Extended Hamilton-Jacobi Theory, symmetries and integrability by quadratures

In this paper, we study the extended Hamilton-Jacobi Theory in the context of dynamical systems with symmetries. Given an action of a Lie group $G$ on a manifold $M$ and a $G$-invariant vector field $X$ on $M$, we construct complete solutions of the Hamilton-Jacobi equation (HJE) related to $X$ (and a given fibration on $M$). We do that along each open subset $U\subseteq M$ such that $\pi\left(U\right)$ has a manifold structure and $\pi_{\left|U\right.}:U\rightarrow\pi\left(U\right)$, the restriction to $U$ of the canonical projection $\pi:M\rightarrow M/G$, is a surjective submersion. If $X_{\left|U\right.}$ is not vertical with respect to $\pi_{\left|U\right.}$, we show that such complete solutions solve the"reconstruction equations"related to $X_{\left|U\right.}$ and $G$, i.e., the equations that enable us to write the integral curves of $X_{\left|U\right.}$ in terms of those of its projection on $\pi\left(U\right)$. On the other hand, if $X_{\left|U\right.}$ is vertical, we show that such complete solutions can be used to construct (around some points of $U$) the integral curves of $X_{\left|U\right.}$ up to quadratures. To do that we give, for some elements $\xi$ of the Lie algebra $\mathfrak{g}$ of $G$, an explicit expression up to quadratures of the exponential curve $\exp\left(\xi\,t\right)$, different to that appearing in the literature for matrix Lie groups. In the case of compact and of semisimple Lie groups, we show that such expression of $\exp\left(\xi\,t\right)$ is valid for all $\xi$ inside an open dense subset of $\mathfrak{g}$.


Introduction
In the last few years, several generalizations of the classical Hamilton-Jacobi equation (HJE) have been developed for Hamiltonian systems on different contexts: on symplectic, cosymplectic, contact, Poisson and almost-Poisson manifolds, and also on Lie algebroids. The resulting Hamilton-Jacobi theories were applied to nonholonomic systems, dissipative and time-dependent Hamiltonian systems, reduced systems by symmetries and Hamiltonian systems with external forces [5,7,8,9,10,11,12,15]. In all of them, the following ingredients are present: (1) a fibration Π:M → N (i.e. a surjective submersion) defined on the phase space M of each system; (2) the solutions of the generalized HJE, which we shall call Π-HJE, given by sections σ :N → M of such a fibration Π; (3) the complete solutions Σ:N × Λ → M , given by local diffeomorphisms such that, for each λ ∈ Λ, σ λ := Σ(·, λ) is a solution of the Π-HJE. This clearly generalizes the classical situation [1,16], where the involved fibration is the cotangent projection π Q : T * Q → Q of a manifold Q and the solutions σ :Q → T * Q are exact 1-forms on Q.
In Ref. [18], an extension to general (i.e. not necessarily Hamiltonian) dynamical systems, of the previously mentioned Hamilton-Jacobi theories, was carried out, focusing on the connection between complete solutions and the integrability by quadratures of the involved systems.
The main aim of the present paper is to further study such an extended theory in the context of dynamical systems with symmetry. Concretely, given a general action ρ : G × M → M (not necessarily free or proper) of a Lie group G on a manifold M and a G-invariant vector field X on M (with respect to ρ), we investigate how to use ρ to construct (local) fibrations Π of M and related solutions of the Π-HJE for X. We first show that, around almost every point of M (depending on the isotropy subgroups of G), there exists a neighborhood U such that the canonical projection π : M → M/G restricted to U , namely π |U : U → π (U ), defines a fibration (even though ρ is neither free nor proper). Then we consider two kinds of vector fields: (a) those for which X |U is not vertical with respect to π |U , which we call horizontal, and (b) the vertical ones. For the horizontal vector fields we show that, related to the action ρ, there exists a submersion Θ transverse to π |U (which plays the role of a flat principal connection) such that Σ := π |U , Θ −1 : π (U ) × Λ → U, (1.1) with Λ a submanifold of G, is a complete solution of the π |U -HJE for X |U . Such a Σ can be seen as a solution of a reconstruction problem, in the sense that, if we know the integral curves γ (t) of the projected vector field Y of X |U on π (U ), then the integral curves of X |U are given by Γ (t) = Σ (γ (t) , λ), with λ ∈ Λ. For the vertical vector fields, we show that we can construct up to quadratures a submersion Θ transverse to π |U such that Σ := Θ, π |U −1 : N × π (U ) → U, (1.2) with N a submanifold of G, is a complete solution of the Θ-HJE for X |U . Moreover, we prove that the integral curves of X |U also can be constructed up to quadratures around some points of M . To do that, we first show that the exponential curves t → exp (ξ t) of G, for some elements ξ of its Lie algebra g, can be constructed up to quadratures. As it is well-known, there exists several explicit expressions of exp (ξ t), unless for matrix Lie groups. What we are giving here is an alternative expression for such curves, valid also for non-matrix Lie groups. In the case in which G is semisimple or compact, we show that such an expression is valid for all ξ in an open dense subset of g.
The paper is organized as follows. In Section 2, we make a brief review of the extended Hamilton-Jacobi Theory appearing in [18,19]. We also present a result that ensures, in the presence of a complete solution and in the context of symplectic manifolds, the integrability by quadratures of general vector fields. It extends a result proved in [18] for Hamiltonian vector fields only. In Section 3, given a dynamical system with symmetry, we construct the complete solutions (1.1) and (1.2) for horizontal and vertical vector fields, respectively. In Section 4, we show the intimate relationship that there exists between the complete solutions of a horizontal (and invariant) vector field and the so-called reconstruction processes. Finally, in Section 5, using the above mentioned result of Section 2, we show that the exponential curves t → exp (ξ t) of G, for some points ξ ∈ g, can be constructed up to quadratures. Then, based on that, we state sufficient conditions under which a vertical (and invariant) vector field can also be integrable up to quadratures.
We assume that the reader is familiar with the main concepts of Differential Geometry (see [6,21,29]) and with the basic ideas related to Hamiltonian systems, Symplectic Geometry and Poisson Geometry in the context of Geometric Mechanics (see for instance [1,3,24,27]). We shall work in the smooth (i.e. C ∞ ) category, focusing exclusively on finite-dimensional smooth manifolds. Regarding the terminology associated to the concept of "integrability by quadratures," we shall adopt the following convention. We shall say that "a function F : P → Q can be constructed up to quadratures," or simply "can be constructed," if its domain P and its values F (p) (for all p ∈ P ): • are simply known; • they can be determined by making a finite number of arithmetic operations (as the calculation of a determinant) and/or solving a finite set of linear equations (which actually can be reduced to arithmetic operations); • or they can be expressed in terms of the derivatives, primitives (i.e. quadratures) and/or lateral inverses (by using the Implicit or Inverse Function Theorem) of another known functions.
When the function F above is an integral curve Γ of a vector field and such a curve can be constructed up to quadratures, we shall say that Γ can be integrated up to quadratures, or by quadratures.
2 Preliminaries: complete solutions, first integrals and integrability

