Un-Reduction of Systems of Second-Order Ordinary Differential Equations

In this paper we consider an alternative approach to ''un-reduction''. This is the process where one associates to a Lagrangian system on a manifold a dynamical system on a principal bundle over that manifold, in such a way that solutions project. We show that, when written in te...

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Datum:	2016
Hauptverfasser:	García-Toraño Andrés, E., Mestdag, T.
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Veröffentlicht:	Інститут математики НАН України 2016
Schriftenreihe:	Symmetry, Integrability and Geometry: Methods and Applications
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spelling	irk-123456789-1485512019-02-19T01:24:07Z Un-Reduction of Systems of Second-Order Ordinary Differential Equations García-Toraño Andrés, E. Mestdag, T. In this paper we consider an alternative approach to ''un-reduction''. This is the process where one associates to a Lagrangian system on a manifold a dynamical system on a principal bundle over that manifold, in such a way that solutions project. We show that, when written in terms of second-order ordinary differential equations (SODEs), one may associate to the first system a (what we have called) ''primary un-reduced SODE'', and we explain how all other un-reduced SODEs relate to it. We give examples that show that the considered procedure exceeds the realm of Lagrangian systems and that relate our results to those in the literature. 2016 Article Un-Reduction of Systems of Second-Order Ordinary Differential Equations / E. García-Toraño Andrés, T. Mestdag // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 19 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 34A26; 37J15; 70H33; 70G65 DOI:10.3842/SIGMA.2016.115 http://dspace.nbuv.gov.ua/handle/123456789/148551 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	In this paper we consider an alternative approach to ''un-reduction''. This is the process where one associates to a Lagrangian system on a manifold a dynamical system on a principal bundle over that manifold, in such a way that solutions project. We show that, when written in terms of second-order ordinary differential equations (SODEs), one may associate to the first system a (what we have called) ''primary un-reduced SODE'', and we explain how all other un-reduced SODEs relate to it. We give examples that show that the considered procedure exceeds the realm of Lagrangian systems and that relate our results to those in the literature.
format	Article
author	García-Toraño Andrés, E. Mestdag, T.
spellingShingle	García-Toraño Andrés, E. Mestdag, T. Un-Reduction of Systems of Second-Order Ordinary Differential Equations Symmetry, Integrability and Geometry: Methods and Applications
author_facet	García-Toraño Andrés, E. Mestdag, T.
author_sort	García-Toraño Andrés, E.
title	Un-Reduction of Systems of Second-Order Ordinary Differential Equations
title_short	Un-Reduction of Systems of Second-Order Ordinary Differential Equations
title_full	Un-Reduction of Systems of Second-Order Ordinary Differential Equations
title_fullStr	Un-Reduction of Systems of Second-Order Ordinary Differential Equations
title_full_unstemmed	Un-Reduction of Systems of Second-Order Ordinary Differential Equations
title_sort	un-reduction of systems of second-order ordinary differential equations
publisher	Інститут математики НАН України
publishDate	2016
url	http://dspace.nbuv.gov.ua/handle/123456789/148551
citation_txt	Un-Reduction of Systems of Second-Order Ordinary Differential Equations / E. García-Toraño Andrés, T. Mestdag // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 19 назв. — англ.
series	Symmetry, Integrability and Geometry: Methods and Applications
work_keys_str_mv	AT garciatoranoandrese unreductionofsystemsofsecondorderordinarydifferentialequations AT mestdagt unreductionofsystemsofsecondorderordinarydifferentialequations
first_indexed	2025-07-12T18:54:26Z
last_indexed	2025-07-12T18:54:26Z
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 12 (2016), 115, 20 pages Un-Reduction of Systems of Second-Order Ordinary Differential Equations Eduardo GARCÍA-TORAÑO ANDRÉS † and Tom MESTDAG ‡ † Departamento de Matemática, Universidad Nacional del Sur, CONICET, Av. Alem 1253, 8000 Bah́ıa Blanca, Argentina E-mail: egtoranoandres@gmail.com ‡ Department of Mathematics and Computer Science, University of Antwerp, Middelheimlaan 1, B–2020 Antwerpen, Belgium E-mail: mestdagtom@gmail.com Received August 12, 2016, in final form November 29, 2016; Published online December 07, 2016 http://dx.doi.org/10.3842/SIGMA.2016.115 Abstract. In this paper we consider an alternative approach to “un-reduction”. This is the process where one associates to a Lagrangian system on a manifold a dynamical system on a principal bundle over that manifold, in such a way that solutions project. We show that, when written in terms of second-order ordinary differential equations (SODEs), one may associate to the first system a (what we have called) “primary un-reduced SODE”, and we explain how all other un-reduced SODEs relate to it. We give examples that show that the considered procedure exceeds the realm of Lagrangian systems and that relate our results to those in the literature. Key words: reduction; symmetry; principal connection; second-order ordinary differential equations; Lagrangian system 2010 Mathematics Subject Classification: 34A26; 37J15; 70H33; 70G65 1 Introduction One of the much-discussed aspects of Lagrangian systems with a symmetry group is their re- duction to the so-called Lagrange–Poincaré equations [6] (but see also [9] for an approach using Lie algebroids, or [13] for an approach that is relevant for this paper). The idea is that one may cancel out the symmetry group, and, once a principal connection has been invoked, arrive at two sets of coupled equations on a quotient manifold, the so-called horizontal and vertical equa- tions. The horizontal equation looks again a bit like a Lagrangian equation (for a reduced-type Lagrangian function), but it has extra non-conservative force terms and, equally important, it is in general not decoupled from the vertical equation. In recent years, there has been some interest in so-called un-reduction [1, 2, 5, 7]. The motivation behind this paper is related to the un-reduction theorem (Theorem 5.1) of the pa- per [5]. Given a Lagrangian ` on a quotient manifold M̄ = M/G and a principal connection ω on π : M → M/G, the theorem tells one how to associate a “Lagrangian system with extra non-conservative forces” on M which has the property that its solutions project to those of the Euler–Lagrange equations of `. The methodology and reasoning of [5] is almost entirely based on Lagrange–Poincaré reduction, and on the fact that one may choose the external forces in such a way that the inconvenient curvature terms which appear in the horizontal Lagrange–Poincaré equation vanish. In our opinion, it may be more advantageous to think of this problem outside of the Lagrange–Poincaré framework. The Lagrange–Poincaré equations give essentially a dy- namical system on the manifold (TM)/G (a Lie algebroid), while the Euler–Lagrange equations mailto:egtoranoandres@gmail.com mailto:mestdagtom@gmail.com http://dx.doi.org/10.3842/SIGMA.2016.115 2 E. Garćıa-Toraño Andrés and T. Mestdag of ` are defined on T (M/G) (a tangent bundle). One of the purposes of this paper is to show that for a comparison of dynamics one may remain in the category of tangent bundles. The Euler–Lagrange equations of ` are but a particular example of a SODE, a system of second-order ordinary differential equations. The un-reduced equations one finds in [5] are no longer Euler–Lagrange equations, since the presence of extra non-conservative forces breaks the variational character of these equations, but they are still a SODE. We will show that the un- reduction process is very natural in the context of SODEs, and that one may identify a (what we call) “primary un-reduced SODE”, to