Unified formalism for non-autonomous mechanical systems

UNIFIED FORMALISM FOR NON-AUTONOMOUS MECHANICAL SYSTEMS Mar´ ıa Barbero-Li˜ n´ an∗ , Arturo Echeverr´ ıa-Enr´ ıquez† , Departamento de Matem´atica Aplicada IV Edificio C-3, Campus Norte UPC C/ Jordi Girona 1. 08034 Barcelona. Spain David Mart´ ın de Diego‡ Instituto de Ciencias Matem´aticas (CSIC-UAM-UCM-UC3M) C/ Serrano 123. 28006 Madrid. Spain Miguel C. Mu˜ noz-Lecanda§ ,Narciso Rom´ an-Roy¶ , Departamento de Matem´atica Aplicada IV Edificio C-3, Campus Norte UPC C/ Jordi Girona 1. 08034 Barcelona. Spain February 29, 2008 Abstract We present a unified geometric framework for describing both the Lagrangian and Hamiltonian formalisms of regular and non-regular time-dependent mechanical systems, which is based on the approach of Skinner and Rusk [18]. The dynamical equations of motion and their compatibility and consistency are carefully studied, making clear that all the characteristics of the Lagrangian and the Hamiltonian formalisms are recovered in this formulation. As an example, it is studied a semidiscretization of the nonlinear wave equation proving the applicability of the proposed formalism. Key words: Lagrangian and Hamiltonian formalisms; autonomous mechanics, symplectic and presymplectic manifolds. AMS s. c. (2000): 37J05, 53D05, 55R10, 70H03, 70H05 1 Introduction In 1983 Skinner and Rusk introduced a representation of the dynamics of an autonomous mechanical system which combines the Lagrangian and Hamiltonian features [18]. The aim of this formulation was to obtain a common framework for both regular and singular dynamics, obtaining simultaneously the Hamiltonian and Lagrangian formulations of the dynamics. Over the ∗e-mail: mbarb[email protected] †e-mail: [email protected]c.edu ‡e-mail: [email protected] §e-mail: [email protected]c.edu ¶e-mail: [email protected]c.edu M. Barbero-Li˜ n´ an et al,Skinner-Rusk formalism for non-autonomous systems 2 years, however, Skinner and Rusk’s framework was extended in many directions. So, Cantrijn et al [2] extended this formalism for explicit time-dependent systems using a jet bundle language. In [6] an extension of this formalism to other kinds of more general time-dependent singular differential equations was given. Cort´es et al [3] used the Skinner and Rusk formalism to consider vakonomic mechanics and the comparison between the solutions of vakonomic and nonholonomic mechanics. Finally, in [4, 9, 15] the Skinner-Rusk model was developed for classical field theories. The aim of this paper is to continue the study of the the Skinner-Rusk formalism for time dependent mechanical systems (Section 3), now, carefully studying the dynamical equations of motion and the submanifolds where they are consistent, and showing how the Lagrangian and Hamiltonian descriptions are recovered from this unified framework (Sections 4,5). The case of field theories was independently developed in [4, 9], and improves the construction given in [2], as it is discussed in Section 7. As a new application, we