The extended Hamilton-Jacobi equation
Consider a manifold M , a vector field X ∈ X (M ) and a surjective submersion Π : M → N (ipso facto an open map). Related to this data (see [18]) we have the Π-Hamilton-Jacobi equation (Π-HJE) for X: whose unknown is a section σ : N → M of Π (ipso facto a closed map). If σ solves the equation above, we shall say that σ is a (global) solution of the Π-HJE for X. On the other hand, given an open subset U ⊆ M , we shall call local solution of the Π-HJE for X along U to any solution of the Π |U -HJE for X |U . (Here, we are seeing Π |U as a submersion onto Π (U ) and X |U as an element of X (U )). Note that σ is a solution of the Π-HJE for X if and only if i.e. the vector fields X ∈ X (M ) and X σ ∈ X (N ) are σ-related. (Moreover, it can be shown that σ is a solution of (2.1) if and only if its image is an X-invariant closed submanifold). This means that, in order to find the trajectories of X along the image of σ, we can look for the integral curves of X σ . Given another manifold Λ such that dim Λ + dim N = dim M , a complete solution of the Π-HJE for X is a surjective local diffeomorphism Σ : N × Λ → M such that, for all λ ∈ Λ, is a solution of the Π-HJE for X. The local version is obtained by replacing M , X, Π and N by U , X |U , Π |U and Π (U ), respectively, being U an open subset of M . Each section σ λ is called a partial solution. We showed in [18] that a (local) complete solution Σ exists around every point m ∈ M for which X (m) / ∈ KerΠ * ,m . Denoting by p N and p Λ the projections of N × Λ onto N and Λ, respectively, it is easy to prove that a surjective local diffeomorphism Σ is a complete solution if and only if being X Σ ∈ X (N × Λ) the unique vector field on N × Λ satisfying Note that X Σ (p, λ) = (X σ λ (p) , 0), with X σ λ := Π * • X • σ λ ∈ X (N ), so, in particular, Also, the fields X and X Σ are Σ-related. This implies that all the trajectories of X can be obtained from those of X Σ . More precisely, since each integral curve of X Σ is clearly of the form t → (γ (t) , λ) ∈ N × Λ, for some fixed point λ ∈ Λ (see (2.7)), those of X are given by So, for each λ, we only need to find the curves γ, which are the integral curves of the vector field X σ λ ∈ X (N ).

The "complete solutions -first integrals" duality
Consider again a manifold M , a vector field X ∈ X (M ) and a surjective submersion Π : M → N . We shall say that a submersion F : M → Λ is a first integrals submersion if Remark 2.1. Note that, if Λ = R l , the components f 1 , ..., f l : M → R of F define a set of l (functionally) independent first integrals, in the usual sense.
Also, we shall say that F is transverse to Π if It was shown in [18] that, given a complete solution Σ : N × Λ → M of the Π-HJE for X, we can construct around every point of M a neighborhood U and a submersion F : U → Λ such that • ImX |U ⊆ KerF * (first integrals), In other words, from Σ we have, around every point of M , a first integrals submersion transverse to Π. The subset U and the function F are given by the formulae where V ⊆ N × Λ is an open subset for which Σ |V is a diffeomorphism onto its image.
Reciprocally (see also [18]), from a submersion F : M → Λ satisfying (2.9) and (2.10), we can construct, around every point of M , a neighborhood U and a local complete solution Σ of the Π-HJE. The involved subset U is one for which (Π, F ) |U is a diffeomorphism onto its image, and Σ is given by Summarizing, a complete solution gives rise to local first integrals via (2.11), and first integrals give rise to a local complete solution via (2.12).