which all other un-reduced SODEs easily relate. There is, in our setting, no need to invoke Lagrange–Poincaré equations or external forces. The issues related to what is called “coupling distortion” and “curvature distortion” in [5] are, in our opinion, side-effects from relying on a Lagrange–Poincaré-based approach to un-reduction (as opposed to a SODE-based approach). In Section 3 we identify the two lifted principal connections that lie at the basis of our un-reduction. After the definition of the primary un-reduced SODE in Section 4 we state a proposition about all other SODEs whose base integral curves project on those of the given SODE. The canonical connection of a Lie group gives, in Section 5, a natural environment where all the introduced concepts can be clarified. Since it is not always variational (i.e., its geodesics are not always solutions of some Euler–Lagrange equations) it shows that our discussion is a meaningful generalization of the one in [5]. In Section 6 we specify to the case of a Lagrangian SODE, and we discuss the example of Wong’s equations and the effect of curvature distortion in Section 7. In Section 8 we say a few words about a second un-reduction procedure. We end the paper with some possible lines of future research. The advantage of un-reduction remains unaltered in our setting, as it is explained in [5]: If one knows that solutions of a second-order system on M̄ are but the projection of those of a system on M , one may compute these solutions by making use of any coordinates on M , not necessarily those adapted to the bundle structure of π. This may be beneficial when, e.g., the equations on M are more convenient to deal with numerically, which is precisely one of the main motivations to study un-reduction. We refer the interested reader to [5] and references therein for a more detailed discussion. 2 Preliminaries Consider a manifold M and its tangent bundle τM : TM → M . The vector fields XC and XV on TM stand, respectively, for the complete lift and the vertical lift of a vector field X on M . In natural coordinates ( xA, ẋA ) on TM and for X = XA∂/∂xA we get XC = XA ∂ ∂xA + ẋB ∂XA ∂xB ∂ ∂ẋA and XV = XA ∂ ∂ẋA . The Lie brackets of these vector fields are [ XC , Y C ] = [X,Y ]C , [ XC , Y V ] = [X,Y ]V and[ XV , Y V ] = 0. The notion of complete and vertical lifts also extends to functions and (1, 1) tensor fields, as follows. For a function f on M , its lifts are the functions fV = f ◦ τM and fC = ẋA ( ∂f/∂xA ) on TM . Let A be a (1, 1) tensor field on M . We may lift it to two (1, 1) tensor fields AC and AV on TM , as follows AC ( XV ) = (A(X))V , AC ( XC ) = (A(X))C , AV ( XV ) = 0, AV ( XC ) = (A(X))V . More details on this can be found in [19], such as the following immediate properties: LXV AC = (LXA)V , LXCAC = (LXA)C , LXV AV = 0, LXCAV = (LXA)V . Un-Reduction of Systems of Second-Order Ordinary Differential Equations 3 Definition 1. A vector field Γ on TM is a second-order differential equations field (SODE in short) on M if all its integral curves γ : I → TM are lifted curves, that is of the type γ = ċ, for base integral curves c : I →M . As such, Γ takes the form Γ = ẋA ∂ ∂xA + fA(x, ẋ) ∂ ∂ẋA . As a vector field on TM , a SODE is characterized by the property that TτM ◦ Γ = idM . A SODE Γ on M always exists, since a Riemannian metric on a (paracompact) manifold always exists, and we may consider its geodesic spray. In what follows we will often express a SODE in terms of a frame of vector fields ZA = ZBA∂/∂x B, which are not necessarily coordinate vector fields. In that context, we say that the quasi-velocities ( vA ) of a vector vm ∈ TmM are the components of vm with respect to that basis, i.e., vm = vAZA(m). Their relation to the standard fibre coordinates ẋA is ẋA = ZAB(m)vB. In terms of the frame {ZA}, the SODE Γ takes the form Γ = vAZCA + FAZVA , meaning that two SODEs Γ1 and Γ2 only differ in their coefficients FA1 and FA2 . Let c(t) =( xA(t) ) be a base integral curve of Γ. The fibre coordinates of integral curves ċ(t) of Γ in TM may also be expressed in quasi-velocities as ċ(t) = vA(t)ZA(c(t)). The functions ( xA(t), vA(t) ) are then solutions of the equations ẋA(t) = ZAB(x(t))vB(t), v̇A = FA(x(t), v(t)). Assume that M comes equipped with a free and proper (left) action Φ: G×M →M of a Lie group G, such that π : M → M̄ = M/G is a principal fibre bundle. For each element ξ in the Lie algebra g we may obtain a fundamental vector field ξM on M , defined by ξM (m) = TΦm(ξ) where, as usual Φm : G→M denotes the map Φ(·,m). In what follows it will be more convenient to write ξM = ξ̃ for a fundamental vector field on M . We will also assume that G is connected. In that case, a vector field X on M is G-invariant if and only if [X, ξ̃] = 0, for all ξ ∈ g. There exist two ways to lift the action Φ to an action on TM . The first action is the G- action ΦTM on TM , defined by ΦTM g = T (Φg) where again we write Φg = Φ(g, ·) : M → M . With this action πTM : TM → (TM)/G, the so-called Atiyah bundle, is a principal G-bundle. Fundamental vector fields of this action are vector fields on TM of the type ξ̃C . The tangent manifold TG of a Lie group G is also a Lie group. It may be identified with the semidirect product G× g, and its Lie algebra with g× g. The second action on TM is the TG-action, given by TΦ: TG × TM → TM . In the current trivialization, this action may be written as (g, ξ) · vm = TΦg ( vm + ξ̃(m) ) . With this action (TM)/(TG) = T (M/G) and Tπ : TM → T (M/G) is also a principal bun- dle, but with structure group TG (see, e.g., [11]). The fundamental vector fields that corre- spond to TΦ are linear combinations of the vector fields on TM given by (ξ, 0)TM = ξ̃C and (0, ξ)TM = ξ̃V . To clarify the concepts we introduce later on, we will often use local coordinates (xi) on M̄ = M/G and coordinates ( xA ) = (xi, xa) on M that are adapted to the bundle π. On the tangent manifold, however, we will use quasi-velocities with respect to a specific frame. If {Ea} is a basis for the (left) Lie algebra g, we will denote the fundamental vector fields (for the 4 E. Garćıa-Toraño Andrés and T. Mestdag G-action on M) by Ẽa. These vector fields span the vertical distribution of π, and their Lie brackets are given by [Ẽa, Ẽb] = −CcabẼc, if we denote the structure constants of g by Ccab. We now assume that we have chosen a principal connection on π (such a connection al- ways exists). Let Xi be the horizontal lifts, with respect to this connection, of the coordinate fields ∂/∂xi on M/G. The fact that the horizontal lifts are G-invariant vector fields may be expressed as [Xi, Ẽa] = 0. Given that[ XC i , Ẽ {C,V } a ] = [ Xi, Ẽa ]{C,V } = 0, [ XV i , Ẽ {C,V } a ] = 0, we see that both XC i and XV i are invariant vector fields, for both the G-action on TM , and the TG-action on TM . The Lie bracket [Xi, Xj ] = RaijẼa of these horizontal vector fields represents the curvature of the connection. We will denote the quasi-velocities with respect to the frame { Xi, Ẽa } as (vi, va). Actually, since the vector fields Xi project, we may conclude that the quasi-velocities vi can be identified with the natural fibre coordinates on M/G, i.e., vi = ẋi. Some further immediate properties are included in the list below (see, e.g., [8, 13] for more details): XC i (vj) = 0, XV i (vj) = δji , XC i (va) = −Raijvj , XV i (va) = 0, ẼCa (vi) = 0, ẼVa (vi) = 0, ẼCa (vb) = Cbacv c, ẼVa (vb) = δba. A SODE Γ takes the form Γ = viXC i + vaẼCa + F iXV i + F aẼVa when expressed in this frame. It will be G-invariant when[ Γ, ẼCa ] = 0 ⇔ ẼCa (F i) = 0, ẼCa (F b) = F cCbac. It is important to realize, however, that a SODE can never be TG-invariant since, besides the previous properties, it would also have to satisfy [Γ, ẼVa ] = 0, which is impossible because of the first term in[ Γ, ẼVa ] = −ẼCa − ẼVa (F i)XV i − ( ẼVa ( F b ) + vcCbca ) ẼVb . For this reason a SODE on M can never be the horizontal lift of a principal connection on the TG-bundle Tπ : TM → T (M/G). 