Euler–Lagrange–Herglotz equations on Lie algebroids

Abstract

We introduce Euler–Lagrange–Herglotz equations on Lie algebroids. The methodology is to extend the Jacobi structure from $TQ\times \mathbb {R}$ and $T^{*}Q \times \mathbb {R}$ to $A\times \mathbb {R}$ and $A^{*}\times \mathbb {R}$, respectively, where A is a Lie algebroid and $A^{*}$ carries the associated Poisson structure. We see that $A^*\times \mathbb {R}$ possesses a natural Jacobi structure from where we are able to model dissipative mechanical systems on Lie algebroids, generalizing previous models on $TQ\times \mathbb {R}$ and introducing new ones as for instance for reduced systems on Lie algebras, semidirect products (action Lie algebroids) and Atiyah bundles.

1 Introduction

Since the works of Weinstein [40] and Martinez [34] during the last 20 years, there has been a lot of applications of Lie algebroids in theoretical physics and other related sciences, more precisely in Classical Mechanics, Classical Field Theory and their applications. The main point is that Lie algebroids provide a general framework for systems with different features as systems with symmetries, systems over semidirect products, Hamiltonian and Lagrangian systems, systems with constraints (nonholonomic and vakonomic), higher-order mechanics and optimal control [1, 6, 7, 10,11,12, 17, 18, 29,30,31,32,33,34,35,36,37] among many others.

In [16] de León et al. have developed a Hamiltonian description for the mechanics on Lie algebroids and they have shown that the dynamics is obtained solving an equation for the Hamiltonian section (Hamiltonian vector field) in the same way than in Classical Mechanics (see also [33]). Moreover, for a Lie algebroid A, they have shown that the Legendre transformation $\mathbb {F}L:A\rightarrow A^{*}$ associated to a Lagrangian $L:A\rightarrow \mathbb {R}$ induces a Lie algebroid morphism and when the Lagrangian is regular both formalisms are equivalent. Marrero and collaborators also have studied non-holonomic mechanics on Lie algebroids [11]. In other direction, in [22] Iglesias et al. have studied singular Lagrangian systems and vakonomic mechanics from the point of view of Lie algebroids obtained through the application of a constrained variational principle, and J. Grabowski, K. Grabowska, and collaborators have also developed the geometric formalism for mechanics on Lie algebroids in a serie of papers [18, 19] including optimal control applications [20].

Contact Hamiltonian and Lagrangian systems have deserved a lot of attention in recent years (see [2, 4, 5, 15], and references therein). Throughout this paper, a contact Hamiltonian or Lagrangian system should be understood as the Hamiltonian (respectively, Lagrangian) vector field associated with a particular Jacobi structure on the product $A^{*}\times \mathbb {R}$ (respectively, $A\times \mathbb {R}$), where A is a Lie algebroid, designed to extend standard contact systems on $T^{*}Q\times \mathbb {R}$ (respectively, $TQ\times \mathbb {R}$) and model dissipative phenomena. One of the most relevant features of contact dynamics is the absence of conservative properties contrarily to the conservative character of the energy in symplectic dynamics; indeed, we have a dissipative behavior. This fact suggests that contact geometry may be the appropriate framework to model many physical and mathematical problems with dissipation we find in thermodynamics, statistical physics, quantum mechanics, gravity or control theory, among many others.

Jacobi manifolds were introduced by A. Lichnerowicz as a rich geometrical notion extending several important geometrical structures, including among others, symplectic, Poisson, contact and cosymplectic structures [26]. In this paper we extend the Jacobi structure from $TQ\times \mathbb {R}$ and $T^{*}Q \times \mathbb {R}$ to $A\times \mathbb {R}$ and $A^{*}\times \mathbb {R}$, respectively, where A is a Lie algebroid and $A^{*}$ carries the associated Poisson structure. We see that $A^*\times \mathbb {R}$ possesses a natural Jacobi structure from where we are able to model dissipative mechanical systems, generalizing previous models on $TQ\times \mathbb {R}$. In particular, when A is the standard Lie algebroid TQ, the Jacobi structure on $T^{*}Q\times \mathbb {R}$ is just the canonical contact structure.

One of the main contributions of the paper is the identification of the contact dynamics on the extended Lie algebroid $A\times \mathbb {R}$, generalizing contact dynamics on $TQ\times \mathbb {R}$. The results in this paper, will be of use to all those who wish to study contact dynamics on Lie groups, homogeneous spaces, action algebroids, Atiyah algebroids or any other phase space different than the tangent bundle. In addition, it also successess in identifying the Jacobi structure hidden behind the dynamics. The other main goal of this paper is to give for first time in the literature the notion of Euler–Lagrange–Herglotz equations on Lie algebroids, extending the Euler–Lagrange–Herglotz equations on $TQ \times mathbb{R}$, and providing the corresponding equations for reduced systems on Lie algebras, action Lie algebroids, and Atiyah algebroids, that is, the Euler–Poincaré–Herglotz and Lagrange–Poincaré–Herglotz equations, respectively. Moreover, in the presence of a Hamiltonian function $h:A^{*}\times \mathbb {R}\rightarrow \mathbb {R}$, we consider the corresponding dynamical system on $A^{*}\times \mathbb {R}$ called Lie–Poisson–Jacobi equations (respectively, Hamilton–Poincaré–Herglotz equations on the Atiyah algebroid). In addition, if the Hamiltonian function is regular then a Lagrangian dynamics, induced by a Lagrangian function $l:A\times \mathbb {R}\rightarrow \mathbb {R}$, may be considered on the space $A\times \mathbb {R}$. Some particular examples are discussed: the dissipative Wong’s equations on Atiyah algebroids and the dynamics of a triaxial attitude control testbed as Euler–Poincaré–Herglotz equations on action Lie algebroids.

The outline of the paper is the following: in Sect. 2 we review the definition of the Hamiltonian equations in the canonical Poisson structure of $A^{*}$, where A is a Lie algebroid. In Sect. 3, we introduce a Jacobi structure on $A^{*}\times \mathbb {R}$, from where we derive the equations of motion for a contact Hamiltonian system and use it to obtain the contact Lagrangian systems on the bundle $A\times \mathbb {R}$. In Sect. 4, we particularize our construction to the case of the action Lie algebroid and the Atiyah algebroid and we derive, from a geometric viewpoint, the dynamics for a Triaxial attitude control testbed and the contact Wong’s equations.

2 Mechanics on Lie algebroids

Let $\tau _{A}:A\rightarrow Q$ be a vector bundle with base manifold Q together with a fiber-preserving map $\rho :A\rightarrow TQ$ called the anchor map. Let $[\cdot ,\cdot ]_{A}:\Gamma (A)\times \Gamma (A)\rightarrow \Gamma (A)$ be a Lie bracket on the set of sections $\Gamma (A)$ (that is, a skew-symmetric bilinear map satisfying the Jacobi identity) satisfying the Leibniz rule

$$\begin{aligned}{}[X,fY]_{A}=f[X,Y]_{A}+\rho (X)(f)Y, \ \ \text {for} \ X,Y\in \Gamma (A) \ \text {and} \ f\in C^{\infty }(Q). \end{aligned}$$

The vector bundle $\tau _{A}:A\rightarrow Q$ equipped with the anchor map $\rho$ and the bracket of sections $[\cdot ,\cdot ]_{A}$ is called a Lie algebroid (see [27] for instance).