Integrability by quadratures on symplectic manifolds
Let (M, ω) be a symplectic manifold. Given a distribution V ⊆ T M (resp. m ∈ M and a linear subspace V ⊆ T m M ), by V ⊥ we shall denote, as usual, the symplectic orthogonal of V w.r.t. ω. The following result is a slightly extension to general dynamical systems of a result given in [18] (only valid for Hamiltonian systems).
Theorem 2.2. Let F : M → Λ be a surjective submersion and X ∈ X (M ) a vector field such that: III. and L X β = 0, with β := i X ω; 1 then the trajectories of X can be integrated up to quadratures.
Proof. We shall proceed in four steps.
a. Given a point m ∈ M , consider an open neighborhood U of m and a surjective submersion Π : U → Π (U ) transverse to F |U : U → F (U ). (As it is well-known, such Π can be constructed just by fixing a coordinate chart and solving linear equations). Using the point (I) above and the results of the last section, it is clear (shrinking U if necessary) that Σ := Π, F |U −1 (see (2.12)), which can be constructed by using the Inverse Function Theorem, is a local complete solution of the Π-HJE for X. According to Theorem 3.12 of [18] (replacing there dH by β), this implies that (recall (2.6)) omitting the restrictions to U of ω and β. b. Using (2.5) and the fact that Σ is a diffeomorphism, the point (III) is equivalent to For each n ∈ Π (U ), let us define β n ∈ Ω 1 (F (U )) such that Then, along an integral curve (γ (t) , λ) of X Σ , it can be shown from (2.14) that and, consequently, Let us prove it. From (I) and (II) we have that ). On the other hand, given a vector field z ∈ X (F (U )), for Z = (0, z) it is easy to see that X Σ , Z is of the form (y, 0). Then, from that and (2.14), Hence, (2.16) follows by combining (2.15) and the last equation. c. Using that ω is closed, we can assume (without loss of generality) that ω = −dθ, with θ ∈ Ω 1 (U ).
In particular, since X Σ ∈ KerF * , it follows that Therefore, we deduce that Then, combining (2.13) and (2.17), As a consequence, in terms of the functions ϕ λ : Π (U ) → T * λ F (U ), given by the Equation (2.18) along an integral curve (γ (t) , λ) of X Σ translates to (using similar calculations as in the previous step) or equivalently, ϕ λ (γ (t)) = ϕ λ (γ (0)) + t β γ(0) (λ) . (2.20) d. Finally, since each ϕ λ is an immersion (see Proposition 3.16, [18]), from the above equation we can construct the curves γ (by using the Implicit Function Theorem), from which all the integral curves of X |U can be obtained. In fact, the latter are given by the formula Γ (t) = Σ (γ (t) , λ), as explained at the end of Section 2.1 (see (2.8)). Since all that can be done around every m ∈ M , then all the integral curves of X can be constructed in the same way.
Given a surjective submersion G : M → Υ and a 1-form φ ∈ Ω 1 (Υ), the vector field satisfies the point (I) above, for another submersion F : M → Λ, if and only if

If in addition
then i X • F * = i X • G * = 0 and consequently So, given a symplectic manifold (M, ω), examples of dynamical systems satisfying the points (I)-(III) of Theorem 2.2 are given by submersions F and G satisfying (2.23), being F isotropic, 1-forms φ satisfying (2.22) and vector fields given by (2.21). These examples can be seen as a generalization of the non-commutative integrable systems, as we show below, and they will appear in the last section of the paper.

Non-commutative integrability and Casimir 1-forms
A Mishchenko-Fomenko or non-commutative integrable (NCI) system [28] (see also [20] and references therein) can be defined as a triple given by a symplectic manifold (M, ω), a Hamiltonian vector field X H = ω ♯ •dH and a surjective submersion F : M → Λ such that: When KerF * = (KerF * ) ⊥ , i.e. F is Lagrangian, the third point is automatic. In such a case, we have a so-called Liouville-Arnold or commutative integrable (CI) system [3,26]. It is well-known that all these systems are integrable by quadratures. The traditional way of proving that relays on the Lie theorem on integrability by quadratures [4,25] (see also [17]).
Usually, in the definition of NCI and CI systems, one more requirement is included: F has compact and connected leaves. In such a case, beside integrability by quadratures, action-angle coordinates can be found for such systems (see [14] and [22]). We do not consider this case here.
3. An alternative definition can be given in terms of subsets of functions F ⊆ C ∞ (M ). The conditions 3 and 2 above are replaced by asking F to be a Poisson sub-algebra and complete (see [20]), respectively, and 1 is replaced by asking that the elements of F Poisson commute with H.
To analyze the relationship between NCI systems and the systems given at the end of the last section, let us consider an arbitrary surjective submersion F : M → Λ. On the one hand, it can be shown that a Hamiltonian vector field X H belongs to ( and F is a Poisson morphism. In such a case, the condition ImX H ⊆ KerF * , for X H given by (2.25), is equivalent to (compare to (2.22)) which says precisely that dh is a Casimir 1-form for Ξ (and h is a Casimir function). In the case when F is Lagrangian, then Ξ = 0, and consequently every 1-form on Λ is a Casimir. Thus, the NCI systems are a subclass of the systems given at the end of the last section, where Λ is a Poisson manifold, G = F : M → Λ is a Poisson morphism and φ = dh is an exact Casimir 1-form with respect to the Poisson structure on Λ.

Complete solutions and symmetries
Given a general action ρ : G × M → M (not necessarily free or proper) of a Lie group G on a manifold M and a G-invariant vector field X on M (with respect to ρ), we shall construct in this section, based on ρ and the canonical projection π : M → M/G, local fibrations Π of M and related complete solutions of the Π-HJE for X. Let us begin with the local fibrations Π based on π.