3 Two lifted connections Assume that a principal connection on π is given. There are many equivalent ways to represent this principal connection. For example, we may either consider it as being given by a (1, 1)-tensor field ω on M (its “vertical projection operator”) or by a connection map $ : TM → g. These two approaches are related as ω(X)(m) = ˜$(X(m))(m). A third way to define a connection makes use of its horizontal lift. The horizontal lift of a vector field X̄ on M̄ is the unique vector field X̄h on M that projects on X̄ and that is such that ω ( X̄h ) = 0. In what follows we will use these three definitions simultaneously. The principal connection lifts to a principal connection on each of the principal bundles πTM and Tπ. In [13, Proposition 3], it is shown that the complete lift ωC (a (1, 1) tensor field on TM) represents a connection on the fibre bundle Tπ : TM → T (M/G). Moreover, since Lξ̃ω = 0, the properties of complete lifts lead to Lξ̃CωC = 0 and Lξ̃V ωC = 0, from which we may conclude that this connection is principal with respect to the structure group TG. The idea of using the complete lift of a connection on a bundle Y → X to define a connection on TY → TX comes from the paper [18] by Vilms, who used it in the context of vector bundles. For that reason, we will refer to ωC as the Vilms connection. Un-Reduction of Systems of Second-Order Ordinary Differential Equations 5 Definition 2. The Vilms connection of a principal connection ω on π : M → M̄ is the TG- principal connection on Tπ : TM → TM̄ whose vertical projection operator is given by the tensor field ωC . The Vilms connection is characterized by its action on vector fields on TM , and therefore by ωC ( XC i ) = 0, ωC ( XV i ) = 0, ωC ( ẼCa ) = ẼCa , ωC ( ẼVa ) = ẼVa . Proposition 1. Let ω be a principal connection on π. The horizontal lift H of the Vilms connection can be characterized in terms of the horizontal lift h of ω, by means of the relations( X̄C )H = ( X̄h )C and ( X̄V )H = ( X̄h )V , for any vector field X̄ on M̄ . Proof. The horizontal lift of the Vilms connection maps a vector field Ȳ on T (M/G) onto a vector field Ȳ H on TM . Vector fields on T (M/G) are functional combinations of the coordinate vector fields ∂/∂xi and ∂/∂ẋi. One may easily check that their horizontal lifts to vector fields on TM , for the Vilms connection, are( ∂ ∂xi )H = XC i , ( ∂ ∂ẋi )H = XV i . From this, one can deduce the statement in the proposition. � If we are given a SODE on M/G, Γ̄ = ẋi∂/∂xi+f i∂/∂ẋi, its horizontal lift is the vector field on TM given by Γ̄H = ẋiXC i + f iXV i . Since L Ẽ {C,V } a ωC = 0, we have [ Ẽ {C,V } a , Γ̄H ] = 0, which expresses that Γ̄H is TG-invariant. Its TG-reduced vector field is, of course, Γ̄. However, Γ̄H is not a SODE on M : Essentially we are missing the term “vaẼCa ”. To understand the part we are missing, we need to invoke a second connection. The second connection is the “vertical lift” of ω. By this we do not mean the (1, 1) tensor field ωV we had defined above. We now need the interpretation of the connection on π as the connection map $ : TM → g. Definition 3. The vertical connection of a principal connection $ on π : M → M̄ is the G- principal connection on πTM : TM → (TM)/G whose connection map $V : TTM → g is given by $V = τ∗M$, with τM : TM →M . For the corresponding interpretation of $V as a (1, 1)-form Ω on TM one may calculate that Ω ( XC i ) = 0, Ω ( XV i ) = 0, Ω ( ẼCa ) = ẼCa , Ω ( ẼVa ) = 0. It may easily be verified that this defines a principal connection on the G-bundle πTM : Among other properties, LẼC a Ω = 0. The tensor field Ω clearly differs from the tensor field ωV , since, for example, Ω ( ẼCa ) = ẼCa , but ωV ( ẼCa ) = ẼVa . Assume that Γ is a G-invariant SODE which reduces to the vector field Γ̂ on (TM)/G. The vertical lift connection $V can be used to reconstruct the integral curve γ (through v0) of Γ, starting from the integral curve γ̂ (through πTM (v0)) of Γ̂. This procedure is in fact valid for any principal bundle N → N/G with a given principal connection, not necessarily N = TM (see, e.g., [13, Proposition 1], where it is explained in detail). Let γ̂Hor be the horizontal lifted curve 6 E. Garćıa-Toraño Andrés and T. Mestdag (by means of the principal connection $V ) of γ̂ through v0. It is defined by the properties that it remains horizontal everywhere, that it goes through v0 and that it projects on γ̂. Let θ be the Maurer–Cartan form on G, i.e., θ(vg) = TLg−1vg for vg ∈ TgG. The reconstruction theorem states that if one solves the equation θ(ġ) = $V ( Γ ◦ γ̂Hor ) (with g(0) = e) for a curve g(t) in G, then the relation between the integral curves is given by γ(t) = ΦTM g(t) γ̂ Hor(t). If we write the SODE Γ as before as Γ = viXC i + vaẼCa +F iXV i +F aẼVa , then the reconstruction equation is of the form (θ(ġ))a = va, where va are the vertical quasi-velocities of γ̂Hor. It is clear that, in this respect, the reconstruction equation is essentially related to the term vaẼCa of the SODE Γ. In what follows, we will often make use of this vector field, and we will denote it by Xω. Proposition 2. Let Γ0 be an arbitrary SODE on M . The vector field Xω = Ω(Γ0) on TM is independent of the choice of Γ0. It is Tπ-vertical, G-invariant, but not TG-invariant. Proof. Since Γ0 must take the form vaẼCa + · · · , the vector field Ω(Γ0) = vaẼCa is independent of the choice of the SODE Γ0. It is clearly Tπ-vertical since Tπ ◦ Ω = 0. It is also easy to see that this vector field is G-invariant, but not TG-invariant. Indeed, we have[ Xω, Ẽ C b ] = va [ ẼCa , Ẽ C b ] − ẼCb (va)ẼCa = 0, but, on the other hand,[ Xω, Ẽ V b ] = va [ ẼCa , Ẽ V b ] − ẼVb (va)ẼCa = − ( vaCcabẼ V c + ẼCb ) . � From its expression in quasi-velocities, it can be seen that a pointwise definition for the vector field Xω is Xω(v) = (̃$(v)) C (v), for all v ∈ TM . 4 Un-reducing second-order systems We start with some general considerations for an arbitrary principal bundle µ : P → P/G. Definition 4. A vector field X ∈ X (P ) is an un-reduction of a vector field X̄ ∈ X (P/G) if all its integral curves project on those of X̄. This is actually equivalent with saying that X and X̄ are µ-related, since for any integral curve c(t) of X the property in the definition means that (µ ◦ c)(t) = (c̄ ◦ µ)(t), which, after differentiating, becomes Tµ ◦X = X̄ ◦ µ. For each principal connection $ on µ, the horizontal lift X̄h is clearly an un-reduction. This shows that if we fix a principal connection, any un-reduction X is of the type X = X̄h + W , where W may be any µ-vertical vector field on P . We now turn to the case of interest, where µ = Tπ and P = TM with structure group TG. For any vector field X̄ on T (M/G), the Vilms connnection ωC on Tπ generates an un-reduc- tion X̄H . We will be interested in the case where either X̄ or X is a SODE. Proposition 3. Let Γ̄ be a vector field on T (M/G), and let Γ be an un-reduction of Γ̄ on TM . (1) If Γ is a SODE on M , then Γ̄ is a SODE on M/G. (2) If Γ̄ is a SODE on M/G, then Γ is a SODE on M if, and only if, $V ◦ Γ = $. (3) If Γ̄ is a SODE on M/G, then Γ is a SODE on M if, and only if, Ω(Γ) = Xω. Un-Reduction of Systems of Second-Order Ordinary Differential Equations 7 Proof. (1) The first statement follows from the commutative diagrams below. If Γ is an un- reduction and a SODE it satisfies TTπ ◦ Γ = Γ̄ and TτM ◦ Γ = IdTM . After applying Tπ on the last relation, we get T (π ◦ τM ) ◦ Γ = IdT (M/G) and, from the diagram, the left hand side becomes TτMG ◦ TTπ ◦ Γ = TτMG ◦ Γ̄, from which it follows that Γ̄ is a SODE. TM M T (M/G) M/G τM Tπ π τ(M/G) TTM TM TT (M/G) TM/G TτM TTπ Tπ Tτ(M/G) (2) The condition can be rewritten as $ ◦ TτM ◦ Γ = $. If Γ is a SODE then TτM ◦ Γ = IdTM and the statement follows. Assume now that $ ◦ TτM ◦ Γ = $. We may use the principal connection to decompose each wm ∈ TmM into its horizontal and vertical part as (m,Tπ(wm))h + ($(wm))M (m). For wm = TτM (Γ(vm)) we get, in view of the assumption that Γ is an un-reduction of a SODE Γ̄ on M/G, wm = (m,T (π ◦ τM )(Γ(vm))))h + ($(TτM (Γ(vm))))M (m) = (m,T (τM/G ◦ Tπ)(Γ(vm))))h + ($(vm))M (m) = (m,T (τM/G)(Γ̄(Tπ(vm))))h + ($(vm))M (m) = (m,Tπ(vm))h + ($(vm))M (m). From this it follows that TτM (Γ(vm)) = wm = vm, and that Γ is a SODE on M . (3) The relation between the connection map $ of a connection, and its vertical projection ω is given by ω(X)(m) = ˜$(X(m))(m). In the case of the vertical connection this becomes Ω(Γ)(vm) = ( ˜$V (Γ(vm)) )C (vm). If Γ is an un-reduction of a SODE Γ̄ then it will be SODE as well, in view of (2), if and only if Ω(Γ)(vm) = ( $̃(vm) )C (vm) = Xω(vm). � We may now introduce the following definition. Definition 5. Let Γ̄ be a SODE on M̄ and ω be a principal connection on π. The primary un-reduced SODE of (Γ̄, ω) is the SODE Γ1 = Γ̄H +Xω on M . In the frame { Xi, Ẽa } , Γ1 takes the form Γ1 = Γ̄H + Γω = viXC i + f iXV i + vaẼCa + 0ẼVa . The reason for calling Γ1 an “un-reduced” SODE is given in the third observation below. Proposition 4. Let Γ̄ be a SODE on M̄ and ω be a principal connection on π. (1) The primary un-reduced SODE Γ1 of (Γ̄, ω) is a G-invariant vector field on TM , but is not TG-invariant. (2) The SODE Γ1 is the unique un-reduced SODE on M that satisfies ωC(Γ) = Xω. (3) The base integral curves c of Γ1 project, via π : M → M̄ , on base integral curves c̄ of Γ̄. (4) Γ1 is the unique SODE that projects on Γ̄ and that has the property that it is tangent to the horizontal distribution of ω. 8 E. Garćıa-Toraño Andrés and T. Mestdag Proof. The first observation follows from the fact that Γ̄H is TG-invariant, and therefore[ Γ1, ξ̃ {C,V }] = [ Xω, ξ̃ {C,V }]. The second observation follows from the fact that the vector fields XC and Y C are Tρ-related if and only if X and Y are ρ-related. A similar observa- tion holds for vertical lifts. One can easily prove this by considering the flows of the involved vector fields. In our case, we know that Γ̄H and Γ̄ are Tπ-related by definition, and that Ẽa is π-related to zero, whence Xω is Tπ-related to zero as well. The observation that the SODE Γ1 is Tπ-related to the SODE Γ̄ means that its integral curves γ in TM project, via Tπ, on integral curves γ̄ of Γ̄ in TM̄ . Since both vector fields are SODEs, their integral curves are lifted curves. We conclude from this that the base integral curves c of Γ1 project, via π : M → M̄ , on base integral curves c̄ of Γ̄. Horizontal vectors, expressed in quasi-velocities as vm = viXi(m) + vaẼCa (m), have the property that va = 0. Let’s write Γ1 for now as Γ1 = viXC i + f iXV i + vaẼCa + F aẼVa . For the last property we need to show that Γ1(vb) = 0 is equivalent with F a = 0. This follows easily from the properties XC i (vb) = −Rbijvj , XV i (va) = 0, ẼCa (vb) = Cbacv c and ẼVa (vb) = δba. � Integral curves of Γ̄ satisfy ẍi = f i(x, ẋ). For Γ1 = Γ̄H +Xω = viXC i + f iXV i + vaẼCa + 0ẼVa , the integral curves satisfy, among other, ẍi = f i(x, ẋ) and v̇a = 0. The remaining equation is the “reconstruction equation” (θ(ġ))a = va, where θ is the Maurer–Cartan form (see the paragraph above Proposition 2). From the first statement in Proposition 4, the one which says that Γ1 is G-invariant, we may conclude that Γ1 can be reduced to a vector field on (TM)/G. This vector field is, obviously, not the same as the SODE Γ̄, which is a vector field on T (M/G). Vector fields on (TM)/G play, however, an essential role in so-called Lagrange–Poincaré reduction. The Euler–Lagrange equations of a G-invariant regular Lagrangian produce such a G-invariant SODE on M , and the integral curves of its reduced vector field on (TM)/G satisfy what are called the Lagrange– Poincaré equations. In this sense, one may think of the un-reduction of [5], which heavily relies on Lagrange–Poincaré reduction, as a process that compares the reduced vector field on (TM)/G with the vector field Γ̄ on T (M/G). Let c̄ be a curve in M/G which goes at t = 0 through a point m̄. Consider a point m ∈ M with π(m) = m̄. The horizontal lift of c̄ at m is the unique curve c̄hm which is defined by the following three properties: (1) it projects on c̄ for all t, (2) it satisfies c̄hm(0) = m, (3) it has only horizontal tangent vectors ˙̄c h m(t). Proposition 5. The base integral curves of the primary un-reduced SODE Γ1 of (Γ̄, ω) through horizontal initial vectors are horizontal lifts of base integral curves of Γ̄. Proof. Consider an initial value m̄ in M/G and an initial tangent vector v̄ = vi0∂/∂x i\|m̄, and their corresponding base integral curve c̄m̄,v̄(t) of Γ̄. Let m ∈ M be such that π(m) = m̄, and consider the vector v := h(m, v̄), the ω-horizontal lift of v̄ to m. In the frame {Xi, Ẽa}, we may write v = vi0Xi(m) + 0Ẽa(m). With the initial values (m, v), the base integral curve cm,v(t) of Γ satisfies π(cm,v(t)) = c̄m̄,v̄(t) and also va(t) = 0 (with this particular initial value). The solution of the remaining equation is then θ(ġ) = 0, from which we see that g(t) remains constant. From the fact that va(t) = 0, we conclude that the velocity ˙̄c h m(t) remains horizontal through- out. Given that also (1) and (2) are satisfied, we may conclude that cm,v = (c̄m̄,v̄) h m. � If we had taken a different initial condition, say va(0) = ξa, we would get va(t) = ξa, and the corresponding curve c would not be a horizontal lift. From this proposition, we may derive an “un-reduction algorithm”. Consider a SODE Γ̄ on M̄ , a principal connection on M → M̄ and their corresponding “lifted” primary un-reduced SODE Γ1 = Xω + (Γ̄)H . Calculate its integral curves c(t) in M , but only for the very specific initial values we gave above: the initial velocity has to be horizontal. We have just shown that Un-Reduction of Systems of Second-Order Ordinary Differential Equations 9 these curves are in fact the horizontal lifts of the curves c̄ that we really want to know. If we project them down, we get the desired curves c̄. Note that any field of initial velocities works as well: we could compute the integral curves of Γ1 for any set of initial conditions, and then project them in view of Proposition 4. Other than the primary un-reduced SODE, there are many more SODEs that have the property that (some of) their integral curves project onto those of Γ̄. All other un-reduction SODEs are of the form Γ2 = Γ1 + V , where V is Tτ -vertical and Tπ-vertical. Since Γ1, XV i and ẼVa are Tπ-related to Γ̄, ∂/∂ẋi and 0, respectively, this means that V may be any vector field on TM of the type V = V aẼVa , which means that it is a vector field with Ω(V ) = 0. Proposition 6. Let Γ̄ be a SODE on M/G. Any vector field Γ2 = Γ1 + V , where V is Tπ- vertical and such that Ω(V ) = 0, is a SODE which has the property that its base integral curves project on those of Γ̄. The full expression of Γ2 is Γ2 = viXC i + f iXV i + vaẼCa + V aẼVa . Its integral curves satisfy ẍi = f i and v̇a = V a. If we only consider vector fields Γ2 that are G-invariant, then, since we know that Γ1 is G-vertical, the coefficients V a need to satisfy ẼCa (V b) = CbadV d. With the above, we have characterized all G-invariant SODEs Γ2 which have the property that all their integral curves project to those of Γ̄. The systems that appear in [5] therefore belong to the SODEs we have discussed in this section, see Section 6 for more details. In Section 8, we mention a different class of SODEs for which only a subclass of its integral curves project. 