analyze the case of semidiscretizations of field theories in Section 6. These methods are designed by numerical schemes that respect physical principles preserved by the continuous systems, specially those described by partial differential equations (PDEs). In this case, there are not only a time dependence (as in ordinary differential equations) but also posses an spatial dependence. Many integration methods, in particular in Hamiltonian dynamics, starts by discretizing the spatial structure (spatial truncation) obtaining a finite dimensional system of ordinary differential equations (ODEs) retaining some physical properties of the original system (see [8]). For simplicity, we restrict ourselves to a particular semidiscretization of the nonlinear wave equation [10, 13] obtaining a unique solution of the dynamics on the secondary constraint submanifold. All the manifolds are real, second countable and C∞. The maps are assumed to be C∞. Sum over repeated indices is understood. 2 Non-autonomous Lagrangian and Hamiltonian systems (See [5, 7, 12, 14, 16] for more details). In the jet bundle description of non-autonomous dynamical systems, the configuration bundle is π:E//R, where Eis a (n+ 1)-dimensional differentiable manifold endowed with local coordinates (t, qi), and Rhas tas a global coordinate. The jet bundle of local sections of π,J1π, is the velocity phase space of the system, with natural coordinates (t, qi, vi), adapted to the bundle π:E//R, and natural projections are π1:J1π//E , ¯π1:J1π//R. (If E≡R×Q, where Qis a n-dimensional differentiable manifold, then J1π≃R×TQ). A Lagrangian density L ∈ Ω1(J1π) is a ¯π1-semibasic 1-form on J1π, and it is usually written as L=Ldt, where L∈C∞(J1π) is the Lagrangian function determined by L. Throughout this paper we denote by dtthe volume form in R, and its pull-backs to all the manifolds. The Poincar´e-Cartan forms associated with the Lagrangian density Lare defined using the vertical endomorphism Vof the bundle J1π(see [5, 17]) ΘL=i(V)dL+L ∈ Ω1(J1π) ; ΩL=−dΘL∈Ω2(J1π). A Lagrangian Lis regular if ΩLhas maximal rank; elsewhere Lis singular. In natural coordinates M. Barbero-Li˜ n´ an et al,Skinner-Rusk formalism for non-autonomous systems 3 we have V= (dqi−vidt)⊗∂ ∂vi⊗∂ ∂t, and ΘL=∂L ∂vidqi−∂L ∂vivi−Ldt ΩL=−∂2L ∂vj∂vidvj∧dqi−∂2L ∂qj∂vidqj∧dqi +∂2L ∂vj∂vividvj∧dt+∂2L ∂qj∂vivi−∂L ∂qj+∂2L ∂t∂vjdqj∧dt . The regularity condition is equivalent to det ∂2L ∂vi∂vj(¯y)6= 0, for every ¯y∈J1π. Geometrically, Lis regular if and only if (ΩL,dt) is a cosymplectic structure on J1π. This means that ΩLand dtare closed and Ωn L∧dtis a volume form (see [11]). The Lagrangian problem consists in finding sections φ:R//Eof π, characterized by (j1φ)∗i(X)ΩL= 0 ,for every X∈X(J1π) where j1φ:R//J1πis the 1-jet extension of φ. In natural coordinates, if φ(t) = (t, φi(t)), this condition is equivalent to demanding that φsatisfies the Euler-Lagrange equations ∂L ∂qij1φ−d dt ∂L ∂vij1φ= 0 ,(for i= 1, . . . , n) where j1φ(t)=(t, φi(t),˙ φi(t)). Assuming that these sections are integral curves