General actions and regular points
Let ρ : G × M → M be an action of a Lie group G on M . Let us introduce some basic notation and recall some well-known facts. As usual, given g ∈ G and m ∈ M , by ρ g and ρ m we shall denote the maps ρ g : M → M and ρ m : G → M such that ρ g (m) = ρ m (g) = ρ (g, m). Also, we shall denote by g the Lie algebra of G and by G m the isotropy subgroup of m.
For latter convenience, let us note that where e ∈ G is the identity element and g m is the Lie algebra of G m . And recall that the fundamental vector field associated to η ∈ g is given by Let π : M → M / G be the canonical projection and consider on M / G the quotient topology. Recall that, since each ρ g : M → M is a homeomorphism for all g ∈ G, then π is open (besides continuous). Recall also the identities When ρ is free (i.e. if G m = {e} for all m ∈ M ) and proper, then, as it is well-known (see [1]), M / G has a unique manifold structure such that π : M → M / G is a surjective submersion. For more general actions we shall show a similar result, but at a local level around a regular point.
We shall call such neighborhood U admissible if in addition U is connected. The (open) subset of all the ρ-regular points will be denoted R ρ .
defines a vector subbundle of the trivial vector bundle pr 1 : U × g → U for each admissible neighborhood U .
To show it, note that given m, m ′ ∈ M such that m ′ = ρ (g, m) for some g, we have that g · G m · g −1 = G m ′ , and consequently Then, given any admissible neighborhood V of m 0 , it is clear that includes m 0 , is open, G-invariant and admissible. As a consequence, the set R ρ is G-invariant.
If the action ρ is free, then every element of M is ρ-regular and M (if connected) is an admissible neighborhood. For G = SO (3) acting on M = R 3 with the natural action ρ nat , we have that dim G m = 1 for m ∈ R 3 − {0} and dim G 0 = dim G = 3. Thus, all the points of R 3 except 0 are ρ nat -regular. In general, we have the following result.
Proof. We already saw that R ρ is G-invariant. We shall prove that: 1. if k is the minimum dimension of the isotropy subgroups and dim G m0 = k, then m 0 is a ρ-regular point; 2. the complement of R ρ has empty interior.
For the first point, define and take m 0 such that dim G m0 = k. Consider the Lie algebra g m0 of G m0 and a linear complement g c m0 of it.
whereρ is the action of g on M induced by ρ. Then, by continuity, there exists a neighborhood U of m 0 such thatρ (v, m) = 0, ∀m ∈ U.
This means that dim g m ≤ dim g m0 = k for all m ∈ U . But k is the minimum dimension, hence dim g m = k for all m ∈ U . This says precisely that m 0 is a ρ-regular point.
For the second point, suppose that the complement R c ρ has interior, i.e., for some m 1 ∈ R c ρ there exists an open subset U 1 such that m 1 ∈ U 1 ⊆ R c ρ . Consider the Lie algebra g m1 of G m1 and a linear complement g c m1 of it. Proceeding as above, we can ensure that there exists a neighborhood Otherwise, m 1 would be a ρ-regular point (with admissible neighborhood U 2 ). Repeating this reasoning for m 2 , we can ensure the existence of a neighborhood U 3 ⊆ U 1 of m 2 for which dim g m ≤ dim g m2 for all m ∈ U 3 , and consequently the existence of a point m 3 ∈ U 3 such that dim g m3 < dim g m2 . In this way, since the dimension of g is finite, in some step of this procedure we shall find m 0 ∈ U 1 ⊆ R c ρ such that dim g m0 = k. Since such m 0 must belong to R ρ , we have arrived at a contradiction.

The submersions π |U
Now, let us construct smooth local versions of the canonical projection π.
Proposition 3.4. Given m 0 ∈ R ρ , there exists a neighborhood U for m 0 such that the open subset π (U ) has a manifold structure and the restriction π |U : U → π (U ) is a submersion satisfying Ker π |U * ,m = Im (ρ m ) * ,e , ∀m ∈ U, (3.6) and, consequently, Moreover, U can be taken G-invariant.
Proof. Let U 1 be an admissible neighborhood of m 0 and consider the distribution given by Since F 1 is clearly generated by the fundamental vector fields η M (see Eq. (3.2)), with η ∈ g, then F 1 is involutive (see for instance [1]). And from the same reason, which is constant and equal to r 1 := dim G − dim G m0 for all m ∈ U 1 (because of (3.4)). Then, defining r := dim M and using the Frobenius Theorem, we can find a local chart 2 open subsets in R r1 and R r−r1 , respectively, and It is clear that m 0 ∈ U 2 ⊆ U and, moreover, U/G ∼ = V ′ 2 and the canonical projection π |U : From now on, by admissible we shall mean any admissible neighborhood U of m 0 for which the last proposition holds.
The following result will be useful later.
Proposition 3.5. Given m 0 ∈ R ρ and an admissible neighborhood U of m 0 , the subset , and the surjective map is smooth and also a submersion.
Proof. Consider the admissible G-invariant open subsetŨ = g∈G ρ g (U ) (see Eq. (3.5)), the related subset RŨ and the related surjective map ΦŨ : G ×Ũ → RŨ (given as in (3.8) and (3.9)). If we prove the proposition forŨ , since R U = RŨ ∩ (U × M ) and Φ U = ΦŨ | G×U , then we would proved it for U . Using that the space of orbits π(Ũ ) =Ũ /G is a quotient smooth manifold and a classical result (see, for instance, [23]), we deduce that RŨ ⊆Ũ ×Ũ is a closed submanifold ofŨ ×Ũ (and also ofŨ × M ). As a consequence, since (g, m) ∈ G ×Ũ −→ (m, ρ g (m)) ∈Ũ ×Ũ is smooth, the same is true for the surjection ΦŨ : G ×Ũ → RŨ . To find the dimension of RŨ and show that ΦŨ is a submersion, it is enough to calculate the rank of ΦŨ and show that is constant (since ΦŨ is surjective). Let us do that. From (3.9) and the identity ρ m • L g = ρ g • ρ m , it follows that and in particular, for g = e, Then, from (3.10) and (3.11) we have that Consequently, for all (g, m) ∈ G ×Ũ , which ends our proof.