Suppose that $(q^{i})$ are local coordinates on Q and that $\{e_{a}\}$ is a local basis of sections of A. The local functions $C^{d}_{a b}$ and $\rho ^{i}_{a}$ defined by

$$\begin{aligned}{}[e_{a}, e_{b}]_{A}=C^{d}_{a b} e_{d}, \quad \rho (e_{a})=\rho ^{i}_{a}\frac{\partial }{\partial q^{i}}, \end{aligned}$$

are called structure functions of the Lie algebroid. In the following sections, we will also considered local coordinates $(q^{i},y^{a})$ on A, adapted to the local basis of sections $\{e_a\}$.

We will see that the Lie algebroid structure on A naturally induces a Poisson structure on its dual bundle $A^{*}$. First, we will review the definition of a linear Poisson structure on the vector bundle $\pi _{A^{*}}:A^{*}\rightarrow Q$.

Definition 2.1

A linear Poisson structure on $A^{*}$ is a bracket of functions $\{\cdot ,\cdot \}_{A^{*}}:C^{\infty }(A^{*})\times C^{\infty }(A^{*}) \rightarrow C^{\infty }(A^{*})$ such that:

• 1. $\{\cdot ,\cdot \}_{A^{*}}$ is a skew-symmetric bilinear map satisfying the Jacobi identity;

• 2. $\{\cdot ,\cdot \}_{A^{*}}$ satisfies the Leibniz rule

  $$\begin{aligned} \{f,g h\}_{A^{*}}=g \{f, h\}_{A^{*}}+ \{f,g\}_{A^{*}}h, \quad f,g,h\in C^{\infty }(A^{*}). \end{aligned}$$

• 3. $\{f,g\}_{A^{*}}$ is a fiberwise linear function if f and g are also fiberwise linear functions on $A^{*}$.

The Poisson bracket is associated with a bi-vector $\Lambda _{A^{*}}$ on $A^{*}$ such that

$$\begin{aligned} \{f,g\}_{A^{*}}=\Lambda _{A^{*}}(df,dg), \quad \forall f,g \in C^{\infty }(A^{*}). \end{aligned}$$

There is a correspondence between sections of A and fiberwise linear functions on $A^{*}$. If $X\in \Gamma (A)$ then the corresponding linear function is denoted by $\hat{X}:A^{*}\rightarrow \mathbb {R}$ and given by

$$\begin{aligned} \hat{X}(\alpha _{q})=\langle \alpha _{q},X(q) \rangle , \end{aligned}$$

for any $\alpha _{q}\in A^{*}_{q}$.

Now, the following theorem (see [13, 17] for the proof) contains the correspondence between Lie algebroid and linear Poisson structures.

Theorem 2.2

There exists a one-to-one correspondence between Lie algebroid structures on the vector bundle $\tau _{A}:A\rightarrow Q$ and linear Poisson structures on the dual bundle $A^{*}$. The correspondence is determined by the relations

$$\begin{aligned} \widehat{[X,Y]}_{A}=\{\hat{X},\hat{Y}\}_{A^{*}}, \quad \rho (X)(f)\circ \pi _{A^{*}}=\{\hat{X}, f\circ \pi _{A^{*}}\}_{A^{*}} \end{aligned}$$

for $X, Y\in \Gamma (A)$ and $f\in C^{\infty }(Q)$.

If $\{e_a\}$ is a local basis of sections for $A\rightarrow Q$, the dual basis $\{e^{a}\}$ is a local basis of sections for the dual bundle. In local coordinates $(q^{i},p_{a})$ on $A^{*}$, adapted to the dual basis $\{e^{a}\}$, the Poisson structure associated to a Lie algebroid $\tau _{A}:A\rightarrow Q$ is given by the relations

$$\begin{aligned} \{p_{a},p_{b}\}_{A^{*}}=C_{a b}^{d} p_{d}, \quad \{p_{a},q^{i}\}_{A^{*}}=\rho _{a}^{i}, \quad \{q^{i},q^{j}\}_{A^{*}}=0. \end{aligned}$$

(1)

Having fixed a Poisson structure we can provide Hamiltonian equations as follow. Let $h:A^{*}\rightarrow \mathbb {R}$ be a Hamiltonian function. Then the Hamiltonian vector field is the vector field $X_{h}$ characterized by

$$\begin{aligned} X_{h}(f)=\{ h,f \}_{A^{*}}, \quad \forall \ f \in C^{\infty }(A^{*}). \end{aligned}$$

Alternatively, using the Poisson structure $\Lambda _{A^{*}}$,

$$\begin{aligned} X_{h}=i_{dh} \ \Lambda _{A^{*}}. \end{aligned}$$

Thus, in local coordinates,

$$\begin{aligned} X_{h}=\rho _{a}^{i}\frac{\partial h}{\partial p_{a}}\frac{\partial }{\partial q^{i}}-\left( \rho _{a}^{i}\frac{\partial h}{\partial q^{i}}+C_{a b}^{d} p_{d}\frac{\partial h}{\partial p_{b}} \right) \frac{\partial }{\partial p_{a}}. \end{aligned}$$

Therefore, the Hamilton equations are

$$\begin{aligned} \dot{q^i}=\rho ^i_{a} \frac{\partial h}{\partial p_{a}},\qquad \dot{p}_{a}=- \rho ^i_{a} \frac{\partial h}{\partial q^i}- p_{c} C^{c}_{ab} \frac{\partial h}{\partial p_{b}}. \end{aligned}$$

(2)

Example 2.3

Consider the tangent bundle of a manifold Q. The sections of the bundle $\tau _{TQ}:TQ\rightarrow Q$ are the set of vector fields on Q. The anchor map $\rho :TQ\rightarrow TQ$ is the identity function and the Lie bracket defined on $\Gamma (\tau _{TQ})$ is induced by the Lie bracket of vector fields on Q.

Note that in this case, Hamilton equations (2) become in the usual Hamilton equations

$$\begin{aligned} \dot{q}^{i}=\frac{\partial h}{\partial p_i},\quad \dot{p}_{i}=-\frac{\partial h}{\partial q^{i}}. \end{aligned}$$

Example 2.4

Given a finite dimensional real Lie algebra $\mathfrak {g}$ and $Q=\{q\}$ be a unique point, we consider the vector bundle $\tau _{\mathfrak {g}}:\mathfrak {g}\rightarrow Q.$ The sections of this bundle can be identified with the elements of $\mathfrak {g}$ and therefore we can consider as the Lie bracket the structure of the Lie algebra induced by $\mathfrak {g}$, and denoted by $[\cdot ,\cdot ]_{\mathfrak {g}}$. Since $TQ=\{0\}$ one may consider the anchor map $\rho \equiv 0$. The triple $(\mathfrak {g},[\cdot ,\cdot ]_{\mathfrak {g}},0)$ is a Lie algebroid over a point.

Note that in this case, Hamilton equations (2) become in the Lie-Poisson equations

$$\begin{aligned} \dot{p}_{a} = \frac{d p_{a}}{d t}=- p_{c} C^{c}_{ab} \frac{\partial h}{\partial p_{b}}. \end{aligned}$$

3 Dissipative mechanical systems on Lie algebroids

Let us recall first the definition of a Jacobi structure (see [25, 26]).

Definition 3.1

(Jacobi structure) A Jacobi structure on a manifold M is a pair $(\Lambda ,E)$, where $\Lambda$ is a bi-vector field and E is a vector field, satisfying the following equations

$$\begin{aligned}{}[\Lambda ,\Lambda ]=2E\wedge \Lambda , \quad [E,\Lambda ]=0, \end{aligned}$$

with $[\cdot ,\cdot ]$ the Schouten-Nijenhuis bracket.

