# On the intrinsic approaches in mechanics

Jean Lerbet

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Jean Lerbet. On the intrinsic approaches in mechanics. 2025. hal-05079917

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# On the intrinsic approaches in mechanics

J. Lerbet

LaMME, UFRST-UEVE, 40, rue du Pelvoux CE 1455 91020 Evry Courcouronnes cedex, France

## Introduction

This paper aims to illustrate the interest of so-called intrinsic or coordinate-free or geometric approaches in mechanics and to start a discussion about their positions with respect to the opposite aspects carried by computational mechanics. Due to recent developments of pure computational approaches using IA that could finally suggest that equations and formal developments are not necessarily so important in mechanics, it could obviously lead to philosophical and epistematical issues. Our main point of view can be summarized by the well-known sentence of René Thom: "To predict is not to explain". Even if we consider that stressing these questions in the community of mechanics should be fundamental and essential, it will not be however the theme of the present work. On the contrary, we will remain technic in order to defend the idea that intrinsic approaches are actually complementary to computational approaches in mechanics and that, even if these approaches may involve more abstract objects and less usual mathematical concepts, they can lead in fact to practically more efficient developments in applications. This thesis then defends the unity of mechanics and the non-sense to hermetically separate theoretical, computational and applied aspects in mechanics.

Let us start by an example and let us consider a mechanical system constituted for example by 5 bodies moving in the space and subjected by a family of kinematic constraints (joints between the bodies) so that the degree of freedom is for example 21. Suppose that some elements of the system are not well known and that some parametric investigations will be necessary regarding for example the inertia operators of some of the 5 bodies. Suppose also that some external given actions are not conservative and that internal actions (joint interactions) are not perfectly known. Which way can be used for studying this problem? The first remark is that it seems impossible to write by hand the 21 independent second order differential equations of the dynamics by Newton-Euler equations ($5 \times 6 = 30$ scalar equations + Actions-reactions principle + constitutive laws of the joints) or the 21 Euler-Lagrange equations as well. For these last equations, the expression of the kinetic energy will be too big and unusable by mixing all the inertia of the different bodies and the loss of a potential energy will remove one of the greatest advantages consisting in concentrating the dynamics in a unique scalar function (the Lagrangian of the system). Since the system is not necessarily a usual robot, the machinery developed in robotics to parametrize the system cannot be directly used.

The approach usually taught (Newton-Euler or Lagrange as well) consists first by introducing a lot of coordinate systems in the (affine) space $\mathcal{E}$ to describe the geometry of the mechanical system (positions of some points of $\mathcal{E}$ and orientations of some bases of the vector space $\mathcal{E}$ associated to the affine space $\mathcal{E}$). In a second time a lot a derivative computations are performed to build the quantities that are introduced in a next and last time in the equations translating the principles of mechanics and leading to very expanded expressions. The additional mechanical characteristics of the system that have to be taken account like the constitutive laws, are then translated into very expanded expressions due to the initial coordinates representations and calculations. Moreover, when "models" of the mechanical system are presented they are often
presented as equations like