5 A non-Lagrangian example We will now consider an example where the dynamics is not of Lagrangian nature, and therefore falls out of the scope of [5]. This section intends to clarify some of the concepts we have introduced in the previous ones. First, we recall some well known facts. In general, if {ZA} is a frame on a manifold M , with quasi-velocities vA, and if ∇ is a linear affine connection with coefficients ∇ZA ZB = γDABZD, then its quadratic spray is the SODE given by vAZCA − γDABv AvBZVD . The base integral curves of this spray satisfy: v̇D = −γDABvAvB. This construction applies in particular to the case where the connection is the Levi-Civita connection of some (pseudo-)Riemannian metric on M . Consider a connected Lie group G, with its Lie algebra g given by the Lie algebra of left- invariant vector fields. The canonical affine connection on a Lie group G can be defined by its action on left-invariant vector fields: ∇ζLGη L G = 1 2 [ζ, η]LG, where ζLG, ηLG stand for the left-invariant vector fields of ζ, η ∈ g. Among other properties, ∇ has vanishing torsion; see [14] for a detailed discussion of this and other aspects of ∇. We denote its quadratic spray by ΓG. If the Lie-algebra g is (semi-simple or) compact then its Killing form defines a (pseudo-)Riemannian metric on G for which the canonical connection is its Levi-Civita connection. In other cases, although the canonical spray is a quadratic spray, the canonical connection may not be metrical and, moreover, its spray need not even be variational (see, e.g., [17] for some occurrences of this situation). It is also well-known that if the canonical connection is the Levi-Civita of some left-invariant Riemannian metric, then that metric is necessarily also right-invariant (see [15]). The base integral curve of ΓG through g and ζ ∈ g (when TG is left trivialized as G× g) is given by t 7→ g exp(tζ). This is easy to see, as follows. 10 E. Garćıa-Toraño Andrés and T. Mestdag The connection coefficients with respect to the frame {(EA)LG} of left-invariant vector fields are γDAB = 1 2C D AB, which are skew-symmetric. The spray is then ΓG = vAL ( (EA)LG )C (with no( (EA)LG )V component) and the equation for its base integral curves is v̇AL = 0, and therefore vAL (t) = ζA, for some constants ζA. Since vAL are the quasi-velocities in the frame of left-invariant vector fields, they are precisely the components of ġ left-translated to the Lie algebra, i.e., we have g−1ġ = ζ. The solution of ( g−1ġ )A = ζA with g(0) = e is precisely the one-parameter group of ζ. The rest of the argument relies on the symmetry of ΓG. Note that, from the expression ΓG = vAL (EA)CL , we may also write that ΓG(vg) = ( ζLG )C (vg), where ζ = TLg−1vg = vAEA. Consider now a (closed) normal subgroup N of G, and consider its right action on G. Then K = G/N is again a Lie group, where multiplication is given by (g1N)(g2N) = (g1g2)N , where gN denotes the left coset. We will write π : G → K = G/N for the projection and g, n and k for the corresponding Lie algebras. Since K is a Lie group, it comes with its own canonical connection ∇K . We wish to show that the sprays ΓK and ΓG of the two connections are related by means of an un-reduction process. Recall that, if that is the case, we only need to introduce coordinates on G (and N) to be able to write down integral curves on K. To start the un-reduction process we need a principal N -connection on π : G → K. Such a connection is, for example, available if we consider on g an AdN -invariant inner product. (Remark that we do not assume that the inner product is AdG-invariant. But, if that were the case, then it would generate a bi-invariant metric on G, whose Levi-Civita connection was ∇G.) The orthogonal complement m of n with respect to this inner product is then AdN -invariant. In effect, this means that N is reductive. We will write g = m⊕ n. In these notations Teπ(g) = k. Consider the short exact sequence 0→ n→ g→ k→ 0. Its splitting k→ g with image m will be denoted by s. From the property that N is normal we get that [n, n] ⊂ n and [n,m] ⊂ n (i.e., n is an ideal of g). From the fact that m is AdN -invariant, we also get that [n,m] ⊂ m, and therefore [n,m] = {0}. Remark that we do not know much about [m,m] (its vertical part is related to the curvature of the connection we will introduce next). Let Pm : g → m and Pn : g → n denote the projections on m and n, respectively. We may associate an N -principal connection on π : G → K to the decomposition g = m ⊕ n by means of the connection map $(vg) = Pn(TLg−1vg). To see that this is a connection, recall that the infinitesimal generators of the N -action are given by the left-invariant vector fields ηLG on G, associated to η ∈ n. Given that Adn ◦Pn = Pn ◦ Adn (where n ∈ N) we may easily verify that indeed $(ηLG(g)) = η, for η ∈ n, and $(TRnvg) = Adn−1($(vg)). Consider now, for ξ ∈ k, the vector field ξLK on K. We will show that its horizontal lift, with respect to the connection $, is the left-invariant vector field (sξ)LG. It is easy to see that this vector field is N -invariant, since for all infinitesimal generators of the action, [ ηLG, (sξ) L G ] = [η, sξ]LG = 0 for all η ∈ n. Moreover, (sξ)LG projects on ξKL , since Tπ ( (sξ)LG(g) ) = Tπ(TLg(sξ)) = TLgN ((π ◦ s)(ξ)) = TLgN (ξ) = ξKL (π(g)). Finally, (sξ)LG is horizontal, since $ ( (sξ)LG(g) ) = Pn(TLg−1(TLg(sξ))) = Pn(sξ) = 0. Proposition 7. Let N be a normal subgroup of a Lie group G and consider the principal connec- tion $ associated to an AdN -invariant inner product on g. Then, the canonical spray ΓG on G is the primary un-reduced SODE of the canonical spray ΓK on K = G/N and the connection $. Un-Reduction of Systems of Second-Order Ordinary Differential Equations 11 Proof. We first derive an expression for the Vilms-horizontal lift of the spray ΓK . Consider a given vg ∈ TgG, and let vk = Tπ(vg) ∈ Tπ(g)K. We will also use the notations ζ = TLg−1vg and ξ = TLk−1vk. By construction, ΓK(vk) = ( ξKL )C (vk). Since the complete lifts on G and K commute with the two connections (on the one hand the Vilms-horizontal lift, and on the other hand the horizontal lift, see Proposition 1) we easily get that (ΓK)H(vg) = (( (ξ)LK )h)C (vg) = ( (sξ)LG )C (vg) = (( s ( TLk−1Tπ(vg) ))L G )C (vg) = (( s ( T ( π ◦ Lg−1 ) (vg) ))L G )C (vg) = (( (s ◦ Tπ) ( Lg−1(vg) ))L G )C (vg) = ( (Pmζ)LG )C (vg). On the other hand, we have already stated that, in the current notations, Xω(vg) = ( ($(vg)) L G )C (vg) = ( (Pnζ)LG )C (vg). Together, we get (ΓK)H(vg) +Xω(vg) = ( (Pmζ)LG )C (vg) + ( (Pnζ)LG )C (vg) = ( (ζ)LG )C (vg), which is exactly ΓG(vg). � Example: The general linear group as a bundle over R. In the set Mn×n(R) of n× n real matrices we consider the Lie groups GL+(n) = {A ∈Mn×n(R) : det(A) > 0}, SL(n) = {A ∈Mn×n(R) : det(A) = 1}. The group SL(n) is a normal subgroup of GL(n)+. This follows from the fact that it is the kernel of the Lie group homomorphism det : GL+ → R+, where R+ is the multiplicative group of positive real numbers. In particular, the determinant induces an isomorphism: ϕ : GL+(n)/SL(n)→ R+. The situation is summarized in the following diagram: GL+(n) GL+(n)/SL(n) R+ GL+(n) GL+(n)/SL(n) R+ A 7→A·(SL(n)) A 7→A·B A·(SL(n))7→det(A) A·(SL(n))7→(A·B)·(SL(n)) x 7→xdet(B) A 7→A·(SL(n)) A·(SL(n))7→det(A) We can therefore think of GL+(n) as a principal SL(n)-bundle over R+, where SL(n) acts on the right on GL+(n) and with the projection given by π(A) = det(A). We also observe that the tangent of the projection π at Id (the n× n identity matrix) is TIdπ = trace. The Lie algebra of GL+(n) is gl(n) and the Lie algebra sl(n) of SL(n) consists of the real n×n traceless matrices. We consider the inner product on gl(n) given by 〈A,B〉gl(n) = trace ( AB−1 ) . Since the trace is invariant under conjugation, 〈·, ·〉gl(n) defines and AdSL(n)-invariant inner product on gl(n). Note that the identity matrix Id is orthogonal to sl(n), and therefore 〈·, ·〉gl(n) 12 E. Garćıa-Toraño Andrés and T. Mestdag induces the splitting gl(n) = sl(n)⊕ 〈Id〉. The associated map s : R→ gl(n) is given by s(λ) =( λ n ) Id. The situation is as follows: 0 → sl(n) gl(n) R → 0i trace λ 7→ (λ n) Id The horizontal space of the connection $ is therefore given by left translation of the identity. A vector vA in TA(GL+(n)) is horizontal if it is of form vA = µA, where µ ∈ R (we identify the tangent space to a vector space with the vector space itself). The horizontal lift of the tangent vector λ ∈ TxR+ to A ∈ π−1(x) is given by the vector vA = ( λ xn ) A. Indeed, this vector is horizontal and it projects back, since π ◦ LA−1 = Lx−1 ◦ π, and thus Tπ(vA) = TLx ◦ Tπ ( A−1vA ) = x trace ( A−1vA ) = λ. We will use x for the coordinate on R+. When written in terms of vector fields, we may see from the above that, e.g., the