Definition 3.2

(Jacobi bracket) A Jacobi bracket on a manifold M is a bilinear, skew-symmetric map $\{\cdot ,\cdot \}:C^{\infty }(M)\times C^{\infty }(M) \rightarrow C^{\infty }(M)$ satisfying the Jacobi identity and the following weak Leibniz rule

$$\begin{aligned} \text {supp}(\{f,g\})\subseteq \text {supp}(f)\cap \text {supp}(g). \end{aligned}$$

A Jacobi manifold is a manifold possessing either a Jacobi structure or a Jacobi bracket since these two definitions are equivalent (see [21, 25, 26, 28]). However, it is much more convenient to introduce a Jacobi structure for practical purposes.

Now, from the Jacobi structure we can define an associated Jacobi bracket as follows:

$$\begin{aligned} \{f, g\}=\Lambda (df, dg)+f E(g)-g E(f), \quad f, g\in C^{\infty }(M, \mathbb {R}) \end{aligned}$$

In this case, the weak Leibniz rule is equivalent to the generalized Leibniz rule

$$\begin{aligned} \{f, gh\} = g\{f, h\} + h\{f, g\} + ghE(f), \end{aligned}$$

(3)

In this sense, this bracket generalizes the well-known Poisson brackets. Indeed, a Poisson manifold is a particular case of Jacobi manifold in which $E=0$.

Given a Jacobi manifold $(M,\Lambda ,E)$, we consider the map $\sharp _{\Lambda }:\Omega ^{1}(M)\rightarrow \mathfrak {X}(M)$ defined by

$$\begin{aligned} \sharp _{\Lambda }(\alpha )=\Lambda (\alpha ,\cdot ). \end{aligned}$$

We have that $\sharp _{\Lambda }$ is a morphism of $C^{\infty }$-modules, though it may fail to be an isomorphism. Given a function $f: M \rightarrow \mathbb {R}$ we define the Hamiltonian vector field $X_f$ by

$$\begin{aligned} X_f=\sharp _{\Lambda } (df) + f E. \end{aligned}$$

3.1 Contact Hamiltonian equations on Lie algebroids

Suppose that $\tau _{A}:A\rightarrow Q$ is a Lie algebroid and $\pi _{A^{*}}:A^{*}\rightarrow Q$ is its dual vector bundle equipped with the associated Poisson structure.

To formulate our contact Hamiltonian equations we will consider the vector bundle $\pi _{1}:A^{*}\times \mathbb {R}\rightarrow Q$ given by $\pi _{1}(\mu ,z)=\pi _{A^{*}}(\mu )$, where z is a global coordinate on $\mathbb {R}$, so that the following diagram commutes

Next we will introduce two special vector fields: the Reeb vector field on $A^{*}\times \mathbb {R}$ which is given in local coordinates by

$$\begin{aligned} \mathcal {R}(q^{i},p_{a},z)=\frac{\partial }{\partial z} \end{aligned}$$

and the Liouville vector field on $A^{*}\times \mathbb {R}$ which is given by

$$\begin{aligned} \Delta ^{*} (\mu _{q},z)=\left. \frac{d}{dt}\right| _{t=0}(\mu _{q}+t\mu _{q},z) \end{aligned}$$

or, in local coordinates, by

$$\begin{aligned} \Delta ^{*} (q^{i},p_{a},z)=p_{a}\frac{\partial }{\partial p_{a}}. \end{aligned}$$

Theorem 3.3

Let $\Lambda _{A^{*}}$ be the Poisson structure on $A^{*}$ corresponding to the Lie algebroid structure on A. Then the pair $(\Lambda _{A^{*}\times \mathbb {R}}, E)$, where $\Lambda _{A^{*}\times \mathbb {R}}$ is the bi-vector defined by

$$\begin{aligned} \Lambda _{A^{*}\times \mathbb {R}}=(\text {pr}_{1})^{*}\Lambda _{A^{*}}+\Delta ^{*} \wedge \mathcal {R}, \end{aligned}$$

with $\text {pr}_{1}:A^{*}\times \mathbb {R}\rightarrow A^{*}$ the projection onto the first factor, and $E = -\mathcal {R}$ is a Jacobi structure on $A^{*}\times \mathbb {R}$.

Proof

Using the fact that

$$\begin{aligned} \Lambda = \text {pr}_{1}^{*}\Lambda _{A^{*}} +\Delta ^{*} \wedge \mathcal {R}, \end{aligned}$$

where $\Lambda _{A^{*}}$ is the Poisson structure on $A^{*}$, $\text {pr}_{1}:A^{*}\times \mathbb {R}\rightarrow A^{*}$ is the projection onto the first factor, $\Delta ^{*}$ is the Liouville vector field and $\mathcal {R}=\frac{\partial }{\partial z}$, we may deduce after some computations involving the Schouten-Nijenhuis bracket and interior products that (see [28])

$$\begin{aligned}{}[\Lambda ,\Lambda ] = 2[\Lambda _{A^{*}},\Delta ^{*} \wedge \mathcal {R}] = 2\Lambda _{A^{*}}\wedge \mathcal {R}, \end{aligned}$$

where we used that $[\Delta ^{*} \wedge \mathcal {R}, \Lambda _{A^{*}}]=[\Lambda _{A^{*}}, \Delta ^{*} \wedge \mathcal {R}]$ (since $\Lambda _{A^{*}}$ and $\Delta ^{*} \wedge \mathcal {R}$ are both (2, 0)-tensors) and the fact that $[\Lambda _{A^{*}},\Delta ^{*}]= \Lambda _{A^{*}}$. The previous equality is equivalent to

$$\begin{aligned}{}[\Lambda ,\Lambda ] = 2\Lambda \wedge \mathcal {R}, \end{aligned}$$

using linearity and skew-symmetry of the wedge product.

In addition, we also have that $[\mathcal {R},\Lambda ]$ vanishes:

$$\begin{aligned}{}[\mathcal {R},\Lambda ] = [\mathcal {R}, \Lambda _{A^{*}}] + [\mathcal {R}, \Delta ^{*} \wedge \mathcal {R}]. \end{aligned}$$

The first term vanishes since $\Lambda _{A^{*}}$ is pulled-back from $A^{*}$ and so $[\mathcal {R}, \Lambda _{A^{*}}] = \mathcal {L}_{\mathcal {R}} \Lambda _{A^{*}}=0$. The second term also vanishes since $[\mathcal {R}, \Delta ^{*} \wedge \mathcal {R}] = [\mathcal {R}, \Delta ^{*}]\wedge \mathcal {R}+ \Delta ^{*} \wedge [\mathcal {R}, \mathcal {R}] = [\mathcal {R}, \Delta ^{*}]\wedge \mathcal {R}$ and the Lie bracket of $[\mathcal {R}, \Delta ^{*}]$ is zero. Hence, $(\Lambda ,E=-\mathcal {R})$ is indeed a Jacobi structure. $\square$

Remark 3.4

The Jacobi structure proposed in the previous theorem is a particular case of the Jacobi structure introduced in [23, 24] using the cocycle $\phi =(0,1)\in \Gamma (A^{*}\times \mathbb {R})$ in their construction.