Symplectic actions and momentum maps
Suppose that M is a symplectic manifold, with symplectic form ω, and ρ is a symplectic action, i.e. (3.12) Proposition 3.6. Under the above conditions, for each admissible neighborhood U we have that: 1. The manifold π (U ) has a Poisson structure Ξ U , characterized by the condition with respect to which π |U is a Poisson morphism.
2. Let X be a G-invariant vector field, i.e. (3.14) Then there exists a unique vector field Y ∈ X (π (U )) such that Proof.
(1) This result is proved in [27] under the hypothesis that U is G-invariant and that the G-action on U is free and proper. But, in that proof the key point is that the space of orbits π (U ) is a quotient manifold, as in our case.
(2) It is also a well-known result (see, for instance [31]) that if U is a principal G-bundle over U/G, then every G-invariant vector field over U is projectable over U/G. But, as in (1), the key point in order to prove this fact is that U/G is a quotient manifold. So, proceeding in a similar way as in [31], we deduce (2).
Suppose that ρ has a (global) momentum map, i.e. a function K : M → g * such that Proposition 3.7. For each admissible neighborhood U , For a proof of this result see, for instance, [1].
Suppose in addition that K can be chosen Ad * -equivariant, i.e.
Here, as usual, Ad : G × g → g : (g, η) → Ad g η denotes the adjoint action and the co-adjoint one.
Proof. Let m 1 ∈ R ρ be such that K (m 1 ) ∈ R Ad * ⊆ g * , and let U 1 be an admissible neighborhood of m 1 . Given a G-invariant admissible neighborhood V ⊆ R Ad * of K (m 1 ) (with respect to the co-adjoint action), define because of the G-invariance of V , and consequently (see (3.17)) This completes our proof.
The previous result will be useful at the end of the paper.

The horizontal submersions
In this subsection, for each admissible neighborhood U , we shall construct submersions Θ transverse to the restricted canonical projection π |U . In terms of such submersions Θ, we shall present at the end of the section the complete solutions we are looking for.

Trivializations and (local) flat connections for principal bundles
Suppose that ρ : G× M → M is free and proper and consider the associated principal G-bundle π : given by Ψ (ρ (g, s (λ))) = (λ, g), or equivalently (Ψ is well-defined and invertible because ρ is free). Note that the map ψ : U → G satisfies Also, ψ (ρ (g, m)) = ψ (ρ (g, ρ (ψ (m) , s (π (m))))) = ψ (ρ (g ψ (m) , s (π (m)))) = g ψ (m) , and consequently On the other hand, it is easy to show that the map A : T U → g given by is a local principal connection for π. In fact, for all m ∈ U , it follows from (3.20) and (3.21) that and In addition, since KerA = Kerψ * , the horizontal distribution is integrable, i.e. A is a flat connection. In the next section, we shall construct an object similar to A, but related to an arbitrary action and its regular points.

A flat-connection-like object for π |U
Now, suppose that ρ is a general Lie group action. For each ρ-regular point m 0 , we shall construct a family of submersions transverse to π |U (being U an admissible neighborhood of m 0 ). To do that, we need the next results.
Lemma 3.9. Let G : P → Q be a submersion, p 0 ∈ P and W ⊆ T p0 P a linear complement of KerG * ,p0 . Then, there exists a neighborhood V of G (p) ∈ Q and a local section S : V → P of G such that S (G (p 0 )) = p 0 and ImS * ,G(p0) = W.
Proof. Let ϕ = (x 1 , ..., x n ) : U → ϕ (U ) be a coordinate system of P around p 0 . Consider the annihilator W 0 ⊆ T * p0 P of W and suppose that the co-vectors give a basis for W 0 . Define F : U → R k as It is clear that KerF * ,p0 = W. Then, since KerG * ,p0 and W are complementary, (G, F ) is a diffeomorphism onto its image G (U ) × F (U ), shrinking U if needed. As a consequence, the function S : G (U ) → P such that is a smooth local section of G and satisfies S (G (p 0 )) = p 0 . Also, given w ∈ W, S * ,G(p0) (G * ,p0 (w)) = w. In particular, since G * ,p0 is surjective, even restricted to W, then ImS * ,G(p0) = W. So, the wanted result follows for V = G (U ).
Note that the construction of the section S has been made just by using algebraic manipulations and the Inverse Function Theorem.
For the rest of the section, fix a ρ-regular point m 0 , an admissible neighborhood U and a section s : π (U ) → U of π |U such that s (π (m 0 )) = m 0 . (3.24) is an open map around (e, π(m 0 )).
Now, the main result of the section. We shall call s-horizontal, or simply horizontal, to such submersions Θ.
Proof. First, let us make some observations about the submersion Φ U of Proposition 3.5.