horizontal lift of the coordinate vector field ∂/∂x on R+ is the vector field defined by the map X : GL+(n)→ GL+(n) : A 7→ 1 n detAA. We may also write X = 1 nx id, where id stands for the vector field given by the identity map on GL+(n). We will write Γ̄ and Γ for the canonical sprays on R+ and GL+(n), respectively. The element 1 ∈ R is a basis for the Lie algebra of R+. Its left-invariant vector field on R+ is then x∂/∂x. If we use natural coordinates (x, ẋ) ∈ TR+, the quasi-velocity with respect to this vector field is then w = ẋ/x. Therefore, the canonical spray Γ̄ on R+ is Γ̄ = w ( x ∂ ∂x )C = ẋ ∂ ∂x + ẋ2 x ∂ ∂ẋ . Due to the properties in Proposition 1, its Vilms horizontal lift is Γ̄H = ẋ (( ∂ ∂x )h)C + ẋ2 x (( ∂ ∂x )h)V = ẋXC + ẋ2 x XV . But, X = 1 nx id. Given that in general (fY )C = fY C + ḟY V , we may also write Γ̄H = ẋ nx (id)C − ẋ2 nx2 (id)V + ẋ2 nx2 (id)V = ẋ nx (id)C . Consider now a basis {Ea} ∪ Id of gl(n) (a = 1, . . . , n2 − 1). It is clear that the fundamental vector field (Id)GL+(n) is simply the vector field id on GL+(n). If {va, v} denote the quasi- velocities w.r.t. the frame { Ẽa, id } then we have that v = ẋ nx . In the spray Γ, Γ = vaẼCa + v(id)C , we recognize therefore in the first term the vector field Xω, and in the last term the Vilms horizontal lift Γ̄H . 6 Un-reducing Lagrangian systems We now specialize to the case where the SODE Γ̄ is the Lagrangian vector field of a regular Lagrangian ` (a smooth function on TM̄). The vector field Γ̄ on TM̄ is then completely determined by two facts: (1) it is a SODE, (2) it satisfies Γ̄ ( ∂` ∂ẋi ) − ∂` ∂xi = 0. Un-Reduction of Systems of Second-Order Ordinary Differential Equations 13 Written in coordinates, this characterizations says that if we write, in general, Γ̄ = Xi∂/∂xi + f i∂/∂ẋi for a vector field on TM̄ , then, from (1) we know that Xi = ẋi, and from (2) we get that f i is determined by the Euler–Lagrange equations, when written in normal form. We will translate property (2) into one that the primary un-reduced SODE Γ1 = Xω + Γ̄H satisfies. First, we recall the following observations for vector fields on a principal K-bundle p : Q → Q/K. When a vector field W on Q is K-invariant, the relation W̄ ◦p = Tp◦W uniquely defines its reduced vector field W̄ on Q/K. Likewise, if F : Q → R is a K-invariant function on Q it can be reduced to a function f : Q/K → R with f ◦ p = F . The relation between these objects can easily be seen to be W (F ) = W (f ◦ p) = W̄ (f) ◦ p, which says that the function W̄ (f) is the reduced function on Q/K of the invariant function W (F ) on Q. We will use this for the case where Q = TM and K = TG, so that Q/K = TM̄ and p = Tπ. Given `, we may define a function LH on TM by LH(v) := `(Tπ(v)). This function is not a regular Lagrangian on M . But it is a TG-invariant function, which means that ẼCa ( LH ) = 0 = ẼVa ( LH ) . Its TG-reduced function is `. The vector fields XC i , XV i , Γ̄H are all TG-invariant (since [ Ẽ {C,V } a , XC i ] = 0 and similar for XV i and Γ̄H), and their TG-reduced vector fields on M̄ are ∂/∂xi, ∂/∂ẋi and Γ̄, respectively. From the relation above, we get that XC i ( LH ) = ∂` ∂xi ◦ Tπ, XV i ( LH ) = ∂` ∂ẋi ◦ Tπ, and therefore, if Γ̄ satisfies the Euler–Lagrange equation, we obtain that Γ̄H satisfies: Γ̄H ( XV i ( LH )) −XC i ( LH ) = 0. We also know that XV i ( LH ) is a G-invariant function (this follows from [ ẼCa , X V i ]( LH ) = 0), and therefore Xω ( XV i ( LH )) = 0. So, we may also write Γ1 ( XV i ( LH )) −XC i ( LH ) = 0, or, if X̄h = X̄iXi is an arbitrary ω-horizontal lift of a vector field X̄ = X̄i∂/∂xi on M̄ , Γ1 (( X̄h )V ( LH )) − ( X̄h )C( LH ) = 0, or, equivalently, Γ1 (( X̄V )H( LH )) − ( X̄C )H( LH ) = 0, where H now stands for the horizontal lift with respect to the Vilms connection ωC . The above equation completely determines the coefficients f i in Γ1 = vaẼCa +viXC i +f iXV i + 0ẼVa . We may thus conclude that the primary un-reduced SODE of a couple (`, ω) is the unique SODE that satisfies any of the above equivalent expressions, and also ωC(Γ1) = Xω (This last property ensures that the coefficient in ẼVa is zero. See also Proposition 4). Remark that if Γ1 satisfies this equation, then so does Γ2 = Γ1 + V , where V is any vector field of the type V aẼVa , since Γ2 ( XV i ( LH )) = Γ1 ( XV i ( LH )) + V aẼVa ( XV i ( LH )) = Γ1 ( XV i ( LH )) + V aXV i ( ẼVa ( LH )) = Γ1 ( XV i ( LH )) , because [ Ẽa, Xi ] = 0 and ẼVa ( LH ) = 0. In light of Proposition 6, we conclude: 14 E. Garćıa-Toraño Andrés and T. Mestdag Proposition 8. Let ` be a regular Lagrangian on M̄ and Γ̄ its corresponding SODE. Let ω be a principal connection on π : M → M̄ . Any vector field Γ2 on M for which (a) Γ2 − Γ1 is Tπ-vertical, (b) Ω(Γ2 − Γ1) = 0, (c) Γ2 satisfies Γ2 (( X̄h )V ( LH )) − ( X̄h )C( LH ) = 0, for any vector field X̄ on M̄ , has the property that its base integral curves project on solutions of the Euler–Lagrange equations of `. Remark that the statement does not require the use of a specific vertical equation, as in [5]. Each choice of Γ2 leads to a different vertical equation. When written in quasi-velocities, we get that Γ2 is of the form Γ2 = viXC i + vaẼCa + f iXV i + V aẼVa . As noted above, the first two terms indicate that Γ2 is a SODE, and the coefficients f i in the third one are completely determined by the Euler–Lagrange equations of `. The only freedom left is therefore the choice of the coefficients V a of the fourth term, which represents the choice of a vertical equation. In the next section we discuss an example where a specific choice for that freedom is naturally available. But the advantage of our approach is the same as the one that is claimed in [5]. All the conditions can be checked on the level of the manifold M , and we may do so in any coordinates on M , not necessarily those that are adapted to M → M̄ . 7 Curvature distortion Proposition 8 showed that there are many un-reductions Γ2 of the same Γ̄. The best choice for Γ2 may depend on the specific example one considers. In some situations, there is a natural choice. This natural choice may lead to the introduction of what was called “coupling distortion” and “curvature distortion” in [5]. We will concentrate here on the latter. To see its relation to the concepts we have introduced above, we may immediately restrict ourselves to the case of a quadratic Lagrangian, without much loss of generality. We consider again a principal fibre bundle M → M̄ = M/G, but now also a metric ḡ on M̄ and its corresponding geodesic spray Γ̄ḡ. The metric defines a quadratic Lagrangian ` on M̄ . The goal of this section is to devise a method to obtain geodesics of ḡ, without ever using coordinates on M̄ . Integral curves of Γ̄ḡ satisfy ẍi + Γ̄ijkẋ j ẋk = 0. The idea of [5] is to use a reasonable construction of a metric g on the un-reduced manifold M and its geodesic spray Γg. That spray, however, does not have the property that its geodesics project on those of ḡ. For that reason, we need to subtract some “distortion” terms from Γg to get a SODE Γ2 which does have that property. In what follows, we wish to obtain a relation between this Γ2 and our primary un-reduced SODE Γ1. The construction of the metric g goes as follows. Let B : g × g → R be an Ad-invariant symmetric and non-degenerate bilinear form on g (i.e., a bi-invariant metric on G). Consider again a principal connection on π. Together with ḡ we can form a (quadratic) Lagrangian on M , L = LH + LV , where LH(v) = `(Tπ(v)) = 1 2gijv ivj and LV (v) = 1 2B($(v), $(v)) = 1 2Babv avb. The coefficients Bab are constants that satisfy BacC d ac +BbdC d ac = 0. This construction of a regular Lagrangian L can also be found in [6], and our notations are chosen in such a way that they match with those in [5]. Un-Reduction of Systems of Second-Order Ordinary Differential Equations 15 Let us denote the (pseudo-)metric on M , associated to L, by g. By construction the mechan- ical connection of this metric is the connection we have started from, i.e., gai = 0. The equations for the