The bi-vector $\Lambda _{A^{*}\times \mathbb {R}}$ naturally generates a Jacobi bracket of functions following the usual definition

$$\begin{aligned} \{f,g\}_{A^{*}\times \mathbb {R}}=\Lambda _{A^{*}\times \mathbb {R}}(df, dg) - f\mathcal {R}(g) + g\mathcal {R}(f), \quad f,g \in C^{\infty }(A^{*}\times \mathbb {R}). \end{aligned}$$

(4)

In local coordinates, we deduce

$$\begin{aligned} \begin{aligned}&\{p_{a},p_{b}\}_{A^{*}\times \mathbb {R}}=C_{a b}^{d}p_{d}, \quad \{p_{a},q^{i}\}_{A^{*}\times \mathbb {R}}=\rho ^{i}_{a} \\&\{q^{i},q^{j}\}_{A^{*}\times \mathbb {R}}=0, \quad \{p_{a},z\}_{A^{*}\times \mathbb {R}}=0 \quad \{q^{i}, z\}_{A^{*}\times \mathbb {R}}=-q^{i}. \end{aligned} \end{aligned}$$

(5)

Definition 3.5

Given a Hamiltonian function $h:A^{*}\times \mathbb {R}\rightarrow \mathbb {R}$, the contact Hamiltonian vector field is given by the relation

$$\begin{aligned} X_{h}(f)=\{h,f\}_{A^{*}\times \mathbb {R}}-fR(h), \quad \forall f \in C^{\infty }(A^{*}\times \mathbb {R}). \end{aligned}$$

(6)

Hence, the local expression of $X_{h}$ is

$$\begin{aligned} X_{h}=\rho _{a}^{i}\frac{\partial h}{\partial p_{a}}\frac{\partial }{\partial q^{i}}-\left( \rho _{a}^{i}\frac{\partial h}{\partial q^{i}}+C_{a b}^{d} p_{d}\frac{\partial h}{\partial p_{b}}+ p_{a}\frac{\partial h}{\partial z} \right) \frac{\partial }{\partial p_{a}}+\left( p_{a}\frac{\partial h}{\partial p_{a}}-h \right) \frac{\partial }{\partial z}.\nonumber \\ \end{aligned}$$

(7)

Example 3.6

When $A=TQ$ is equipped with the Lie brackets and the anchor map is just the identity, then the Jacobi structure is the canonical one in $T^{*}Q\times \mathbb {R}$. In that case, we recover the contact Hamiltonian equations in [15]

$$\begin{aligned} \dot{q}^i=\frac{\partial h}{\partial p_{a}},\,\,\,\dot{p}_a=-\frac{\partial h}{\partial q^{i}}-p_{a}\frac{\partial h}{\partial z},\,\,\,\dot{z}=p_{a}\frac{\partial h}{\partial p_{a}}-h. \end{aligned}$$

Example 3.7

When A is a Lie algebra, say $A=\mathfrak {g}$, we find that the Hamiltonian vector field on $\mathfrak {g}^{*}\times \mathbb {R}$ is just

$$\begin{aligned} X_{h}=-\left( C_{a b}^{d} p_{d}\frac{\partial h}{\partial p_{b}}+ p_{a}\frac{\partial h}{\partial z} \right) \frac{\partial }{\partial p_{a}}+\left( p_{a}\frac{\partial h}{\partial p_{a}}-h \right) \frac{\partial }{\partial z}. \end{aligned}$$

(8)

which gives rise to the Lie–Jacobi equations

$$\begin{aligned} \dot{p}_a=-C_{a b}^{d} p_{d}\frac{\partial h}{\partial p_{b}}- p_{a}\frac{\partial h}{\partial z},\,\,\,\dot{z}=p_{a}\frac{\partial h}{\partial p_{a}}-h. \end{aligned}$$

3.2 Herglotz equations on Lie algebroids

The Lagrangian function $l:A\times \mathbb {R}\rightarrow \mathbb {R}$ is said to be regular if the fiber derivative map given by

$$\begin{aligned} \begin{aligned} \mathbb {F} l: A \times \mathbb {R}&\rightarrow A^{*}\times \mathbb {R}\\ (\alpha _{q},z)&\mapsto (\mu _{q}(\alpha _{q},z),z), \end{aligned} \end{aligned}$$

(9)

where

$$\begin{aligned} \langle \mu _{q}(\alpha _{q},z), \Omega _{q} \rangle =\left. \frac{d}{dt} \right| _{t=0} l (\alpha _{q}+t\Omega _{q},z), \quad \forall \ \Omega _{q}\in A_{q} \end{aligned}$$

(10)

is a diffeomorphism.

We also define the Liouville vector field on $A\times \mathbb {R}$ to be

$$\begin{aligned} \Delta (\alpha _{q},z)=\left. \frac{d}{dt}\right| _{t=0}(\alpha _{q}+t\alpha _{q},z) \end{aligned}$$

or, in local coordinates, by

$$\begin{aligned} \Delta (q^{i},y^{a},z)=y^{a}\frac{\partial }{\partial y^{a}}. \end{aligned}$$

The Lagrangian energy is the function $E_{l}:A\times \mathbb {R}\rightarrow \mathbb {R}$ given by

$$\begin{aligned} E_{l}(\alpha _{q},z)=\Delta (\alpha _{q},z)(l)-l(\alpha _{q},z), \end{aligned}$$

(11)

whose local expression is

$$\begin{aligned} E_{l}(q^{i},y^{a},z) = y^{a}\frac{\partial l}{\partial y^{a}} - l. \end{aligned}$$

Theorem 3.8

If $l:A\times \mathbb {R}\rightarrow \mathbb {R}$ is a regular contact Lagrangian function and $h:A^{*}\times \mathbb {R}\rightarrow \mathbb {R}$ is the contact Hamiltonian function defined by $h=E_{l}\circ (\mathbb {F} l)^{-1}$, then the curve $(\mu ,z):I\rightarrow A^{*}\times \mathbb {R}$ is an integral curve of the Hamiltonian vector field $X_{h}$ if and only if the curve $(\alpha ,z)=(\mathbb {F} l)^{-1} \circ (\mu ,z)$ satisfies the Herglotz equations

$$\begin{aligned} \begin{aligned}&\frac{d}{dt}\frac{\partial l}{\partial y^{a}}-\rho _{a}^{i}\frac{\partial l}{\partial q^{i}}+C_{a b}^{d}y^{b}\frac{\partial l}{\partial y^{d}}=\frac{\partial l}{\partial z}\frac{\partial l}{\partial y^{a}}, \\&\dot{q}^{i} = \rho ^{i}_{a}y^{a} \\&\dot{z}=l. \end{aligned} \end{aligned}$$

(12)

Proof

Consider local coordinates in $A\times \mathbb {R}$ and $A^{*}\times \mathbb {R}$ given by $(q^{i},y^{a},z)$ and $(q^{i},p_{a}, z)$, respectively. In theses coordinates the fiber derivative has the local expression

$$\begin{aligned} \mathbb {F}l(q^{i},y^{a},z) = \left( q^{i},\frac{\partial l}{\partial y^{a}},z\right) . \end{aligned}$$

Also, suppose that the image of the curve $(q^{i}(t), y^{a}(t), z(t))$ under $\mathbb {F}l$ is the curve $(q^{i}(t),p_{a}(t), z(t)$ and that the former satisfies the contact Hamiltonian equations for the function h.

Let us pullback the Jacobi structure on $A^{*}\times \mathbb {R}$ to the Lie algebroid $A\times \mathbb {R}$ using the fiber derivative. The Jacobi bracket $\{\cdot , \cdot \}$ of the pullback structure is determined by the relations

$$\begin{aligned} \begin{aligned}&\Big \{ \frac{\partial l}{\partial y^{a}},\frac{\partial l}{\partial y^{b}} \Big \} = C_{a b}^{d}\frac{\partial l}{\partial y^{d}}, \quad \Big \{ \frac{\partial l}{\partial y^{a}}, q^{i} \Big \} = \rho _{a}^{i} \\&\{q^{i}, q^{j} \}= 0 \quad \Big \{ \frac{\partial l}{\partial y^{a}}, z \Big \} = 0, \quad \{q^{i}, z \}= -q^{i}. \end{aligned} \end{aligned}$$

and the corresponding Reeb vector field R is the pullback of $\mathcal {R}= \displaystyle {\frac{\partial }{\partial z}}$ on $A^{*}\times \mathbb {R}$. Note that $R(q^{i})=\mathcal {R}(q^{i}) = 0$, $R(\frac{\partial l}{\partial y^{a}}) = \mathcal {R}(p_{a}) = 0$ and $R(z)=\mathcal {R}(z)=1$.