Vertical and horizontal vector fields
Fix again a point m 0 ∈ R ρ .
Definition 3.14. We shall say that X ∈ X (M ) is vertical around m 0 if X (m) ∈ Ker π |U * ,m , ∀m ∈ U, and that X is horizontal at m 0 if X (m 0 ) / ∈ Ker π |U * ,m0 , for some admissible neighborhood U of m 0 . Finally, we shall say that X is Θ-horizontal if ImX |U ⊆ KerΘ * for some horizontal submersion Θ : U → Θ (U ) and some admissible neighborhood U of m 0 .
From (3.6), it is clear that if there exists a function η : U → g such that X(m) = (ρ m ) * e (η(m)), for all m ∈ U, then X is vertical along U. We are interested in vertical fields which are in addition G-invariant (see (3.14)). For them, we have the next result. for some ξ g,m ∈ g ρg (m) . We shall say that η is Ad-equivariant if ξ g,m = 0 for all g, m.
Regarding horizontal fields, note that if X is Θ-horizontal and X (m 0 ) = 0, then X is horizontal at m 0 . Reciprocally, we have the next result.
Proposition 3.16. If X is horizontal at m 0 and G-invariant, then there exist an admissible neighborhood U of m 0 , a section s : π (U ) → U of π |U satisfying (3.24) and a horizontal submersion Θ : U → Θ (U ) such that X is Θ-horizontal.

Local complete solutions from general group actions
From above results and the duality between complete solutions and first integrals, the theorem below easily follows.
1. If X is vertical around m 0 , then there exists an admissible neighborhood U of m 0 such that, for every section s : π (U ) → U of π |U satisfying (3.24) and every s-horizontal submersion Θ : U → Θ (U ), the map is a complete solution of the Θ-HJE for X |U .
2. If X is horizontal at m 0 and G-invariant, then there exist an admissible neighborhood U of m 0 , a section s : π (U ) → U of π |U satisfying (3.24) and a s-horizontal submersion Θ : U → Θ (U ) such that 2 is a complete solution of the π |U -HJE for X |U .
Proof. In the first case we have that ImX |U ⊆ Ker π |U * and that π |U and Θ are transverse. Using the results of Section 2.2, it follows that, shrinking U if needed, Σ := Θ, π |U −1 is a complete solution of the Θ-HJE for The second case can be proved in the same way, but using in addition Proposition 3.16 in order to ensure the existence of the section s and the submersion Θ such that ImX |U ∈ KerΘ * . Remark 3.18. Regarding the objects described in Section 3.2.1, it is clear that the complete solutions Σ given in the last theorem, or more precisely their inverses Σ −1 , define the analogue of a trivialization Ψ : U → π (U ) × G of a principal bundle.
Summarizing, given a vertical vector field X around m 0 ∈ R ρ , an admissible neighborhood U of m 0 and a smooth section s : π(U ) → U of π |U , we have shown that a submersion Θ : U → Θ (U ) and a complete solution of the Θ-HJE for X |U can be constructed up to quadratures. Also, given a horizontal vector field X at m 0 , if X is G-invariant, then there exists a complete solution of the π |U -HJE for X |U . But the latter has not been constructed up to quadratures (the proof of Proposition 4.13 of [18], which is used in Proposition 3.16, is based on the rectification of the field X).

Horizontal dynamical systems and reconstruction
Consider again a manifold M , a vector field X ∈ X (M ) and a group action ρ : G × M → M . Assume by now that ρ is free and proper, what implies that π : M → M / G defines a principal fiber bundle. Assume also that X is G-invariant, and consequently π-related with a unique vector field Y ∈ X ( M / G), i.e. π * • X = Y • π. In many cases, the integral curves of Y are known, and one is interested in constructing the integral curves of X from those of Y . Any procedure that enable us to do that is usually called reconstruction. The purpose of this section is to show that there exists a deep connection between reconstruction procedures and the complete solutions of a horizontal vector field presented in Theorem 3.17, even though the action ρ is neither free nor proper.

The usual reconstruction process
Assume that we are in the setting of the beginning of this section and we want to find the integral curve Γ of X such that Γ (0) = p 0 . Then we can (see, for instance, [30]): 1. consider the integral curve γ (t) of Y such that γ (0) = π (p 0 ); 2. fix a principal connection A : T M → g;
It is easy to show that Γ (t) = ρ (g (t) , d (t)) is the integral curve we are looking for. The four steps above constitute the usual reconstruction process, and (4.1) and (4.2) the related reconstruction problem.
If X is vertical along all of M (in the usual sense), i.e. ImX ⊆ Kerπ * , then Y = 0 and consequently the curves d (t) and ξ (t) are constant. In this case, we only have to solve (4.2), whose solutions are given by the exponential curves. We shall consider this situation in the next section. So, suppose that X (m) / ∈ Kerπ * ,m , for all m ∈ M . In that case we can consider a connection A such that X ∈ KerA, i.e. X is horizontal with respect to A (in the usual sense). Then, ξ (t) = 0 and g (t) = g 0 for all t. Consequently, the reconstruction problem reduces to solve (4.1). In other words, we have the following alternative (three steps) reconstruction process: 1. consider the integral curve γ (t) of Y such that γ (0) = π (p 0 ); 2. find a principal connection A : T M → g such that X is horizontal; 3. find a curve d (t) satisfying (4.1).
Then, the curve Γ (t) = ρ (g 0 , d (t)), with g 0 such that p 0 = ρ (g 0 , d (0)), is the integral curve of X through p 0 . In the next subsection, we shall extend this procedure to Lie group actions which are not necessarily free and proper.