of vector fields in J1πthe corresponding equations for these vector fields are i(XL)ΩL= 0 ,i(XL)dt= 1 (1) where XL∈X(J1π) is holonomic (recall that a vector field in J1πis said to be holonomic, or also asecond order differential equation (SODE for simplicity), if its integral curves are holonomic; that is, canonical liftings of sections ϕ:R//E). In the regular case, there is a unique solution to these equations. In the singular case the existence of a solution is not assured, except perhaps on some submanifold (or subset) of J1π, where the solution is not unique, in general. Consider now the extended momentum phase space T∗E, and the restricted momentum phase space which is defined by J1π∗= T∗E/π∗T∗R. Local coordinates in these manifolds are (t, qi, p, pi) and (t, qi, pi), respectively. Then, the following natural projections are τ1:J1π∗//E , ¯τ1=π◦τ1:J1π∗//R, µ: T∗E//J1π∗, p: T∗E//R. Let Θ ∈Ω1(T∗E) and Ω = −dΘ ∈Ω2(T∗E) be the canonical forms of T∗Ewhose local expressions are Θ = pidqi+pdt , Ω=dqi∧dpi+ dt∧dp . (In the particular case E=R×Q, we have T∗E≃R×R∗×T∗Q, and J1π∗≃R×T∗Q and introducing the projections pr1: T∗(R×Q)//R×R∗,pr2: T∗(R×Q)//T∗Q, we have Θ = pr∗ 1ΘR+pr∗ 2ΘQand Ω = pr∗ 1ΩR+pr∗ 2ΩQ; where ΩR=−dΘR∈Ω2(R×R∗) and ΩQ=−dΘQ∈Ω2(T∗Q) denote the natural symplectic forms of R×R∗and T∗Q). Being ΘL∈Ω1(J1π)π1-semibasic, we have a natural map g FL:J1π//T∗E, given by g FL(¯y) = ΘL(¯y) (2) M. Barbero-Li˜ n´ an et al,Skinner-Rusk formalism for non-autonomous systems 4 which is called the extended Legendre map associated to the Lagrangian density L. The restricted Legendre map is FL =µ◦g FL:J1π//J1π∗. Their local expressions are g FL∗t=t , g FL∗qi=qi,g FL∗pi=∂L ∂vi,g FL∗p=L−vi∂L ∂vi FL∗t=t , FL∗qi=qi,FL∗pi=∂L ∂vi or, in other words, g FL(t, qi,˙qi) = (t, qi, L −vi∂L ∂vi,∂L ∂vi) and FL(t, qi,˙qi) = (t, qi,∂L ∂vi). Moreover, we have g FL∗Θ = ΘL, and g FL∗Ω=ΩL. The Lagrangian Lis regular if, and only if, FL is a local diffeomorphism. As a particular case, Lis a hyper-regular Lagrangian if FL is a global diffeomorphism. If Lis a hyper-regular Lagrangian, then ˜ P=g FL(J1π) is a 1-codimensional, µ-transverse embedded submanifold of T∗E, with natural embedding ˜0:˜ P,→T∗E, which is diffeomorphic to J1π∗. This diffeomorphism is the inverse of µrestricted to ˜ P, and also coincides with the map h=g FL◦FL−1, when it is restricted onto its image (which is just ˜ P). This map his called aHamiltonian section, and is used to construct the Hamilton-Cartan forms in J1π∗by making Θh=h∗Θ∈Ω1(J1π∗),Ωh=h∗Ω∈Ω2(J1π∗). Locally, the Hamiltonian section his specified by h(t, qi, pi) = (t, qi,−H, pi), where His the local Hamiltonian function given by H=pi(FL−1)∗vi−(FL−1)∗L. The local expressions are Θh=pidqi−Hdt , Ωh= dqi∧dpi+ dH∧dt . Of course FL∗Θh= ΘL, and FL∗Ωh= ΩL. The Hamiltonian problem consists in finding sections of ¯τ1,ψ:R//J1π∗, characterized by ψ∗i(X)Ωh= 0 ,for every X∈X(J1π∗) . This condition leads to the Hamilton equations which, if ψ(t) = (t, qi(t), pi(t)), in natural coordinates are dqi dt =∂H ∂piψ ;dpi dt =−∂H ∂qiψ . Assuming that these sections are integral curves of vector fields Xh∈X(J1π∗), the corresponding equations for these vector fields are i(Xh)Ωh= 0 ,i(Xh)dt= 1 . As a final remark, it can be proved that solutions to the Lagrangian and Hamiltonian problems are equivalent, in the sense that they are FL-related; that is, ψ=FL ◦ j1φ; TFL ◦ XL=Xh◦ FL .