Next, we will compute the time derivatives $\dot{q}^{i}$, $\dot{z}$ and $\frac{d}{dt}\frac{\partial l}{\partial y^{a}}$ along the curve $(q^{i}(t), y^{a}(t), z(t))$. In the first place, we obtain that

$$\begin{aligned} \dot{q}^{i} = \{ E_{l}, q^{i} \} - q^{i}R(E_{l}) \end{aligned}$$

We have that

$$\begin{aligned} \{ E_{l}, q^{i} \} = \Big \{ y^{a}\frac{\partial l}{\partial y^{a}} - l, q^{i} \Big \} = y^{a}\Big \{\frac{\partial l}{\partial y^{a}}, q^{i} \Big \} + \frac{\partial l}{\partial y^{a}} \{y^{a}, q^{i} \} + y^{a}\frac{\partial l}{\partial y^{a}} R(q^{i}) - \{l, q^{i}\} \end{aligned}$$

and

$$\begin{aligned} R(E_{l}) = \frac{\partial l}{\partial y^{a}} R(y^{a}) - R(l). \end{aligned}$$

Now, from the definition of Jacobi bracket of two functions in terms of the Jacobi structure—together with the local expression of the Jacobi brackets of coordinate functions—we deduce that

$$\begin{aligned} \{l,q^{i}\} = \{y^{a},q^{i}\}\frac{\partial l}{\partial y^{a}} - q^{i}\frac{\partial l}{\partial y^{a}}R(y^{a}) + q^{i}R(l). \end{aligned}$$

Thus, we have that

$$\begin{aligned} \dot{q}^{i} = \rho _{a}^{i} y^{a}. \end{aligned}$$

Now, we also have that

$$\begin{aligned} \frac{d}{dt}\frac{\partial l}{\partial y^{a}} = \Big \{ E_{l}, \frac{\partial l}{\partial y^{a}} \Big \}-\frac{\partial l}{\partial y^{a}}R(E_{l}) \end{aligned}$$

Analogously,

$$\begin{aligned} \Big \{ E_{l}, \frac{\partial l}{\partial y^{a}} \Big \} = y^{b}\Big \{\frac{\partial l}{\partial y^{b}}, \frac{\partial l}{\partial y^{a}} \Big \} + \frac{\partial l}{\partial y^{b}} \Big \{y^{b}, \frac{\partial l}{\partial y^{a}} \Big \} + y^{b}\frac{\partial l}{\partial y^{b}} R\Big (\frac{\partial l}{\partial y^{a}}\Big ) - \Big \{l, \frac{\partial l}{\partial y^{a}}\Big \} \end{aligned}$$

Using the same reasoning as before,

$$\begin{aligned} \Big \{l, \frac{\partial l}{\partial y^{a}}\Big \} = -\rho _{a}^{i}\frac{\partial l}{\partial q^{i}} + \Big \{y^{b}, \frac{\partial l}{\partial y^{a}}\Big \}\frac{\partial l}{\partial y^{b}} - \frac{\partial l}{\partial y^{a}}\frac{\partial l}{\partial y^{b}} R(y^{b}) -\frac{\partial l}{\partial y^{a}}\frac{\partial l}{\partial z} + \frac{\partial l}{\partial y^{a}}R(l) \end{aligned}$$

Hence,

$$\begin{aligned} \begin{aligned} \frac{d}{dt}\frac{\partial l}{\partial y^{a}} = -C_{a b}^{d}y^{b}\frac{\partial l}{\partial y^{d}} + \rho _{a}^{i}\frac{\partial l}{\partial q^{i}} + \frac{\partial l}{\partial y^{a}}\frac{\partial l}{\partial z}. \end{aligned} \end{aligned}$$

Finally,

$$\begin{aligned} \dot{z} = \{E_{l}, z\} - z R(E_{l}). \end{aligned}$$

Therefore,

$$\begin{aligned} \{E_{l}, z\} = y^{b}\Big \{\frac{\partial l}{\partial y^{b}}, z \Big \} + \frac{\partial l}{\partial y^{b}} \{y^{b}, z \} + y^{b}\frac{\partial l}{\partial y^{b}} R(z) - \{l, z\} \end{aligned}$$

In addition, from

$$\begin{aligned} \{l, z\} = \{y^{b}, z\}\frac{\partial l}{\partial y^{b}} + y^{b}\frac{\partial l}{\partial y^{b}} - z\frac{\partial l}{\partial y^{b}}R(y^{b}) + zR(l) - l, \end{aligned}$$

we conclude that

$$\begin{aligned} \dot{z} = l. \end{aligned}$$

which finishes the proof. $\square$

Example 3.9

Note that in the case of the Lie alebroid $A=TQ$, equations (12) are just Herglotz equations [2, 4, 14, 15]

$$\begin{aligned} \begin{aligned}&\frac{d}{dt}\frac{\partial l}{\partial \dot{q}^{a}}-\frac{\partial l}{\partial q^{a}}=\frac{\partial l}{\partial z}\frac{\partial l}{\partial \dot{q}^{a}}, \\&\quad \dot{z}=l. \end{aligned} \end{aligned}$$

Moreover, in the case of the Lie algebroid $A=\mathfrak {g}$, equations (12) are the Euler–Poincaré–Herglotz equations

$$\begin{aligned} \begin{aligned}&\frac{d}{dt}\frac{\partial l}{\partial y^{a}}+C_{a b}^{d}y^{b}\frac{\partial l}{\partial y^{d}}=\frac{\partial l}{\partial z}\frac{\partial l}{\partial y^{a}}, \\&\quad \dot{z}=l. \end{aligned} \end{aligned}$$

Example 3.10

Let $\phi :Q\times G\rightarrow Q$ be an action of G on the manifold M where G is a Lie group. The induced anti-homomorphism between the Lie algebras $\mathfrak {g}$ and $\mathfrak {X}(Q)$ by the action is determined by $\Phi :\mathfrak {g}\rightarrow \mathfrak {X}(Q)$, $\xi \mapsto \xi _Q$, where $\xi _{Q}$ is the infinitesimal generator of the action for $\xi \in \mathfrak {g}.$

The vector bundle $\tau _{Q\times \mathfrak {g}}:Q\times \mathfrak {g}\rightarrow Q$ is a Lie algebroid over Q. The anchor map $\rho :Q\times \mathfrak {g}\rightarrow TQ$, is defined by $\rho (q,\xi )=\xi _{Q}(q)$ and the Lie bracket of sections is given by the Lie algebra structure on $\Gamma (\tau _{Q\times \mathfrak {g}})$ as

$$\begin{aligned} {[\![\hat{\xi },\hat{\eta }]\!]}_{Q\times \mathfrak {g}}(q)=(q,[\xi ,\eta ])=\widehat{[\xi ,\eta ]} (q) \end{aligned}$$

for $q\in Q$, where $\hat{\xi }(q)=(q,\xi )$, $\hat{\eta }(q)=(q,\eta )$ for $\xi ,\eta \in \mathfrak {g}$. The triple $(Q\times \mathfrak {g}, \rho , {[\![\cdot ,\cdot ]\!]}_{Q\times \mathfrak {g}})$ is called Action Lie algebroid.