Reconstruction from complete solutions
Let us go back to the general setting: a manifold M , a vector field X ∈ X (M ) and a general Lie group action ρ : G×M → M . Assume that X is G-invariant and horizontal at every m 0 ∈ R ρ (see Definition 3.14). According to the second part of Theorem 3.17, there exist an admissible neighborhood U of m 0 , a section s : π (U ) → U of π |U satisfying (3.24) and a s-horizontal submersion Θ : U → Θ (U ) such that is a complete solution of the π |U -HJE for X |U . The related partial solutions are functions σ g : π (U ) → U, g ∈ Θ (U ) , such that σ g (λ) = ρ (g, s (λ)) for all λ ∈ π (U ) (see (2.4)). In other words, Theorem 4.1. Each vector field X σg ∈ X (π (U )) (see (2.3)) is equal, for all g ∈ Θ (U ), to the unique vector field Y ∈ X (π (U )) such that Proof. The Proposition 3.6 ensures the existence of a unique vector field Y ∈ X (π (U )) satisfying (4.4). So, we only must prove that Y = X σg for all g ∈ Θ (U ). But from (2.3), (4.3) and (4.4), as we wanted to show.
According to (2.8), the integral curves Γ of X are given by where γ is an integral curve of Y = X σg . In other words, above formula enable us to construct the integral curves of X from those of a vector field in the quotient. Note that π (Γ (t)) = γ (t) and Θ (Γ (t)) = g for all t. Then, given p 0 ∈ U , in order to find the integral curve Γ of X |U such that Γ (0) = p 0 , we have the following (two steps) reconstruction process: 1. consider the integral curve γ (t) of Y such that γ (0) = π (p 0 ); 2. find a submersion Θ : U → Θ (U ) such that X is Θ-horizontal.

Vertical dynamical systems and integrability by quadratures
In this section, using the integrability result of Section 2 (see Theorem 2.2), we show that the exponential curves t → exp (ξ t) of a Lie group G, for some points ξ of its Lie algebra g, can be explicitly constructed up to quadratures. Moreover, we show that, for compact and for semisimple Lie groups, such a construction works for all ξ inside a dense open subset of g. Then, we state sufficient conditions under which a vertical (and invariant) vector field is integrable up to quadratures.

Invariant and vertical vector fields
Consider again a manifold M , a vector field X ∈ X (M ) and a Lie group action ρ : G × M → M . Assume that X is vertical around every ρ-regular point m 0 (see Definition 3.14) and consider a complete solution as those given in the first part of Theorem 3.17. The related partial solutions are with σ λ (g) = ρ (g, s (λ)) for all g ∈ Θ (U ) (see (2.4)). In other words, σ λ = ρ s(λ) , λ ∈ π (U ) .
In order to consider concrete examples of vertical and G-invariant fields, suppose that M is a symplectic manifold, with symplectic form ω, and ρ is a symplectic action with an Ad * -equivariant momentum map K.

A class of invariant vertical vectors
Given a Lie group G, consider its cotangent bundle T * G with its canonical symplectic structure ω G = −dθ G . Consider also the action ρ : G × T * G → T * G such that, for all g ∈ G and α h ∈ T * h G, Note that ρ is symplectic (see (3.12)) and has an Ad * -equivariant momentum map J : T * G → g * given by . Also, ρ is a free and proper action, the quotient T * G/ G is a manifold diffeomorphic to g * and the canonical projection π can be seen as the submersion In other words, every point of T * G is ρ-regular and the whole of T * G is an admissible neighborhood. Then, according to Proposition 3.6 (see Eq. (3.13)), defines a Poisson bracket on g * and π is a Poisson morphism between (T * G, ω G ) (with its related Poisson structure) and (g * , Ξ). Moreover, it can be shown that Ξ is the Kirillov-Kostant bracket on g * (see [27]), i.e. Ξ ♯ (η) = ad * η α, η ∈ T * α g * ∼ = g.
Remark 5.3. Note that (π G , π) : T * G → G × g * is the left trivialization of T * G. Thus, T * G may be identified with G× g and, under this identification, the projections π G : T * G → G and π : T * G → g * are just the canonical projections pr 1 : G × g * → G and pr 2 : G × g * → g * on the first and second factor, respectively. Moreover, the canonical symplectic structure ω G on T * G is the 2-form on G × g * given by [1]). In addition, the action ρ : G × (G × g * ) → G × g * is just the left translation on the first factor, that is, ρ (g, (g ′ , α)) = (gg ′ , α) for g, g ′ ∈ G and α ∈ g * , and the momentum map J : G × g → g * is just the co-adjoint action of G on g * J (g, α) = Ad * g −1 α (for more details, see [1]).

Construction of the exponential curves up to quadratures
In this subsection, we are going to show that X φ (see (5.15)) is integrable up to quadratures (on a dense subset of T * G) and, consequently, the exponential curves exp (φ (α) t) can be explicitly obtained, also up to quadratures. The proof will be based on Theorem 2.2.

Moreover,
KerF * = Ker J |U * ∩ Ker π |U * (5.18) and Proof. It is easy to see that the composition of (J, π) : T * G → g * × g * and (the inverse of the right trivialization) gives Then, given an Ad * -regular point α 0 and an admissible neighborhood V ⊆ g * of α 0 , we have from Proposition 3.5 (applied to the action Ad * ) that (J, π) • R −1 restricted to G × V is a submersion onto the closed submanifold F (U ) ⊆ V × g * . As a consequence, since R −1 is a diffeomorphism, the first affirmation of the proposition follows. On the other hand, (5.18) follows straightforwardly and (5.19) is a direct consequence of the identity Because of the form of X φ , it is clear that ImX φ ⊆ (Kerπ * ) ⊥ . As a consequence (recall (3.16)) So, using the last proposition and combining (5.18), (5.19) and (5.20), it follows that, for each Ad * -regular point α 0 , we can construct a neighborhood U of α 0 and a submersion F : U → F (U ) (given by (5.17)) such that ImX φ |U ⊆ KerF * and KerF * ⊆ (KerF * ) ⊥ .
Remark 5.6. It can be shown that (KerF * ) ⊥ is an integrable distribution. Then, if φ is an exact 1-form, X φ |U and F define a NCI system on U (see Section 2.4).
In addition, since KerF * ⊆ Ker π |U * , we have that as we saw at the end of Section 2.3 (recall (2.23) and (2.24)). This enable us to apply Theorem 2.2 to X φ |U .
Proposition 5.8. Given a Casimir 1-form φ : g * → g and a point α ∈ R Ad * , the exponential curve exp (φ (α) t) can be constructed up to quadratures. More explicitly, it is given by the formula
It is natural to ask, given ξ ∈ g, if we can construct exp (ξ t) up to quadratures. In the following subsection, we shall give a partial answer to that question.