$$M(q)\ddot{q} + N(q, \dot{q})\dot{q} + G(q) + F(q, \dot{q}) = 0 \quad (1)$$

with $M(q)$ the mass matrix, $N(q, \dot{q})$ the matrix containing Coriolis and centrifugal forces, $G(q)$ the vector of positional forces and $F(q, \dot{q})$ a vector describing for example damping forces. A similar expression is the basic relation coming from FE methods to investigated dynamics issues for continuous systems as non linear modes computation. Such a formal expression is however only numerically calculated at each step (of the time interval or of the loading path) before solving it for the step thanks to a linear development. In fact, these objects ($M(q)$, $N(q, \dot{q})$, $G(q)$, $F(q, \dot{q})$) are not explicitly calculated nor used as functions of the variables $q$. In fact, such an explicit expression is rather used in the dual problem of the control of the mechanical system when the trajectory $t \mapsto q(t)$ is prescribed and when motor actions in the links must be calculated to follow as good as possible the prescribed trajectory. Moreover, for such problems, the sensibility of the system with respect to some parameters is a fundamental question that lead for example to make derivatives of $M(q)$, $N(q, \dot{q})$, ... with respect to these parameters. Doing it explicitly necessitates to have in hand the most compact but explicit expressions of these quantities. Unfortunately, the usual approaches do not provide such compact but explicit expressions since they carry with them the artificial complexity due to the use of coordinates systems to represent any quantity by a family of real numbers. Roughly speaking, a coordinate-free approach proposes to report this use of coordinate systems at the end of the process and not at its beginning and to process to all derivative calculations (kinematics, dynamics, sensibility, etc) before introducing coordinate. Of course, the first contribution of intrinsic approaches is to identify the basic concepts linked to the investigated problems and to focus on the properties of these concepts without being diverted by useless computational developments that often could hidden the keys of the investigated problem. A second direct contribution (and often the most remarkable) is to solve a question that pure computational approaches could not definitively deal with. The third contribution most rarely emphasized is the practical aspects of the intrinsic approaches and the aid they can bring to computational methods.To illustrate our position we will consider two very different kinds of applications of geometric concepts in mechanics. The first one deals with calculations on Lie groups whereas the second one with calculations in vector bundles. We will give some (recent and "old") results provided by such approaches and calculations in such spaces motivated by multi-body or continuous system dynamics for the Lie group approaches and by granular mechanics for the vector bundle concept.

# 1 Lie group approaches

## 1.1 An adapted coordinate-free approach

We first highlight the main elements of a coordinate-free approach by using calculations on Lie groups. These elements are completely developed in several publications but synthesized in the book [4].

1. the set $D(\mathcal{E})$ of affine euclidean displacements of the 3-dimension affine space $\mathcal{E}$ is a 6-dimensional Lie group $D(\mathcal{E})$. Its Lie algebra $D(\mathcal{E})$ is isomorphic to (and identified with) the set of skew-symmetric fields on $\mathcal{E}$ (with values on $E$ the usual 3-dimension vector space associated to $\mathcal{E}$). There is on $D(\mathcal{E})$ a (3, 3) bilinear symmetric form invariant by the adjoint action of the Lie group denoted Ad of $D(\mathcal{E})$. It is the Klein form $[.,]$ that allows to intrinsically identify the dual $D(\mathcal{E})^*$ (describing the force fields acting on a rigid body) with $D(\mathcal{E})$ describing the velocity fields of a rigid body (see hereafter). The set $\mathfrak{T}(\mathcal{E})$ of translations is an ideal of $D(\mathcal{E})$ and is especially involved in the analysis of observation frames.
2. the so-called free and transitive action denoted • of the 6-dimensional Lie group D(E) on the configuration space S_S of any rigid body S allows us to endowed this space to a differential manifold equipped with the structure of homogeneous space. In fact, this structure is exactly equivalent to the rigidity hypothesis for S. This mathematical structure also allows to develop a complete but explicit coordinate-free kinematics for the body S which is entirely described by the Maurer-Cartan form θ : TS_S → D(E).

3. the additional hypothesis necessary to deal with kinetics and dynamics is related with a Riemannian structure on S_S that is equivalent to a given field s ∈ S_S ↦ H_s ∈ D(E) of linear operators of the Lie algebra D(E) of D(E) with some well-defined algebraic properties like symmetry with respect to the Klein form [., .]. This linear operator translates the way to pass from the kinematics object namely the velocity field V(t) = θ(\dot{s}(t)) to the kinetic momentum H_{s(t)}(V(t)) when the motion of S is t ↦ s(t) plotted on S_S.