(3) For regular, but not hyper-regular systems, the results are the same, but only locally on open neighbourhoods at every point, instead of J1π∗. A singular Lagrangian Lis almost-regular if: P=FL(J1π) is a closed submanifold of J1π∗ (let :P,→J1π∗be natural embedding), FL is a submersion onto its image, and for every ¯y∈J1π, the fibres FL−1(FL(¯y)) are connected submanifolds of J1π. If Lis an almost-regular Lagrangian, the submanifold Pof J1π∗is a fibre bundle over E and M. In this case the µ-transverse submanifold ˜:˜ P,→T∗Eis diffeomorphic to P. This M. Barbero-Li˜ n´ an et al,Skinner-Rusk formalism for non-autonomous systems 5 diffeomorphism is denoted by ˜µ:˜ P//P, and is just the restriction of the projection µto ˜ P. Then, taking the Hamiltonian section ˜ h= ˜◦˜µ−1, we define the forms Θ0 h=˜ h∗Θ ; Ω0 h=˜ h∗Ω which verify that FL∗ 0Θ0 h= ΘLand FL∗ 0Ω0 h= ΩL(where FL0is the restriction map of FL onto P). Then, the Hamiltonian problem and the equations of motion are stated as in the hyper-regular case. Now, the existence of a solution to these equations is not assured, except perhaps on some submanifold of P, where the solution is not unique, in general. 3 Unified formalism We define the extended jet-momentum bundle Wand the restricted jet-momentum bundle Wr W=J1π×ET∗E , Wr=J1π×EJ1π∗ with natural coordinates (t, qi, vi, p, pi) and (t, qi, vi, pi), respectively. Natural submersions are ρ1:W//J1π , ρ2:W//T∗E , ρE:W//E , ρR:W//R(4) ρr 1:Wr//J1π , ρr 2:Wr//J1π∗, ρr E:Wr//E , ρr R:Wr//R, with π1◦ρ1=τ1◦µ◦ρ2=ρE. For ¯y∈J1π,p∈T∗E, and [p] = µ(p)∈J1π∗, there is also the natural projection µW:W//Wr (¯y, p)7→ (¯y, [p]) The bundle Wis endowed with the following canonical structures: Definition 1 1. The coupling 1-form in Wis the ρR-semibasic 1-form ˆ C ∈ Ω1(W)defined as follows: for every w= (j1φ(t), α)∈ W (that is, α∈T∗ ρE(w)E) and V∈TwW, then ˆ C(V) = α(Tw(φ◦ρR)V). 2. The canonical 1-form ΘW∈Ω1(W)is the ρE-semibasic form defined by ΘW=ρ∗ 2Θ. The canonical 2-form is ΩW=−dΘW=ρ∗ 2Ω∈Ω2(W). Being ˆ CaρR-semibasic form, there is ˆ C∈C∞(W) such that ˆ C=ˆ Cdt. Note also that ΩWis degenerate, its kernel being the ρ2-vertical vectors; then (W,ΩW) is a presymplectic manifold. The local expressions for ΘW, ΩW, and ˆ Care ΘW=pidqi+pdt , ΩW=−dpi∧dqi−dp∧dt , ˆ C= (p+pivi)dt . Given a Lagrangian density L ∈ Ω1(J1π), we denote ˆ L=ρ∗ 1L ∈ Ω1(W), and we can write ˆ L=ˆ Ldt, with ˆ L=ρ∗ 1L∈C∞(W). We define a Hamiltonian submanifold W0={w∈ W | ˆ L(w) = ˆ C(w)}. So, W0is the submanifold of Wdefined by the regular constraint function ˆ C−ˆ L= 0, which is globally defined in Wusing the dynamical data and the geometry. In local coordinates it is ˆ C−ˆ L=p+pivi−ˆ L(t, qj, vj) = 