The Euler–Lagrange–Herglotz equations on the action Lie algebroid $A=Q\times \mathfrak {g}$ are

$$\begin{aligned} \begin{aligned}&\frac{d}{dt}\frac{\partial l}{\partial \xi ^{a}}-\rho _{a}^{i}\frac{\partial l}{\partial q^{i}}+C_{a b}^{d}\xi ^{b}\frac{\partial l}{\partial \xi ^{d}}=\frac{\partial l}{\partial z}\frac{\partial l}{\partial \xi ^{a}}, \\&\quad \dot{q}=\xi _{Q}(q), \\&\quad \dot{z}=l. \end{aligned} \end{aligned}$$

(13)

4 Examples

4.1 Triaxial attitude control testbed

Motivated by the rigid body attitude dynamics studied in [9, 38] for the triaxial attitude control testbed given in [3], we will obtain Euler–Lagrange–Herglotz equations for this system seen as an action Lie algebroid. The model in a particular case of study is just the heavy top with a linear dissipation term as we describe in the following. Consider the Lie algebra $\mathfrak {g}=\mathfrak {so}(3)$ and the manifold $M=S^2$. An element of M will be considered as an unit vector $\chi$ in $\mathbb {R}^{3}$, representing the direction of the gravity as measured in the body coordinate frame, and an element of $\mathfrak {so}(3)$ will also be considered as a vector $\xi$ in $\mathbb {R}^{3}$, representing the angular velocity in body coordinates.

The anchor map is $\rho (\chi ,\xi )=(\chi ,\xi \times \chi )\in T_{\chi }S^2$ and the Lie bracket is ${[\![\xi _1,\xi _2]\!]}=\xi _1\times \xi _2$. The Lagrangian function is given by

$$\begin{aligned} l(\chi ,\xi )=\frac{1}{2}\langle \mathbb {I}\xi ,\xi \rangle -mg\ell \chi \cdot \rho -\gamma z, \end{aligned}$$

$\mathbb {I}: \mathfrak {so}(3)\rightarrow \mathfrak {so}(3)^{*}$ is the inertia tensor of the top (it is calculated with respect to the pivot, which is not, in general, the center of mass), m is the mass of the body, g is the acceleration due to gravity, $\rho \in \mathbb {R}^{3}$ is the position of the center of mass which is a constant vector in the body coordinate frame, $\chi \in \mathbb {R}^{3}$ is the direction of gravity as seen in the body coordinate frame, $\ell$ is the length of the line segment between the center of mass and the contact point of the heavy top with the ground and $\gamma \in \mathbb {R}$ is a dissipative term. By Example 3.10, the Euler–Lagrange–Herglotz equations on the action Lie algebroid $A=S^2\times \mathfrak {so}(3)$ are

$$\begin{aligned} \mathbb {I}\dot{\xi }+\xi \times \mathbb {I}\xi&=mg\ell \rho \times \chi -\gamma \mathbb {I}\xi ,\\ \dot{\chi }&=\xi \times \chi ,\\ \dot{z}&=l. \end{aligned}$$

In the case where the dissipation matrix C from [38] (see section 3 in [38]) is $\gamma \mathbb {I}$, we obtain the equations for rigid body dynamics studied in [38] and [9] for the triaxial attitude control testbed given in [3].

4.2 Atiyah algebroid and dissipative Wong’s equations

Let G be a Lie group and we assume that G acts freely and properly on Q. We denote by $\pi :Q\rightarrow \widehat{Q}=Q/G$ the associated principal bundle. The tangent lift of the action gives a free and proper action of G on TQ and $\widehat{TQ}=TQ/G$ is a quotient manifold. The quotient vector bundle $\tau _{\widehat{TQ}}:\widehat{TQ}\rightarrow \widehat{Q}$ where $\tau _{\widehat{TQ}}([v_q])=\pi (q)$ is a Lie algebroid over $\widehat{Q}.$ The fiber of $\widehat{TQ}$ over a point $\pi (q)\in \widehat{Q}$ is isomorphic to $T_{q}Q.$

The Lie bracket is defined on the space $\Gamma (\tau _{\widehat{TQ}})$ which is isomorphic to the Lie subalgebra of G-invariant vector fields, that is,

$$\begin{aligned} \Gamma (\tau _{\widehat{TQ}})=\{X\in \mathfrak {X}(Q)\mid X \hbox { is } G-\hbox { invariant }\}. \end{aligned}$$

Thus, the Lie bracket on $\widehat{TQ}$ is the bracket of G-invariant vector fields. The anchor map $\rho :\widehat{TQ}\rightarrow T\widehat{Q}$ is given by $\rho ([v_q])=T_{q}\pi (v_q).$ Moreover, $\rho$ is a Lie algebra homomorpishm satisfying the compatibility condition since the G-invariant vector fields are $\pi$-projectable. This Lie algebroid is called Lie-Atiyah algebroid associated with the principal bundle $\pi :Q\rightarrow \widehat{Q}.$

Let $\mathcal {A}:TQ\rightarrow \mathfrak {g}$ be a principal connection in the principal bundle $\pi :Q\rightarrow \widehat{Q}$ and $B:TQ\oplus TQ\rightarrow \mathfrak {g}$ be the curvature of $\mathcal {A}.$ The connection determines an isomorphism $\alpha _{\mathcal {A}}$ between the vector bundles $\widehat{TQ}\rightarrow \widehat{Q}$ and $T\widehat{Q}\oplus \widetilde{\mathfrak {g}}\rightarrow \widehat{Q}$, where $\widetilde{\mathfrak {g}}=(Q\times \mathfrak {g})/G$ is the adjoint bundle associated with the principal bundle $\pi :Q\rightarrow \widehat{Q}$ (see [8] for example).

We choose a local trivialization of the principal bundle $\pi :Q\rightarrow \widehat{Q}$ to be $U\times G,$ where U is an open subset of $\widehat{Q}.$ Suppose that e is the identity of G, $(q^{i})$ are local coordinates on U and $\{\xi _{A}\}$ is a basis of $\mathfrak {g}.$

Suppose that $\{\xi _{A}\}$ is a basis of $\mathfrak {g},$ and consider the G-invariant vector fields $\{e_{A}\}$ on $U\times G$ given by

$$\begin{aligned} e_{A}(q,g)=(ad_{g} \xi _A)_Q (q,g), \end{aligned}$$

where $ad_{g}: \mathfrak {g}\rightarrow \mathfrak {g}$ is the adjoint action and $\xi _{Q}$ is the fundamental vector on Q associated with $\xi \in \mathfrak {g}$. If

$$\begin{aligned} \mathcal {A}\left( \frac{\partial }{\partial q^{i}}\Big |_{(q,e)}\right) =\mathcal {A}_{i}^{A}(q)\, \xi _{A},\quad \mathcal {B}\left( \frac{\partial }{\partial q^{i}}\Big |_{(q,e)},\frac{\partial }{\partial q^{j}}\Big |_{(q,e)}\right) =\mathcal {B}_{ij}^{A}\,(q)\xi _{A}, \end{aligned}$$

for $i,j= 1,\ldots , \dim (Q)-\dim (G)$, and $q\in U,$ then the horizontal lift of the vector field $\frac{\partial }{\partial q^{i}}$ is the vector field on $\pi ^{-1}(U)\simeq U\times G$ given by

$$\begin{aligned} e_i = \left( \frac{\partial }{\partial q^{i}}\right) ^{h}=\frac{\partial }{\partial q^{i}}-\mathcal {A}_{i}^{A} \, (\xi _{A})_Q. \end{aligned}$$

Therefore, the vector fields $\{e_{i}, e_{A} \}$ on $U\times G$ are G-invariant under the action of G over Q and define a local basis on $\Gamma (\widehat{TQ})=\Gamma (\tau _{T\widehat{Q}\oplus \tilde{\mathfrak {g}}})$, which induces local coordinates $(q^{i},\dot{q}^{i},v^{A})$ on TQ/G.