The case of semisimple and compact Lie groups
Let G be a Lie group with Lie algebra g. Theorem 5.9. Consider ξ ∈ g such that where ad ξ (g) 0 is the annihilator in g * of the subspace ad ξ (g) ⊆ g.
For an important subclass of Lie groups, we have the following result.
Theorem 5.10. Let G be a connected Lie group with Lie algebra g and R Ad the open dense subset of g which consists of the regular points in g with respect to the adjoint action of G on g. Suppose that there exists a non-degenerate ad-invariant symmetric bilinear form B : g × g → R. Then, 1. The linear map B ♭ : g → g * given by B ♭ (ξ) , η = B (ξ, η), for all ξ, η ∈ g, is a isomorphism satisfying B ♭ (R Ad ) = R Ad * , and its inverse B ♯ : g * → g is a Casimir 1-form.
2. For every ξ ∈ R Ad , the curve t → exp(ξ t) can be obtained by quadratures.
Proof. We have that (non-degeneracy) is commutative, for every ξ ∈ g. So, since G is a connected Lie group, we also have that the diagram is commutative for every g ∈ G. Thus, if G ξ (resp. G B ♭ (ξ) ) is the isotropy group of ξ ∈ g (resp. B ♭ (ξ) ∈ g * ) with respect to the adjoint (resp. co-adjoint) action of G on g (resp. g * ), we deduce that This implies that B ♭ (R Ad ) = R Ad * . Therefore, given α ∈ g * , if we write α = B ♭ (ξ) for some ξ ∈ g, we have for B ♯ = B ♭ −1 : g * → g that for all η ∈ g. Then, B ♯ is a Casimir 1-form. This proves the first point. To prove the second point, note that, according to (5.27), for every ξ ∈ R Ad there exists α ∈ R Ad * such that ξ = B ♯ (α). Then, it is enough to use Proposition 5.8 for φ = B ♯ .
Remark 5.11. It can be show that, under the conditions of the theorem above, So, the point 2 of Theorem 5.10 can also be proved by combining the equation above and Theorem 5.9.
Under the conditions of the last theorem, we can use (5.24) for φ = B ♯ and for all ξ ∈ R Ad , which gives Remark 5.12. In particular, for ξ ∈ R Ad and close to 0 (in order for a ξ (t) to be defined when t = 1), we have the following expression of the exponential map: Remark 5.13. It is worth mentioning that B ♯ is an exact 1-form, i.e. B ♯ = dh with h : g * → R given by h (α) = 1 2 α, B ♯ (α) .
Then, according to Remark 5.6, the related vector field X B ♯ |U and the submersion F define a NCI Hamiltonian system on U .
For a semisimple Lie group G with Lie algebra g, the Killing form on g satisfies the conditions in Theorem 5.10 (see for example [32]). On the other hand, a Lie algebra g is the Lie algebra of a compact Lie group if and only if g admits an ad-invariant scalar product (see, for instance, [13]). So, using Theorem 5.10, we have the next corollary.
Corollary 5.14. Let G be a connected Lie group with Lie algebra g and ξ ∈ R Ad ⊆ g. If G is semisimple or compact then t → exp(ξ t) can be obtained by quadratures.
The last two results tell us that the exponential curve exp(ξ t) can be constructed by quadratures for ξ living in an open dense subset of g. Unfortunately, we can not ensure the same for every Lie group.

Integrability conditions for invariant vertical fields
Let us go back to Section 5.1. Consider a manifold M , a vector field X ∈ X (M ) and a Lie group action ρ : G × M → M . Assume that X is vertical around every point m 0 ∈ R ρ and G-invariant. Consider a covering of R ρ given by admissible neighborhoods U , each one of them with an associated complete solution Σ U := ρ • id Θ(U) × s , as those given in Theorem 3.17, and the map η U : π (U ) → g : λ → Θ * ,s(λ) • X (s (λ)) given by (5.3) in Theorem 5.1. From now on, we shall denote g ρ,m the isotropy sub-algebra related to the point m and the action ρ.
More interesting examples can be constructed by using the next lemma.
Lemma 5.18. If h : g * → R is a G-invariant function with respect to Ad * , then dh : g * → g is equivariant and a Casimir 1-form.
For a proof, see [2], Lemma 2.9. Proof. Since each field ω ♯ • K * dh i is G-invariant and vertical, the same is true for X. On the other hand, given (as in the proof of Theorem (5.17)) a G-invariant open subset of V ⊆ R ρ such that K (V ) ⊆ R Ad * , a covering of V by admissible neighborhoods U and the related maps η U , for each λ ∈ π (U ) we have that f i (s (λ)) dh i (K (s (λ))) + ξ λ , for some ξ λ ∈ g ρ,s(λ) . Then, defining φ λ : g * → g by which is a Casimir 1-form, we have that η U (λ) = φ λ (K (s (λ))) + ξ λ .
It is worth mentioning that the vector field X given by (5.29) is not, in general, a Hamiltonian vector field.