V = X i (q) (vertical lift), vi Xc = X i (q) q i + vj Xi q j (complete lift). vi A SODE Γ defines a third horizontal lift (non-linear connection on T M): X X (M) X H X (T M) = 1 2 (Xc + [X V, Γ]), X i (q) ( ) q i X i H i = X i q i Γj i v j, where Γ j i = 1 f j 2 q i.

7 General observation Many objects of interest, although living on T M with dimension 2n, are fully determined by components in dimension n: e.g. 1. ξ X (T M) is a symmetry vector field of Γ if [ξ, Γ] = 0 or ξ = ξ i q i + Γ(ξi ) v i. 2. Cartan 1-form of a Lagrangian system: θ L = L v i dqi. 3. Cartan 2-form dθ L, expected to have a (2n 2n)-coefficient matrix, is completely determined by the (n n)-hessian matrix 2 L/ v i v j. Common feature: tensor fields along the map τ : T M M appear. The idea is: more efficiency in the calculations should come from tools and operations which directly act on forms and vector fields along τ.

8 Calculus of forms along τ : T M M A vector field X along τ (a 1-form α along τ) is a map defined by the following commutative scheme T M T M T M X τ τ M T M α τ π M Coordinate representation: X = X i (q, v) q i α = α i (q, v) dq i Examples: (1) the canonical vf: T = id = v i / q i, (2) basic vfs on M Vector fields along τ are sections of the pullback bundle τ τ : τ T M T M. Notations: X (τ) : set of vector fields along τ (τ): set of scalar forms along τ.

9 Main references: Tools for the calculus along τ : T M M Given Γ. Wanted! exterior derivative covariant derivative curvature E. Martínez, J.F. Cariñena and W. Sarlet, Derivations of differential forms along the tangent bundle projection, Diff. Geom. Appl. 2 (1992) E. Martínez, J.F. Cariñena and W. Sarlet, Derivations of differential forms along the tangent bundle projection II, Diff. Geom. Appl. 3 (1993) 1 29.

10 Tool 1: Exterior derivative There is a canonically defined vertical exterior derivative d V, determined by d V F = V i (F ) dq i, d V α = 0 for α 1 (M). V i = v i, F C (T M)... but the full machinery requires a connection: Any choice of a basis of horizontal vector fields H i = q i Γj i (q, v) v j, allows to construct a horizontal exterior derivative d H : d H F = H i (F ) dq i, F C (T M) d H α = dα for α 1 (M).

11 Tool 2: Linear connection Vertical and horizontal lift procedures extend naturally to vector fields along τ. Every ξ X (T M) has a unique decomposition in a vertical and horizontal part: ξ = ξ V H v + ξ h for some ξ v, ξ h X (τ). With the aid of the SODE connection on T M, one can construct a linear connection on τ τ, i.e. by D : X (T M) X (τ) X (τ), D ξ X = ( [ξ h H, X V ] ) v + ( [ξ v V, X H ] ) h, with ξ X (T M), X X (τ). This is said to be a connection of Berwald type.

12 Let s write, for X X (τ), D X V = D V X, D X H = D H X. If we define, for functions F on T M, D V XF = X V (F ), D H XF = X H (F ), and then further extend by duality, D V arbitrary type. In coordinates, D V XF = X i V i (F ), D H XF = X i H i (F ), and D H extend to tensor fields of D V X D H X q i = 0 q i = Xj V i (Γ k j ) q k

13 Tool 3: operators and Φ Given a SODE Γ, there are two important operators which contain a great deal of information about the dynamics: a degree 0 derivation, called the dynamical covariant derivative a (1,1) tensor Φ V 1 (τ), called Jacobi endomorphism. They appear naturally by decomposing certain vector fields on T M: [Γ, X V ] = X H + ( X) V [Γ, X H ] = ( X) H + Φ(X) V. By setting F = Γ(F ) on functions F C (T M), and by using self-duality, the action of can be extended to arbitrary tensor fields. If R(X, Y ) = [X h, Y h ] v is the curvature of the nonlinear connection, then d V Φ = 3R, d H Φ = R.

14 The inverse problem of the calculus of variations = the search for existence of a non-singular (g ij (q, v)) for a given SODE Γ, st: g ij ( q j f j ) = d dt ( ) L q i L q i for some (regular) Lagrangian L. Answer 1: Iff a function L on T M exists, st L Γ θ L = dl, with θ L = L v i dqi. This condition has in fact only n = dim M components. Condition along τ : θ L = d H L, with now θ L = d V L. Answer 2: Iff a non-singular, symmetric (0,2) tensor field g along τ exists, st g = 0, D V X g(y, Z) = DV Zg(Y, X), g(φx, Y ) = g(φy, X). (Helmholtz conditions)

15 Allowing non-conservative forces of dissipative type Given a SODE Γ, when does a non-singular, symmetric (g ij ) exists such that g ij ( q j f j ) = d dt ( ) L q i for some (regular) Lagrangian L and some D. L q i D q i. Compared to before, now Γ will satisfy: θ L = d H L + d V D. (1) Corollary: The condition (1) is equivalent to d H θ L = 0. PROOF: Use [, d V ] = d H to re-express θ L = d V L in (1) as or d H L = d V G for some G. d H L = 1 2 dv ( (L) D), Using d V d V = 0 and d V d H = d H d V, this is equivalent to d H θ L = 0.