4. the three previous items allow to deal with kinematics and dynamics of a single rigid body provided we give the adapted statements of the Newton-Euler laws namely the principles of mechanics!! In order to extend this full intrinsic calculation from a single rigid body S to a rigid bodies system Σ = {S_i, i = 1, ..., n}, natural hypotheses are necessary. To preserving the Lie group approach leads to suppose that the links between two any pair {S_i, S_j} of Σ in respective configurations s_i and s_j is described by an element D_{ij} of a Lie subgroup L(s_i, s_j) of D(E). Such an assumption describes in fact a large part of usual links in rigid body systems excepted for example pure contact links.

With these tools, we can define an observational frame $\mathcal{F}$ and the way to pass from any frame $\mathcal{F}_1$ to any other $\mathcal{F}_2$. We also may define the class $\mathcal{Gal}$ of Galilean or inertial frames. The principle of mechanics then (time denoted $t$ is supposed to be absolute) has two sides: the first one is a statement valid for any observational frame, the second one deals with the class $\mathcal{Gal}$.

1. if S is a rigid body with S_S as configuration space, if F is any frame of observation of the motion t ↦ s(t) of S, if the external actions observed with respect to F are described by an element F_{ext}(t) ∈ D(E) and if F_{in}(t) denotes the element of D(E) describing the inertial actions on S due to its motion with respect to F then for all t and :

$$F_{in}(t) + F_{ext}(t) = 0 \qquad (2)$$

2. if $\mathcal{F} \in \mathcal{Gal}$ is any inertial frame for the observation of the motion $t \mapsto s(t)$ of S then

$$F_{in}(t) = -\frac{d}{dt}(H_{s(t)}(V(t))) = -H_{s(t)}(\dot{V}(t)) - [V(t), H_{s(t)}(V(t))] \quad (3)$$

where $\theta(\dot{s}(t)) = V(t)$ for all $t$.

## 1.2 Three remarkable results

To illustrate the power of such an intrinsic approach, we now present three results that have been obtained thanks to this Lie group geometric approach.

### 1.2.1 Objectivity of inertial actions

The first result we have chosen to present highlights the deep meaning of the objectivity of inertial forces and the associated properties of the "mass" of a body in order that the associated
inertial force is an objective quantity. The main references are [5] and more recently [4].

Let $\mathcal{F}_1$ and $\mathcal{F}_2$ be two any observation frames of the motion of a rigid body $S$. They induce two maps $t \mapsto s_i(t)$ in $\mathbb{S}_S$ for $i = 1, 2$ so that it defines a $\mathcal{C}^\infty$ map $t \in \mathbb{R} \mapsto A_{12}(t) \in \mathbb{D}(\mathcal{E})$ with $s_1(t) = A_{12}(t) \bullet s_2(t)$ for all $t$. A mechanical quantity attached to $S$ and described by elements $X_i : t \mapsto X_i(t) \in \mathcal{D}(\mathcal{E})$ when the motion of $S$ is observed with respect to the frames $\mathcal{F}_i$ for $i = 1, 2$ is then said to be objective or frame-indifferent if they are related by $X_1(t) = \mathrm{Ad}(A_{12}(t))X_2(t)$ for all $t$. This definition is a mathematical translation of philosophical ideas presented by Noll ( [18] for example) and other great authors but mainly for continuum mechanics and surprisingly never or few investigated for inertial actions. For example, the velocity field $V$ of $S$ is not an objective quantity (in fact a calculation shows that $V_2(t) = \mathrm{Ad}(A_{12}(t))(V_1(t) + U_{12}(t))$ with $U_{12}(t) = \theta_\ell(\dot{A}_{12}(t))$ where $\theta_\ell$ is the left-Maurer-Cartan form of the Lie group $\mathbb{D}(\mathcal{E})$). On the contrary, any force $F$ acting on $S$ is (must be!) an objective quantity. We stress that objectivity is a fundamental property in mechanics and is related to any change of frames and not only of inertial frames. We then now question the objectivity of the inertial forces. The question is then: are the forces defined in Galilean frames by the expression (3) objective quantities?