0 . M. Barbero-Li˜ n´ an et al,Skinner-Rusk formalism for non-autonomous systems 6 The natural embedding is 0:W0,→ W. We have the projections (submersions), see diagram (5): ρ0 1:W0//J1π , ρ0 2:W0//T∗E , ρ0 E:W0//E , ρ0 R:W0//R which are the restrictions to W0of the projections (4), and ˆρ0 2=µ◦ρ0 2:W0//J1π∗. Local coordinates in W0are (t, qi, vi, pi), and we have that ρ0 1(t, qi, vi, pi) = (t, qi, vi), 0(t, qi, vi, pi) = (t, qi, vi, L −vipi, pi) ˆρ0 2(t, qi, vi, pi) = (t, qi, pi), ρ0 2(t, qi, vi, pi) = (t, qi, L −vipi, pi). Proposition 1 W0is a 1-codimensional µW-transverse submanifold of W, diffeomorphic to Wr. (Proof ) For every (¯y, p)∈ W0, we have L(¯y)≡ˆ L(¯y, p) = ˆ C(¯y, p), and (µW◦0)(¯y, p) = µW(¯y, p) = (¯y, µ(p)) . First, µW◦0is injective: let (¯y1,p1),(¯y2,p2)∈ W0, then we have (µW◦0)(¯y1,p1) = (µW◦0)(¯y2,p2)⇒(¯y1, µ(p1)) = (¯y2, µ(p2)) ⇒¯y1= ¯y2, µ(p1) = µ(p2) hence L(¯y1) = L(¯y2) = ˆ C(¯y1,p1) = ˆ C(¯y2,p2). In a local chart, the third equality gives p(p1) + pi(p1)vi(¯y1) = p(p2) + pi(p2)vi(¯y2) but µ(p1) = µ(p2) implies pi(p1) = pi([p1]) = pi([p2]) = pi(p2); then p(p1) = p(p2), and p1=p2. Second, µW◦0is onto, then, if (¯y, [p]) ∈ Wr, there exists (¯y, q)∈0(W0) such that [q] = [p]. In fact, it suffices to take [q] such that, in a local chart of J1π×ET∗E=W pi(q) = pi([p]) , p(q) = L(¯y)−pi([p])vi(¯y). Finally, since W0is defined by the constraint function ˆ C−ˆ Land, as ker µW∗ =n∂ ∂p olocally and ∂ ∂p(ˆ C−ˆ L) = 1, then W0is µW-transversal. As a consequence of this result, the submanifold W0induces a section ˆ h:Wr//Wof the projection µW. Locally, ˆ his specified by giving the local Hamiltonian function ˆ H=−ˆ L+pivi; that is, ˆ h(t, qi, vi, pi) = (t, qi, vi,−ˆ H, pi). In this sense, ˆ his a Hamiltonian section of µW. So we have the following diagram J1π W0 ρ0 1 77 o o o o o o o o o o o o o0// ρ0 2 '' P P P P P P P P P P P P P ˆρ0 2 @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @W ρ1 OO ρ2  µW//Wr ρr 1 ggOOOOOOOOOOOOO ρ2◦ˆ h wwooooooooooooo ρr 2 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ T∗E µ  J1π∗ (5) M. Barbero-Li˜ n´ an et al,Skinner-Rusk formalism for non-autonomous systems 7 Remark: Observe that, from the Hamiltonian µW-section ˆ h:Wr//Win the extended unified formalism, we can recover the Hamiltonian µ-section ˜ h= ˜◦˜µ−1:P//T∗Ein the standard Hamiltonian formalism assuming that Lis almost-regular. In fact, given [p]∈J1π∗, the section ˆ hmaps every point (¯y, [p]) ∈(ρr 2)−1([p]) into ρ−1 2[ρ2(ˆ h(¯y, [p]))]. Now, the crucial point is the projectability of the local function ˆ Hby ρ2. However, ∂ ∂vibeing a local basis for ker ρ2∗,ˆ His ρ2-projectable if, and only if, pi=∂L ∂vi, and this condition is fulfilled when [p]∈ P = Im FL ⊂ J1π∗, which implies that ρ2[ˆ h((ρr 2)−1([p]))] ∈˜ P= Im g FL ⊂ T∗E. Then, the Hamiltonian section ˜ his defined as ˜ h([p]) = (ρ2◦ˆ h)[(ρr 2)−1(([p]))] = (˜◦˜µ−1)([p]) ,for every [p]∈ P. So we have the diagram ˜ P˜// ˜µ  T∗E µ  W ρ2 oo P ˜µ−1 OO ˜ h 77 p p p p p p p p p p p p p//J1π∗Wr ρr 2 oo ˆ h OO For (hyper) regular systems this diagram is the same with P= Im FL =J1π∗. Finally, we can define the forms Θ0=∗ 0ΘW=ρ0∗ 2Θ∈Ω1(W0),Ω0=∗ 0ΩW=ρ0∗ 2Ω∈Ω2(W0) with local expressions Θ0= (L−pivi)dt+pidqi,Ω0= d(pivi−L)∧dt−dpi∧dqi(6) and we have the presymplectic Hamiltonian systems (W0,Ω0) and (Wr,Ωr), with Ωr=ˆ h∗Ω0. 