geodesics of g are the so-called “Wong equations” (see again [6]). An easy way to obtain these equations goes as follows. The geodesic spray Γg is the Euler–Lagrange SODE of L. As such, it is the SODE determined by Γg ( XV (L) ) −XC(L) = 0, for any choice of vector field X on M . If we use X = Ẽa, then we know from the invariance of the Lagrangian that ẼCa (L) = 0. Therefore the corresponding Euler–Lagrange equation is Γg ( Babv b ) = 0, meaning that the momentum Babv b is conserved along geodesics. This is, in essence, the “ver- tical” Wong equation. The “horizontal” Wong equation is the Euler–Lagrange equation we get by making use of X = Xi, Γg ( XV i (L) ) −XC i (L) = 0. Consider now again the SODE Γ̄ḡ on M̄ and its many un-reduced SODEs Γ2, given in Proposition 8. By construction, all these vector fields satisfy Γ2 ( XV i ( LH )) −XC i ( LH ) = 0. Motivated by what we know about Γg, we now choose within that class, the particular SODE Γ2 which has the property that Γ2(Babv b) = 0. It is easy to see that this last property uniquely determines a specific Γ2 within the class of all SODEs that satisfy the conditions of Proposition 8, since it fixes the value of the coefficients V a in Γ2 = viXC i +vaẼCa +f iXV i +V aẼVa . In this case, given that Bab is a non-degenerate constant matrix, we get simply Γ2 ( vb ) = 0, which is exactly the defining relation of the primary un-reduced SODE Γ1 within the class of all Γ2’s, and thus V a = 0. Since Γg satisfies the same equation, it has the same coefficients V a = 0, when written in quasi-velocities. The difference A = Γg − Γ1 is therefore a vector field of the type A = AiXV i . Given that L = LH + LV , and that XV i ( LV ) = 0, its coefficients Ai are completely determined by the difference between the respective horizontal equations, i.e., by the relation A ( XV i ( LH )) −XC i ( LV ) = 0. If we derive the expressions of the curvature coefficients from [Xi, Xj ] = RaijẼa, then XC i (vb) = −Rbijvj , we get that XV i ( LH ) = Aj ḡij and XC i ( LV ) = −BabRaijvjvb and thus is Ak = −ḡikBabRaijvbvj . The term A in Γ1 = Γg −A 16 E. Garćıa-Toraño Andrés and T. Mestdag is what is called “curvature distortion” in [5]. It is the term one needs to subtract from Γg in order to get a SODE with the property that its base integral curves project on those of Γ̄ḡ. The (vertical) vector field A may, of course, also be expressed in a coordinate-free manner, but we will not go into these details here. As we saw, the effect of subtracting A to Γg is that it does not change the term of Γg in ẼVa , but that from the term in XV i it cancels out the curvature term in the right-hand side of the horizontal Wong equation. The point is that both Γg and A, and by the above construction also Γ1, can be computed in any coordinates on M . The above procedure is, in essence, the method that is applied in [5]. The fibration of the rotation group over the sphere. We will illustrate the discussion above by means of the realization of SO(3) as a S1-bundle over the sphere S2 ⊆ R3. More precisely, for the standard metric on the sphere, we will consider the geodesics of the metric determined by L = LH + LV and explicitly compute the “distortion” term. We identify each rotation about the origin of R3 with an orthogonal matrix. Such a rotation R ∈ SO(3) is determined by three consecutive counterclockwise rotations, defined by the Euler angles (ψ, θ, ϕ) (following the convention of [3]): R(ψ, θ, ϕ) = cosψ − sinψ 0 sinψ cosψ 0 0 0 1 1 0 0 0 cos θ − sin θ 0 sin θ cos θ cosϕ − sinϕ 0 sinϕ cosϕ 0 0 0 1  . = cosψ cosϕ− cos θ sinψ sinϕ − cosψ sinϕ− cos θ sinψ cosϕ sinψ sin θ sinψ cosϕ+ cos θ cosψ sinϕ − sinψ sinϕ+ cos θ cosψ cosϕ − cosψ sin θ sinϕ sin θ cosϕ sin θ cos θ . The group S1 acts on SO(3) on the left by rotations about the z-axis, namely as: ( α ∈ S1, R(ψ, θ, ϕ) ∈ SO(3) ) 7→ cosα − sinα 0 sinα cosα 0 0 0 1  ·R(ψ, θ, ϕ) ∈ SO(3). Note that two elements R1, R2 ∈ SO(3) are in the same orbit if their last row, which parameter- izes a sphere S2 ⊂ R3, coincides. This defines a principal S1-bundle π : SO(3)→ S2. In terms of the Euler angles the projection is π(ψ, θ, ϕ) = (θ, ϕ), and the infinitesimal generator (spanning the vertical distribution) is Ẽ = ∂/∂ψ. In what follows, we will use the principal connection on π : SO(3) → S2, given by the con- nection form $ = dψ + cos θdϕ. This is, in fact, the mechanical connection of the invariant Lagrangian on T (SO(3)), given by L̃ = 1 2 ( θ̇2 + ϕ̇2 sin2 θ ) + 1 2 ( ψ̇ + ϕ̇ cos θ )2 . (For the mechanical connection, we regard L̃ as a metric and define the horizontal space of the connection as the space that is orthogonal to the vertical space of π.) The Lagrangian L̃ corre- sponds to a Lagrange top with equal moments of inertial, see [3]. For this principal connection, we may give the following basis of horizontal vector fields on SO(3): X1 = ∂ ∂θ , X2 = ∂ ∂ϕ − cos θ ∂ ∂ψ . Quasi-velocities w.r.t. the frame { X1, X2, Ẽ } will be denoted {v1 = θ̇, v2 = ϕ̇, w = ψ̇+ ϕ̇ cos θ}. Consider now the standard metric ḡ = dθ2 +sin2 θdφ2 on S2. Its geodesic equations are given by the Euler–Lagrange equations of the Lagrangian ` = θ̇2 + sin2 φ̇2 on TS2, or by the geodesic Un-Reduction of Systems of Second-Order Ordinary Differential Equations 17 spray Γ̄ḡ = θ̇ ∂ ∂θ + ϕ̇ ∂ ∂ϕ + sin θ cos θϕ̇2 ∂ ∂θ̇ − 2 cot θθ̇ϕ̇ ∂ ∂ϕ̇ . Its primary un-reduced SODE is the vector field on T (SO(3)) given by Γ1 = Γω + Γ̄Hḡ = ψ̇ ∂ ∂ψ + θ̇ ∂ ∂θ + ϕ̇ ∂ ∂ϕ + sin θ cos θϕ̇2 ∂ ∂θ̇ − 2 cot θθ̇ϕ̇ ∂ ∂ϕ̇ + ( sin θθ̇ϕ̇+ 2 cot θ cos θθ̇ϕ̇ ) ∂ ∂ψ̇ . We now contrast this with the procedure of [5], where the system is un-reduced by subtracting curvature distortion from the Euler–Lagrange equations of a certain L = LH+LV . A bi-invariant metric on the Lie algebra of S1 is specified by a real number B. If we set LV = Bw2, we may write L = θ̇2 + sin2 θϕ̇2 +B ( ψ̇ + ϕ̇ cos θ )2 , which determines a metric g on SO(3). Its geodesic spray is Γg = ψ̇ ∂ ∂ψ + θ̇ ∂ ∂θ + ϕ̇ ∂ ∂ϕ + ( sin θ cos θϕ̇2 −B sin θϕ̇w ) ∂ ∂θ̇ + ( −2 cot θθ̇ϕ̇+ B sin θ θ̇w ) ∂ ∂ϕ̇ + ( sin θθ̇ϕ̇+ 2 cot θ cos θθ̇ϕ̇−B cot θθ̇w ) ∂ ∂ψ̇ , where w is short-hand for ψ̇+ ϕ̇ cos θ. Clearly, Γg is not yet one of the un-reduced SODEs of Γ̄ḡ, and curvature distortion is required. In the previous paragraph, we had concluded that Γ1 is the unique un-reduced SODE within the class of vector fields Γ2 which has the property that Γ1(Bw) = 0. From the expressions above, it follows that the curvature distortion is given by A = Γg − Γ1 = −B sin θφ̇w ∂ ∂θ̇ + Bθ̇w sin θ ( ∂ ∂ϕ̇ − cos θ ∂ ∂ψ̇ ) . This is clearly related to the coefficient of the curvature [Xθ, Xϕ] = sin θẼ, by means of the inverse of the metric matrix (ḡij) = ( 1 0 0 sin2 θ ) . To end this paragraph, we sketch an alternative approach, without invoking coordinates on S2. We refer the reader to [3, 10] for details omitted here. The Lie algebra so(3) of SO(3) is given by the set of skew symmetric matrices. It has the following basis e1 = 0 0 0 0 0 −1 0 1 0  , e2 =  0 0 1 0 0 0 −1 0 0  , e3 = 0 −1 0 1 0 0 0 0 0  . We will identify so(3) with the Lie algebra R3 (given by the cross product “×”): R3 3 ω = ω1 ω2 ω3  7→ [ω] =  0 −ω3 ω2 ω3 0 −ω1 −ω2 ω1 0  . The left and right invariant vector fields corresponding to the basis vectors will be denoted by (ei)` and (ei)r, and we will use (ei)` and (ei)r for the dual basis. Then, on so(3), one may 18 E. Garćıa-Toraño Andrés and T. Mestdag consider the invariant inner product 〈A,B〉so(3) = trace ( AB−1 ) and its associated metric on SO(3), for which {(ei)`} and {(ei)r} are orthonormal moving frames. In terms of the Euler angles, this metric is the one given by the Lagrangian L̃ we had mentioned before. Suppose now that γ ∈ S2. If π(R) = γ, then we have γ̇ = Tπ(Ṙ) = −[ω]γ = γ × ω. The connection form is $ = (e3)r and its curvature 2-form is the area element on S2. Finally, the horizontal lift of a vector γ̇ to R is the element Ṙ for which R−1Ṙ = [γ̇ × γ], see [10]. One might also identify a rotation with an an oriented orthonormal frame (v, w, v × w) in R3 (the rotation needed to move the standard R3 basis onto the new frame). With this in mind, we pick ê1, ê2 ∈ R3 such that {ê1, ê2, γ} is orthonormal. The horizontal lift vh of a vector v = (u1ê1 + u2ê2) to an element R ∈ SO(3) is the tangent vector Ṙ such that R−1Ṙ = [(u1ê1 + u2ê2)× γ] = [u2ê1 − u1ê2]. In other words, vh = u2(ê1)r − u1(ê2)r. In this expression it is understood that (ê1)r, (ê2)r are evaluated at the point R to which we are lifting. Therefore, one has an explicit construction of horizontal vectors in SO(3), and one can compute L = LH + LV without further difficulty. 