The corresponding local structure functions of $\tau _{\widehat{TQ}}:\widehat{TQ}\rightarrow \widehat{Q}$ are

$$\begin{aligned} C_{ij}^{k}= & {} C_{iA}^{j}=-C_{Ai}^{j}=C_{AB}^{i}=0,\quad C_{ij}^{A}=-\mathcal {B}_{ij}^{A},\quad C_{iA}^{C}=-C_{Ai}^{C}=c_{AB}^{C}\mathcal {A}_{i}^{B},\\ C_{AB}^{C}= & {} c_{AB}^{C},\quad \rho _{i}^{j}=\delta _{ij},\quad \rho _{i}^{A}=\rho _{A}^{i}=\rho _{A}^{B}=0, \end{aligned}$$

being $\{c_{AB}^{C}\}$ the constant structures of $\mathfrak {g}$ with respect to the basis $\{\xi _{A}\}$ (see [16] for more details). That is,

$$\begin{aligned}{}[e_{i},e_{j}]_{\widehat{TQ}}= & {} -\mathcal {B}_{ij}^{C}e_{C},\quad [e_{i},e_{A}]_{\widehat{TQ}}=c_{AB}^{C}\mathcal {A}_{i}^{B}e_{C},\quad [e_{A},e_{B}]_{\widehat{TQ}}=c_{AB}^{C} e_{C}, \\{} & {} \qquad \rho _{\widehat{TQ}}(e_{i})=\frac{\partial }{\partial q^{i}},\quad \rho _{\widehat{TQ}}(e_{A})=0. \end{aligned}$$

4.2.1 Lagrange–Poincaré–Herglotz equations

On this section, consider the local coordinates $(q^{i},\dot{q}^{i},v^{B})$ on $\widehat{TQ}=TQ/G$ induced by the basis $\{e_{i},e_{B}\}$.

Given a reduced contact Lagrangian function $\ell :\widehat{TQ}\times \mathbb {R}\rightarrow \mathbb {R}$ associated with the Atiyah algebroid $\widehat{TQ}\rightarrow \widehat{Q},$ the Euler–Lagrange equations for $\ell$ are given by

$$\begin{aligned} \frac{\partial \ell }{\partial q^{j}}-\frac{d}{dt}\left( \frac{\partial \ell }{\partial \dot{q }^{j}}\right)&=\frac{\partial \ell }{\partial v^{A}}\left( \mathcal {B}_{ij}^{A}\dot{q }^{i}+c_{DB}^{A}\mathcal {A}_{j}^{B} v^{B}\right) - \frac{\partial \ell }{\partial z}\frac{\partial \ell }{\partial \dot{q }^{j}} \quad \forall j,\\ \frac{d}{dt}\left( \frac{\partial \ell }{\partial v^{B}}\right)&=\frac{\partial \ell }{\partial v^{A}}\left( C_{DB}^{A} v^{D}-c_{DB}^{A}\mathcal {A}_{i}^{D}\dot{q }^{i}\right) + \frac{\partial \ell }{\partial z}\frac{\partial \ell }{\partial v^{B}}\quad \forall B, \end{aligned}$$

which are the Lagrange–Poincaré–Herglotz equations associated to a G-invariant Lagrangian $L:TQ\times \mathbb {R}\rightarrow \mathbb {R}$ (see [8] and [16] for example).

4.2.2 Dissipative Wong’s equations

To illustrate the theory that we have developed in this section, we will consider Wong’s equations. Wong’s equations arise in the dynamics of a charged particle in a Yang-Mills field and the falling cat theorem (see [39]; and also [8] and references therein). Our framework allows to include dissipative forces in these models.

Let $(M, g_{M})$ be a given Riemannian manifold, G be a compact Lie group with a bi-invariant Riemannian metric $\kappa$ and $\pi : Q \rightarrow M$ be a principal bundle with structure group G. Suppose that ${\mathfrak g}$ is the Lie algebra of G, that $\mathcal {A}: TQ \rightarrow {\mathfrak g}$ is a principal connection on Q and that $B: TQ \oplus TQ \rightarrow {\mathfrak g}$ is the curvature of $\mathcal {A}$. If $q \in Q$ then, using the connection $\mathcal {A}$, one may prove that the tangent space to Q at q, $T_{q}Q$, is isomorphic to the vector space ${\mathfrak g} \oplus T_{\pi (q)}M$. Thus, $\kappa$ and $g_{M}$ induce a Riemannian metric $g_{Q}$ on Q and we can consider the kinetic contact energy $L: TQ \times \mathbb {R}\rightarrow \mathbb {R}$ associated with $g_{Q}$. The Lagrangian L is given by

$$\begin{aligned} L(v_{q}, z) = \displaystyle \frac{1}{2}( \kappa _{e}(\mathcal {A}(v_{q}), \mathcal {A}(v_{q})) + (g_{M})_{\pi (q)}((T_{q}\pi )(v_{q}), (T_{q}\pi )(v_{q}))) - \gamma z, \end{aligned}$$

for $v_{q} \in T_{q}Q$, e being the identity element in G. It is clear that L is hyperregular and G-invariant.

On the other hand, since the Riemannian metric $g_{Q}$ is also G-invariant, it induces a fiber metric $g_{TQ/G}$ on the quotient vector bundle $\tau _{Q}|G: TQ/G \rightarrow M = Q/G$. The reduced contact Lagrangian $\ell : TQ/G \times \mathbb {R}\rightarrow \mathbb {R}$ is just the kinetic energy of the fiber metric $g_{TQ/G}$, that is,

$$\begin{aligned} \ell ([v_{q}], z) = \displaystyle \frac{1}{2} (\kappa _{e}(\mathcal {A}(v_{q}), \mathcal {A}(v_{q})) + (g_{M})_{\pi (q)}((T_{q}\pi )(v_{q}), (T_{q}\pi )(v_{q}))) - \gamma z,\nonumber \\ \end{aligned}$$

(14)

for $v_{q} \in T_{q}Q$.

We have that $\ell$ is hyperregular. In fact, the Legendre transformation associated with $\ell$ is the map $([v_{q}], z)\mapsto (\flat _{g_{TQ/G}}([v_{q}]),z)$, where $\flat _{g_{TQ/G}}$ is the vector bundle isomorphism between TQ/G and $T^{*}Q/G$ induced by the fiber metric $g_{TQ/G}$.

Now, we choose a local trivialization of $\pi : Q \rightarrow M$ to be $U \times G$, where U is an open subset of M such that there are local coordinates $(x^{i})$ on U. Suppose that $\{\xi _{A}\}$ is a basis of ${\mathfrak g}$, that $c_{AB}^{D}$ are the structure constants of ${\mathfrak g}$ with respect to the basis $\{\xi _{A}\}$, that $\mathcal {A}_{i}^{A}$ (respectively, $B_{ij}^{A}$) are the components of $\mathcal {A}$ (respectively, B) with respect to the local coordinates $(x^{i})$ and Lie algebra basis $\{\xi _{A}\}$, and that

$$\begin{aligned} \kappa _{e} = \kappa _{AB} \xi ^{A} \otimes \xi ^{B}, \text{[ }.4cm]{} g_{M} = g_{ij} dx^{i} \otimes dx^{j}, \end{aligned}$$

where $\{\xi ^{A}\}$ is the dual basis to $\{\xi _{A}\}$. Note that since $\kappa$ is a bi-invariant metric on G, it follows that

$$\begin{aligned} c_{AB}^{D}\kappa _{DE} = c_{AE}^{D}\kappa _{DB}. \end{aligned}$$

(15)