Keep in mind that until now the kinetics of a rigid body $S$ is not defined from a model of the body (distribution of particles for example that necessitates to accept an hypothetical object without volume but with a mass) but thanks to a field $s \in \mathbb{S}_S \mapsto H_s \in \mathcal{L}(\mathcal{D}(\mathcal{E}))$ that produces, from the velocity field $V(t)$ at time $t$, the kinetic momentum $H_{s(t)}(V(t))$ of the body $S$ ($t \mapsto s(t)$ is the motion and $\theta(\dot{s}(t)) = V(t)$ is the velocity field of the body). The general properties of this field are

1. for all $s \in \mathbb{S}_S$, $H_s$ is a symmetric definite positive (for the Klein form $[\cdot, \cdot]$) operator of the Lie algebra (mathematical translation of the fundamental property of the kinetic energy)

2. the map $s \mapsto H_s$ is equivariant relatively to the left action $\bullet$: for all $s \in \mathbb{S}_S$ and $A \in \mathcal{D}(\mathcal{E})$, $H_{A \bullet s} = \mathrm{Ad}A \circ H_s \circ \mathrm{Ad}A^{-1}$ (mathematical translation of the fact that the "mass" of the rigid body rigidly follows the body during any motion)

We introduce the following field $s \mapsto C_s$ of bilinear equivariant maps $C_s$ of $\mathcal{L}(\mathcal{D}(\mathcal{E}))$:

$$C_s(X, Y) = [X, H_s(Y)] + [Y, H_s(X)] + H_s([X, Y]) \quad \forall s \in \mathbb{S}_S, \forall X, Y \in \mathcal{D}(\mathcal{E}) \quad (4)$$

The equivariant property of $s \mapsto C_s$ comes from the one of $s \mapsto H_s$ and the fact that the adjoint map is a Lie algebra representation. It reads $C_{A \bullet s}(X, Y) = C_s(\mathrm{Ad}A^{-1}(X), \mathrm{Ad}A^{-1}(Y))$ for all $s \in \mathbb{S}_S$, $A \in \mathcal{D}(\mathcal{E})$ and $X, Y \in \mathcal{D}(\mathcal{E})$.

The deep result is:

**Theorem 1** Inertial forces acting on a rigid body $S$ are objective, if and only if the field of inertia operators $s \in \mathbb{S}_S \mapsto H_s \in \mathcal{L}(\mathcal{D}(\mathcal{E}))$ is such that:

$$\forall X \in \mathcal{T}(\mathcal{E}), Y \in \mathcal{D}(\mathcal{E}), s \in \mathbb{S}_S \quad C_s(X, Y) = 0 \quad (5)$$

and by equivariance property, it is sufficient that this property holds for only one configuration $r \in \mathbb{S}_S$.

A "simple" calculation then shows that, when the field of inertia operators $s \in \mathbb{S}_S \mapsto H_s \in \mathcal{L}(\mathcal{D}(\mathcal{E}))$ comes from a pre-defined mass distribution (for example a pre-defined volume density of mass), the objectivity condition (5) then holds. Obviously, such an analysis is in fact absolutely impossible without using such an intrinsic language allowing to focus only on the essential objects and to highlight the deep rule of the operators $C_s$.
### 1.2.2 Full description of singularities of mechanisms

This second illustration of the power of the intrinsic geometric Lie group approach deals with pure kinematics and more exactly with mechanism theory. Roughly speaking a mechanism is a rigid body system so that the graph of the links of the system is a closed chain. For open chains and for a large class of links, the configuration space of the system is (up to a rigid displacement) a differentiable manifold whose dimension is easily calculable from the nature of each link of the system and especially from the degree of freedom of each joint of the system. For closed chains, any formula for computing the mobility of the closed chain has always exceptions corresponding to "paradoxical" mechanisms. Moreover, even non paradoxical mechanisms may present singular configurations whose the description escapes the classical approach. In fact, the geometric approach mixing Lie group calculations and theory of singularities of maps (and the so-called stratification concept) allowed to definitively solve this very old issue. In this paragraph we not only shortly present the ideas of this solution but we will also explain what the usual approach lacked to do it. All considerations are local even if global questions can also be tackled. There are a lot of references about mechanism theory but for this short presentation, references are [9], [11] and again [4] for a synthetic view.