4 The dynamical equations for sections Now we establish the dynamical problem for the system (W0,Ω0) which, as a consequence of the diffeomorphism stated in Proposition 1, is equivalent to making it for the system (Wr,Ωr). The Lagrange-Hamiltonian problem associated with the system (W0,Ω0) consists in finding sections of ρ0 R,ψ0:R//W0, which are characterized by the condition ψ∗ 0i(Y0)Ω0= 0 ,for every Y0∈X(W0).(7) This equation gives different kinds of information, depending on the type of the vector fields Y0 involved. In particular, using ˆρ0 2-vertical vector fields, denoted by XV(ˆρ0 2)(W0), we have: Lemma 1 If Y0∈XV(ˆρ0 2)(W0), then i(Y0)Ω0is ρ0 R-semibasic. (Proof ) A simple calculation in coordinates leads to this result. In fact, taking ∂ ∂vias a local basis for the ˆρ0 2-vertical vector fields, and bearing in mind (6) we obtain the ρ0 R-semibasic forms i∂ ∂viΩ0=pi−∂L ∂vidt . M. Barbero-Li˜ n´ an et al,Skinner-Rusk formalism for non-autonomous systems 8 As an immediate consequence, when Y0∈XV(ˆρ0 2)(W0), condition (7) does not depend on the derivatives of ψ0: it is a pointwise (algebraic) condition. We can define the submanifold W1={(¯y, p)∈ W0|i(V0)(Ω0)(¯y,p)= 0,for every V0∈V(¯y,p)(ˆρ0 2)} where V(ˆρ0 2) denotes the ˆρ0 2-vertical vectors. W1is called the first constraint submanifold of the Hamiltonian pre-multisymplectic system (W0,Ω0), as every section ψ0solution to (7) must take values in W1. We denote by 1:W1,→ W0the natural embedding. Locally, W1is defined in W0by the constraints pi=∂L ∂vi. Moreover: Proposition 2 W1is the graph of g FL; that is, W1={(¯y, g FL(¯y)) ∈ W | ¯y∈J1π}. (Proof ) Consider ¯y∈J1π, let φ:R//Ebe a representative of ¯y, and p=g FL(¯y). For every U∈T¯π1(¯y)R, consider V= T¯π1(¯y)φ(U) and its canonical lifting ¯ V= T¯π1(¯y)j1φ(U). From the definition of the extended Legendre map (2) we have (T¯yπ1)∗(g FL(¯y)) = (ΘL)¯y, then i(¯ V)[(T¯yπ1)∗(g FL(¯y))] = i(¯ V)(ΘL)¯y. Furthermore, as p=g FL(¯y), we also have that i(¯ V)[(T¯yπ1)∗(g FL(¯y))] = i(T¯π1(¯y)j1φ(U))[(T¯yπ1)∗p] = i((T¯yπ1)∗(T¯π1(¯y)j1φ(U)))p =i(T¯π1(¯y)φ(U))p=i(V)p. Therefore we obtain i(U)(φ∗p) = i(U)[(j1φ)∗(ΘL)¯y] and bearing in mind the definition of the coupling form C, this condition becomes i(U)( ˆ C(¯y, p)) = i(U)[(j1φ)∗ΘL)¯y]. Since it holds for every U∈T¯π1(¯y)R, we conclude that ˆ C(¯y, p) = [(j1φ)∗ΘL]¯y, or equivalently, ˆ C(¯y, p) = ˆ L(¯y, p), where we have made use of the fact that ΘLis the sum of the Lagrangian density Land a contact form i(V)dL(vanishing by pull-back of lifted sections). This is the condition defining W0, and thus we have proved that (¯y, g FL(¯y)) ∈ W0, for every ¯y∈J1π; that is, graph g FL ⊂ W0. Furthermore, graph g FL and W1are defined as subsets of W0by the same local conditions: pi−∂L ∂vi= 0. So we conclude that graph g FL =W1. As W1is the graph of g FL, it is diffeomorphic to J1π. Every section ψ0:R//W0is of the form ψ0= (ψL, ψH), with ψL=ρ0 1◦ψ0:R//J1π, and if ψ0takes values in W1 then ψH=g FL ◦ ψL:R//T∗E. In this way every constraint, differential equation, etc. in the unified formalism can be translated to the Lagrangian or the Hamiltonian formalisms by restriction to the first or the second factors of the product bundle. However, as was pointed out before, the geometric condition (7) in W0, which can be solved only for sections ψ0:R//W1⊂ W0, is stronger than the Lagrangian condition ψ∗ Li(Z)ΩL= 0, (for every Z∈X(J1π)) in J1π, which can be translated to W1by the natural diffeomorphism between them. The reason is that, as ρ0 1is a submersion, and W1is a ρ0 1-transversal submanifold of W0(as a consequence of Proposition 2), we have the splitting ∗ 1TW0= TW1⊕W1∗ 1V(ρ0 1), 1:W1,→ W0being the natural embedding. Therefore the additional information comes from the ρ0 1-vertical vectors, and is just the holonomic condition. In fact: M. Barbero-Li˜ n´ an et al,Skinner-Rusk formalism for non-autonomous systems 9 Theorem 1 Let ψ0:R//W0be a section fulfilling equation (7), ψ0= (ψL, ψH) = (ψL,g FL ◦ ψL), where ψL=ρ0 1◦ψ0. Then: 1. ψLis the canonical lift of the projected section φ=ρ0 E◦ψ0:R//E(that is, ψLis a holonomic section). 2. The section ψL=j1φis a solution to the Lagrangian problem, and the section µ◦ψH= µ◦g FL ◦ ψL=FL ◦ j1φis a solution to the Hamiltonian problem. Conversely, for every section φ:R//Esuch that j1φis a solution to the Lagrangian problem (and hence FL ◦ j1φis a solution to the Hamiltonian problem) we have that the section ψ0= (j1φ, g FL ◦ j1φ), is a solution to (7). (Proof ) 1. Taking n∂ ∂pioas a local basis for the ρ0 1-vertical vector fields: i∂ ∂piΩ0=vidt−dqi so that for a section ψ0we have 0 = ψ∗ 0i∂ ∂piΩ0=vi−∂qi ∂t dt and thus the holonomy condition appears naturally within the unified formalism. So we have that ψ0=t, qi,dqi dt ,∂L ∂vi, since ψ0takes values in W1, and hence it is of the form ψ0= (j1φ, g FL ◦ j1φ), for φ= (t, qi) = ρ0 E◦ψ0. 2. Consider the diagram W ρ1                   ρ2 '' P P P P P P P P P P P P P W0 0 OO ρ0 1 wwooooooooooooo ρ0 2//T∗E J1π π1 '' O O O O O O O O O O O O OW1 1 OO ρ1 1 ooρ1 2// ρ1 E  J1π∗ τ1 wwnnnnnnnnnnnnnT∗E E R ψL=j1φ ``@ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ φ OOψ1 OO ψ0 OO ψH=g FL ◦ j1φ 66 n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n Since sections ψ0:R//W0solution to (7) take values in W1, we can identify them with sections ψ1:R//W1. These sections ψ1verify, in particular, that ψ∗ 1i(Y1)Ω1= 0 holds for every Y1∈X(W1). Obviously ψ0=1◦ψ1. Moreover, as W1is the graph of g FL, denoting by ρ1 1=ρ0 1◦1:W1//J1πthe diffeomorphism which identifies W1with J1π, if we define Ω1=∗ 1Ω0, we have that Ω1=ρ1∗ 1ΩL. In fact; as (ρ1 1)−1(¯y) = (¯y, g FL(¯y)), for every ¯y∈J1π, then (ρ0 2◦1◦(ρ1 1)−1)(¯y) = g FL(¯y)∈T∗E, and hence ΩL= (ρ0 2◦1◦(ρ1 1)−1)∗Ω = [((ρ1 1)−1)∗◦∗ 1◦ρ0∗ 2]Ω = [((ρ1 1)−1)∗◦∗ 1]Ω0= ((ρ1 1)−1)∗Ω1.