16 Theorem 1: The second-order field Γ represents a dissipative system if and only if there exists a function D and a (non-singular) symmetric type (0, 2) tensor g along τ such that g = D V D V D, D V X g(y, Z) = DV Zg(Y, X), Φ g (Φ g) T = d V d H D. (2) SKETCH OF THE PROOF OF EQUIVALENCE BETWEEN (1) AND (2): 1. Assume (1). Proof (2) by making use of commutator relations. 2. Assume (2). The symmetry of g and D V g imply that g is a Hessian, i.e. g = D V D V F for some function F. Such an F is only determined up to a term α i (q)v i + f(q). The further assumptions (2) ensure that F can be modified to provide the appropriate L.

17 First step. From the symmetry of D V g and D V g = D V D V D V D, also D H g is symmetric. Use the commutator [D V, D H ] to show that d H θ F is a basic 2-form. Second step. Manipulate D V D V D = g to show that the 1-form β is basic: β = θ F (d H F + D V D) i T d H θ F. Third step. Show that d H θ F and β are actually closed (take a d H -derivative and use the Φ-condition), hence locally d H θ F = d H α and β = d H f, for some basic 1-form α and basic function f. Last step. L = F i T α + f is a function which has the same Hessian as F and verifies the required relation θ L = d H L + d V D.

18 Remark: putting the dissipation function D equal to zero, we recover the analysis and results of the standard inverse problem, so there is no immediate incentive to push this further. Quite surprisingly, however, one can eliminate the dependence on D all together and arrive at the following necessary and sufficient conditions involving the multiplier g only.

19 Theorem 2: The second-order field Γ represents a dissipative system of type d dt ( ) L q i L q i = D q i if and only if there exists a (non-singular) symmetric type (0, 2) tensor g along τ such that where X,Y,Z D V X g(y, Z) = DV Zg(Y, X), D H X g(y, Z) = DH Zg(Y, X), (3) X,Y,Z g(r(x, Y ), Z) = 0, stands for a cyclic sum over the indicated arguments. What is remarkable in this result is that the symmetry of D H g and the curvature condition X,Y,Z g(r(x, Y ), Z) = 0, which make their appearance here, are known as integrability conditions for the set of Helmholtz conditions in the standard inverse problem.

20 Expressions in quasi-velocities Plug in X = x i,... to get coord. expressions in natural bundle coord. (q i, v i ). In some applications it is appropriate to work with quasi-velocities (i.e. fibre coord. w i in T Q w.r.t. a non-standard basis {X i } of vector fields on Q). E.g. 0 = ( θ L d H L d V D)(X i ) = Γ(θ L (X i )) θ L ( X i ) X H i (L) X V i (D) = Γ(X V i (L)) X C i (L) X V i (D). If X i = X j i / qj and [X i, X j ] = A k ij X k, then X C i = X j i q j Aj ik vk w j and Xi V = w i, and the above becomes ( ) L Γ w i X j i L q j + L Aj ikwk w j = D w i (Boltzmann-Hamel equations) Any of the other conditions can also be recast in terms of quasi-velocities.

21 And what with non-conservative forces of gyroscopic type? i.e. Lagrangian equations of the form d dt ( ) L q i L q i = ω ki(q) q k, ω ki = ω ik. Again, starting from second-order equations in normal form (the SODE Γ) q i = f i (q, q), one may wonder whether a multiplier matrix g ij (q, q) may bring this system into the above form. We have carried out the same analysis here as in the dissipative case. The starting, intrinsic geometrical formulation here requires the existence of a function L and a basic 2-form ω, such that θ L = d H L + i T ω.

22 Theorem 3: Γ represents a gyroscopic system if and only if there exists a basic 2-form ω and a (non-singular) symmetric type (0, 2) tensor g along τ such that g = 0, D V X g(y, Z) = DV Zg(Y, X), Φ g (Φ g) T = i T d H ω. Theorem 4: If Γ represents a gyroscopic system, then there exists a symmetric type (0, 2) tensor g along τ such that g = 0, D V X g(y, Z) = DV Zg(Y, X), ( ) Φ g (Φ g) T (X, Y ) = X,Y,T g(r(x, Y ), T). The converse is true as well, provided Φ g is smooth on the zero section of T M M.

23 References My papers can be downloaded from tmestdag T. Mestdag, W. Sarlet and M. Crampin, The inverse problem for Lagrangian systems with certain non-conservative forces. preprint (2010). M. Crampin, T. Mestdag and W. Sarlet, On the generalized Helmholtz conditions for Lagrangian systems with dissipative forces. Z. Angew. Math. Mech. 90 (2010), G. Kielau and P. Maisser, A generalization of the Helmholtz conditions for the existence of a first-order Lagrangian, Z. Angew. Math. Mech. 86 (2006) U. Jungnickel, G. Kielau, P. Maisser and A. Müller, A generalization of the Helmholtz conditions for the existence of a first-order Lagrangian using nonholonomic velocities, Z. Angew. Math. Mech. 89 (2009)