Denote by $\{e_i, \widehat{\xi }_A\}$ the local basis of G-invariant vector fields on Q, and by $(x^i,\dot{x}^i,v^A, z)$ the corresponding local fibred coordinates on $TQ/G \times \mathbb {R}$. We have that

$$\begin{aligned} \ell (x^i,\dot{x}^i,v^A, z) = \displaystyle \frac{1}{2} (\kappa _{AB} v^A v^B + g_{ij} \dot{x}^i \dot{x}^j) - \gamma z, \end{aligned}$$

(16)

Thus, the Hessian matrix of $\ell$, $W_{\ell }$, is

$$\begin{aligned} \left( \begin{array}{ll} g_{ij}&{}\quad 0\\ 0&{}\quad \kappa _{AB}\end{array}\right) . \end{aligned}$$

The Lie algebroid Lagrange–Poincaré–Herglotz equations for the contact Lagrangian function $\ell$ are given by

$$\begin{aligned} \frac{\partial g_{im}}{\partial q^{j}}\dot{x}^{i}\dot{x}^{m}-\frac{\partial g_{ij}}{\partial q^{k}}\dot{x}^{k}\dot{x}^{i} - g_{ij}\ddot{x}^{i}&=\kappa _{AB} v^{B}\left( \mathcal {B}_{ij}^{A}\dot{x}^{i}+c_{DB}^{A}\mathcal {A}_{j}^{B}v^{B}\right) + \gamma g_{ij}\dot{x}^{i} \end{aligned}$$

(17)

$$\begin{aligned} \kappa _{AB}\dot{v}^{A}&=\kappa _{AE}v^{E}\left( C_{DB}^{A}v^{D}-c_{DB}^{A}\mathcal {A}_{i}^{D}\dot{x}^{i}\right) - \gamma \kappa _{AB}v^{A}. \end{aligned}$$

(18)

4.2.3 Hamilton–Poincaré–Herglotz equations

Given a reduced contact Hamiltonian function $h:T^{*}Q/G \times \mathbb {R}\rightarrow \mathbb {R}$ associated with the Atiyah algebroid $\widehat{TQ}\rightarrow \widehat{Q}$, let $\{e_i, \widehat{\xi }_A\}$ be the local basis of G-invariant vector fields on Q, and $(q^i,\dot{q}^i,v^A)$ be the corresponding local fibred coordinates on TQ/G. Then, denote by $(q^i,p_i,\bar{p}_A)$ the (dual) coordinates on $T^*Q/G$ and $(q^i,p_i,\bar{p}_A,z)$ the corresponding coordinates on $T^*Q/G \times \mathbb {R}$.

In these coordinates, the contact Hamiltonian equations are given by

$$\begin{aligned} \begin{aligned}&\dot{q}^{i} = \frac{\partial h}{\partial p_{i}}, \quad \dot{p}_{i} = -\frac{\partial h}{\partial q^{i}} + B_{ij}^A \bar{p}_A \frac{\partial h}{\partial p_{j}} - c_{AB}^{C}\mathcal {A}_{i}^{B}\bar{p}_C \frac{\partial h}{\partial \bar{p}_{A}} - p_{i}\frac{\partial h}{\partial z} \\&\dot{\bar{p}}_{A} = c_{AB}^{C}\mathcal {A}_{i}^{B}\bar{p}_C \frac{\partial h}{\partial p_{i}} - c_{AB}^{C} \bar{p}_C \frac{\partial h}{\partial \bar{p}_{B}} - \bar{p}_{A}\frac{\partial h}{\partial z}, \quad \dot{z}= p_{i}\frac{\partial h}{\partial p_{i}} + \bar{p}_{A}\frac{\partial h}{\partial \bar{p}_{A}} - h \end{aligned} \end{aligned}$$

Given the reduced Lagrangian (14), the corresponding reduced Hamiltonian $h: T^{*}Q/G \times \mathbb {R}\rightarrow \mathbb {R}$ is given by

$$\begin{aligned} h([\alpha _{q}], z) = E_{\ell }(\flat _{g_{TQ/G}}^{-1}[\alpha _{q}], z), \end{aligned}$$

for $\alpha _{q} \in T^{*}_{q}Q$, where $E_{\ell }$ is the Lagrangian energy function (11). In local coordinates,

$$\begin{aligned} h(x^i,p_i,\bar{p}_A, z) = \displaystyle \frac{1}{2} (\kappa ^{AB} \bar{p}_A \bar{p}_B + g^{ij} p_i p_j) + \gamma z, \end{aligned}$$

(19)

where $(\kappa ^{AB})$ (respectively, $(g^{ij})$) is the inverse matrix of $(\kappa _{AB})$ (respectively, $(g_{ij})$).

The inverse of the Hessian matrix $W_{\ell }$ is

$$\begin{aligned}\left( \begin{array}{ll} g^{ij}&{}0\\ 0&{}\kappa ^{AB}\end{array}\right) . \end{aligned}$$

The contact Hamiltonian equations for the contact Hamiltonian function h are given by

$$\begin{aligned} \begin{aligned}&\dot{q}^{i} =g^{ij}p_j, \quad \dot{p}_{i} = -\frac{1}{2} \frac{\partial g^{jk}}{\partial q^i} p_j p_k + B_{ij}^A \bar{p}_A g^{jk}p_{k} - c_{AB}^{C}\mathcal {A}_{i}^{B}\bar{p}_C \kappa ^{AB} \bar{p}_B - \gamma p_{i}, \\&\dot{\bar{p}}_{A} = c_{AB}^{C}\mathcal {A}_{i}^{B}\bar{p}_C g^{ij}p_j \!-\! c_{AB}^{C} \bar{p}_C \kappa ^{BD} \bar{p}_D\! -\! \gamma \bar{p}_{A}, \!\!\quad \dot{z}\!=\! \frac{1}{2} (\kappa ^{AB} \bar{p}_A \bar{p}_B \!+\! g^{ij} p_i p_j) - \gamma z. \end{aligned} \end{aligned}$$

5 Conclusions

In this paper, we introduce Herglotz equations on Lie algebroids. More precisely, for a Lie algebroid A, we introduce a Jacobi structure on the product manifold $A^{*}\times \mathbb {R}$ (using the linear Poisson structure on the dual bundle $A^{*}$ associated with the Lie algebroid structure on A). When A is the standard Lie algebroid TQ, the Jacobi structure on $T^{*}Q\times \mathbb {R}$ is just the canonical contact structure.

Moreover, in the presence of a Hamiltonian function $h:A^{*}\times \mathbb {R}\rightarrow \mathbb {R}$, we consider the corresponding dynamical system on $A^{*}\times \mathbb {R}$. In addition, if the Hamiltonian function is regular then a Lagrangian dynamics, induced by a Lagrangian function $l: A\times \mathbb {R}\rightarrow \mathbb {R}$, may be considered on the space $A\times \mathbb {R}$. The corresponding dynamical equations are the so-called Herglotz equations for the Lagrangian function l. When A is the standard Lie algebroid TQ, these equations are just the classical Herglotz equations that have been discussed previously in the literature (see for instance [2] and references therein). All these constructions are used to model dissipative mechanical systems on $A\times \mathbb {R}$ and $A^{*}\times \mathbb {R}$.

Some particular examples are discussed. In particular we first particularize our construction to the case of Lie algebras, centered semidirect products Lie groups (action Lie algebroids) and the Atiyah algebroid obtaining the Euler–Poincaré–Herglotz equations and Lagrange–Poincaré–Herglotz equations, respectively (correspondingly Lie–Poisson–Jacobi equations and Hamilton–Poincaré–Herglotz equations in the Hamiltonian side) and we derive, from a geometric viewpoint, the contact Wong’s equations from a Lagrangian and Hamiltonian viewpoint. In addition, we show as an example of action Lie algebroid the dynamical model for a triaxial attitude control testbed.