It is not the object here to present in the details the most general possible theory. Assumptions will be here made only to simplify the understanding and are in fact not restrictive (see [4]). Thus we will suppose that each mechanical joint $L_k$ has only one degree of freedom. The relative motion allowed by this joint is then described by a so-called one-parameter subgroup of $D(\mathcal{E})$ defined by an element $\xi_k$ of the Lie algebra $\mathcal{D}(\mathcal{E})$. The configuration space $\mathcal{V}$ of the mechanism made by the family $L_k, k = 1, \dots, n$ of n such joints is then (locally in a neighborhood $U$ of $(0, \dots, 0)$ in $\mathbb{R}^n$) given by the implicit equation also called the closure equation

$$f(q) = \exp(q_1\xi_1) \dots \exp(q_n\xi_n) = e = \mathrm{id}_{\mathcal{D}(\mathcal{E})} \qquad (6)$$

where $e = \mathrm{id}_{\mathcal{D}(\mathcal{E})}$ is the neutral element or the identity of the group $D(\mathcal{E})$ and $\exp$ is the usual exponential map of a Lie group. $f: \mathbb{R}^n \to D(\mathcal{E})$ is the closure map or closure function of the mechanism. For $q \in \mathbb{R}^n$ and $k \in \{1, \dots, n\}$, we put $A_k(q_k) = \exp q_k \xi_k$ and $\mu_k(q) = \mathrm{Ad}(A_1(q_1) \dots A_k(q_k))(\xi_k)$.

**Regular mechanisms** As mentioned in the introduction, the power of the method lies in the fact that all the derivatives are computed explicitly at this level and do not necessitate to introduce bases making for example vectors $\xi_k$ of the Lie algebra as column vectors in $\mathbb{R}^6$. The first derivative of $f$ is given by:

$$Df(q): x = (x_1, \dots, x_n) \mapsto \delta_{f(q)}^T \sum_{k=1}^{n} x_k \mu_k(q) = \delta_{f(q)}^T f_1(q) \qquad (7)$$

where $f_1: \mathbb{R}^n \to \mathcal{L}(\mathbb{R}^n, \mathcal{D}(\mathcal{E}))$ is the right first differential of $f$. Rank analysis for $Df$ may then be reported to $f_1$. The mechanism is regular at $q_0$ if and only if the rank $r(q_0)$ of $f_1(q_0)$ is maximal and equal to $r_m = \max_{q \in \mathbb{R}^n} \mathrm{rank}(f_1(q))$. In this regular case, usual results of differential geometry hold: $\mathcal{V}$ is a differential manifold at $q_0$ with $\mathrm{dim}\mathcal{V} = n - r_m$, its tangent space at $q_0$ is $T_{q_0}\mathcal{V} = \ker f_1(q) = \{x = (x_1, \dots, x_n) \mid \sum_{k=1}^n x_k \mu_k(q_0) = 0\}$ and we can provide the Taylor expansion of a family of $r_m$ dependent variables as functions of the other $n - r_m$ independent variables.