8 Extending the un-reduction method and further outlook So far we have considered SODEs Γ2 which have the properties that all their base integral curves project on those of Γ̄. We now indicate that there may also exist SODEs Γ3 which have the property that only some of their base integral curves project on those of Γ̄. We consider a principal connection ω given. We will denote its horizontal distribution by H ⊂ TM . Integral curves of Γ̄ satisfy ẍi = f i(x, ẋ). Consider again the primary un-reduced SODE Γ1 = viXC i + f i(xj , vj)XV i + vaẼCa + 0ẼVa . Its integral curves satisfy ẍi = f i(x, ẋ) and v̇a = 0, together with some reconstruction equation that is primarily associated to the part Xω = vaẼCa of Γ1. Recall that we have shown in Proposition 5 that integral curves of Γ1 with a horizontal initial velocity are horizontal lifts of base integral curves of Γ̄ and that Γ1 is tangent to H. Consider now a SODE of the type Γ3 = viXC i +F i(xj , vj , va)XV i + vaẼCa +F a(xj , vj , va)ẼVa , but with the property that F i(xj , vj , va = 0) = f i(xj , vj) and F a(xj , vj , va = 0) = 0, i.e., Γ3\|H = Γ1\|H. For example, besides the property on F i, one could simply have that F a = 0. (This property, that there is no component along ẼVa , can be characterized by saying that the SODE Γ3 satisfies Ω(Γ3) = ωC(Γ3).) Integral curves of such a Γ3 through m0 with initial velocity vi0Xi(m0) + va0Ẽa(m0) satisfy, for sure, va(t) = va0 . As long as we consider integral curves of Γ3 with a horizontal initial velocity, i.e., with va0 = 0, they will satisfy va(t) = 0, and (xi(t), vi(t)) will be solutions of ẍi = F i(xj , vj , 0) = f i(xj , vj). With other words, some integral curves of Γ3, namely the horizontal ones, will project on those of Γ̄. This situation actually also occurs in the example with Wong’s equations. The quadratic spray Γg is not one of the un-reduced SODEs of Γ̄ḡ, but it does have the property that Γg\|H = Γ1\|H, since A\|{va=0} = 0. We can now relate this behaviour to what is called “horizontal shooting” in [4, 7]. We have shown that the geodesic spray Γg satisfies Γg(Bavv b) = 0. This means that its integral curves (i.e., the lifted curves in TM of the base integral curves), whose fibre coordinates when written in quasi-velocities are (vi(t), va(t)), satisfy a conservation law (of momentum-type) Bavv b(t) = µa. Here µa are the components of an element µ ∈ g∗ along the basis Ea. The specific base integral curves which happen to have “zero momentum”, µa = 0, are exactly those whose lifted curves remain horizontal for all t, va(t) = 0. They, therefore, coincide with the horizontal Un-Reduction of Systems of Second-Order Ordinary Differential Equations 19 lifts of base integral curves. Given that we know that Γg\|H = Γ1\|H, it follows that Proposition 5 represents a generalization of the method of “horizontal shooting” for geodesic problems. One may find in the literature some results on what could be called “un-un-reduction”. The inverse procedure of un-reduction should of course be a kind of reduction process. It is, however, not Lagrange–Poincaré reduction, since the corresponding reduced Lagrange–Poincaré equations can not necessarily be associated to a SODE on M̄ . But, for an arbitrary G-invariant SODE Γ on TM , it makes sense to wonder whether it is Tπ-related to a (yet to be determined) SODE Γ̄ on M̄ . An answer to this question is given in the papers on submersive SODEs. A SODE Γ on M is said to be “submersive” in [12, 16] if there exists a projection π : M → N for which it is projectable to a SODE Γ̄ on N . In the theory of [12, 16] the projection π is part of the unknowns, but if we assume that π is a given principal fibre bundle π : M → M̄ from the outset, being submersive means that Γ is an un-reduction of an (unknown) SODE Γ̄ in M̄ . The next proposition can be found as Theorem 3.1 in [12]. Proposition 9. Let M → M̄ be a principal bundle and let Γ be a SODE on M . Under the following two conditions Γ is submersive: (1) [ Γ, ξ̃C ] = 0 (with other words, it is G-invariant), (2) [ ξ̃V ,Γ ] − ξ̃C is tangent to gV . Proof. If {Ea} is a basis for g, then “being tangent to gV ” means that the expression in (2) is of the type AaẼVa . If we write, as before, Γ in the form Γ = viXC i + vaẼCa + F iXV i + F aẼVa with F i, F a functions on TM , then condition (2) implies that ẼVb (F i) = 0, while from condi- tion (1) we get that ẼCb (F i) = 0 (and also that ẼCb (F a) = CabcF c). Therefore, the function F i is TG-invariant, and it defines a function f i on TM̄ . The SODE on M̄ given by Γ̄ = ẋi∂/∂xi + f i∂/∂ẋi is the one for which Γ is an un-reduction. � With the results of [12, 16] in mind, it would be of interest to be able to “un-reduce” a SODE Γ̄ on a manifold M̄ to any bundle π for which M̄ happens to be the base (given a connection on that bundle), not just a principal bundle M → M̄ = M/G. Results which may point in that direction are Theorem 1.5 in [12], or Theorem 2.1 in [16], which give conditions for a SODE Γ on M to be submersive. Moreover, some of the concepts that we have mentioned in this paper transfer to this more general context. For example, one may still define a Vilms connection on the bundle TM → TM̄ , corresponding to a connection on an arbitrary fibre bundle M → M̄ . However, constructing something that is similar to the vertical connection Ω (that has lead us to the definition of Xω and the primary un-reduced SODE Γ1) seems to be a more challenging task. Acknowledgements EGTA thanks the CONICET for financial support through a Postdoctoral Grant. TM is a vi- siting professor at Ghent University: he is grateful to the Department of Mathematics for its hospitality. References [1] Arnaudon A., Castrillón López M., Holm D.D., Covariant un-reduction for curve matching, arXiv:1508.05325. [2] Arnaudon A., Castrillón López M., Holm D.D., Un-reduction in field theory, with applications, arXiv:1509.06919. http://arxiv.org/abs/1508.05325 http://arxiv.org/abs/1509.06919 20 E. Garćıa-Toraño Andrés and T. 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[19] Yano K., Ishihara S., Tangent and cotangent bundles: differential geometry, Pure and Applied Mathematics, Vol. 16, Marcel Dekker, Inc., New York, 1973. http://dx.doi.org/10.1007/978-1-4757-2063-1 http://dx.doi.org/10.1007/s10851-013-0490-z http://arxiv.org/abs/1305.1150 http://dx.doi.org/10.3934/jgm.2011.3.363 http://arxiv.org/abs/1012.0076 http://dx.doi.org/10.1090/memo/0722 http://dx.doi.org/10.3934/jgm.2010.2.51 http://arxiv.org/abs/0911.2205 http://dx.doi.org/10.1063/1.2885077 http://arxiv.org/abs/0802.0528 http://dx.doi.org/10.1088/0305-4470/38/24/R01 http://dx.doi.org/10.1088/0305-4470/38/24/R01 http://arxiv.org/abs/math.DG/0407528 http://dx.doi.org/10.1007/0-8176-4419-9_4 http://arxiv.org/abs/math-ph/0408005 http://dx.doi.org/10.1007/978-3-662-02950-3 http://dx.doi.org/10.1017/S0305004100070262 http://dx.doi.org/10.1017/S0305004100070262 http://dx.doi.org/10.1088/1751-8113/41/34/344015 http://arxiv.org/abs/0802.0146 http://dx.doi.org/10.2307/2372398 http://dx.doi.org/10.3934/jgm.2009.1.209 http://dx.doi.org/10.1016/S0926-2245(02)00161-4 1 Introduction 2 Preliminaries 3 Two lifted connections 4 Un-reducing second-order systems 5 A non-Lagrangian example 6 Un-reducing Lagrangian systems 7 Curvature distortion 8 Extending the un-reduction method and further outlook References

Un-Reduction of Systems of Second-Order Ordinary Differential Equations

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