**Weak singular mechanism** Suppose now that $\mathrm{rank}(f_1(q_0)) = r < r_m$ (oftenly, $r = r_m - 1$) and put $\Sigma_r = \{q \in \mathbb{R}^n \mid \mathrm{rank}f_1(q) = r\}$ called the r-class of $f$ and $E^r = \{u \in \mathcal{L}(\mathbb{R}^n, \mathcal{D}(\mathcal{E})) \mid$
rank $u = r$} (In fact, we must replace $\mathcal{D}(\mathcal{E})$ by the Lie algebra $\mathcal{L}$ generated by $\xi_1, \dots, \xi_n$ which can be strictly included in $\mathcal{D}(\mathcal{E})$ as for example in planar mechanisms). Less known results of differential geometry allows to equip $\Sigma_r(f) = f_1^{-1}(E^r)$ with a submanifold structure when a so-called transversality condition holds that allows to transfert the submanifold structure from $E^r$ to the class $\Sigma_r(f)$. This condition involves the derivative $f_2$ of $f_1$. Calculations give

**Proposition 2** The derivative of $f_1$ is given by

$$f_2(q) = Df_1(q) : x \in \mathbb{R}^n \mapsto f_2(q)(x, y) = \sum_{k=1}^{n} \sum_{\ell<k} x_{\ell}y_k[\mu_{\ell}(q), \mu_k(q)] \quad (8)$$

This map has very nice properties of symmetry and skew-symmetry (see [4]). The transversality condition is technical: it means that $\{y \mapsto f_2(x, y) | x \in \mathbb{R}^n\}$ is enough big to contain all the linear maps from $K_q = \ker f_1(q)$ to a supplementary space of $F_q = \operatorname{Span}(\mu_1(q), \dots, \mu_n(q))$ in the Lie algebra $\mathcal{L}$ and can be brought back to a rank calculation of a matrix. When this transversality condition holds, then $\Sigma_r(f)$ is a submanifold whose tangent space is given by

$$T_{q_0} \Sigma_r(f) = \{x \in \mathbb{R}^n \mid f_2(q_0)(x, y) \in F_{q_0} \quad \forall y \in K_{q_0}\} \quad (9)$$

and if $\mathbb{V}$ is locally included in $\Sigma_r(f)$ at $q_0$ then $\mathbb{V}$ is locally a submanifold ($q_0$ is then called a weak singular configuration) and the tangent space at $q_0$ is given by

$$T_{q_0} \mathbb{V} = \{x \in K_{q_0} \mid f_2(q_0)(x, x) = \sum_{k=1}^{n} \sum_{\ell<k} x_{\ell}x_k[\mu_{\ell}(q_0), \mu_k(q_0)] \in F_{q_0}\} \quad (10)$$

The condition is quadratic and not linear and does not define a priori a vector space but only a cone. In this case, we can prove that the cone is a vector subspace.

**Remarks** In a lot of books of robotics, we can find elements of calculation that start by the map $f$ (the geometric model), its first derivative (the Jacobian $J(f)$ of $f$ namely the so-called kinematic model) and the dynamic model that necessitate the second derivative of $f$ or the derivative of $J(f)$. Unfortunately these expressions are completely general and purely formal as in (1) and does not contain the necessary explicit informations that allows to concretely produce significant results. In the same veine, kinematicians of XIX and XX centuries had in hand a relation of the type

$$\sum_{k=1}^{n} \dot{x}_{k} \xi_{k} = 0$$

describing the first order condition of mobility. It necessitates only the vector space structure of $\mathcal{D}(\mathcal{E})$ (screw theory). Regarding to the second order condition (10) involving Lie brackets, the Lie algebra structure is unavoidable to discover the full conditions of mobility.

**Strong singular mechanisms** It can also occur that the transversality condition fails or more often that $\mathbb{V}$ is not locally included in $\Sigma_r(f)$ at $q_0$. It is the situation when $\mathbb{V}$ is no longer a differential manifold at $q_0$. $q_0$ is then a singular point according to the meaning of singular points for algebraic varieties. In this case, there is no tangent vector space at $q_0$. A beautiful example of such a situation in mechanism theory occurs when we open and/or close a music stand. When it is open we are on one of branch of the algebraic variety and when it is closed we are on the other branch: the intersection point of the two branches is a such strong singular configuration that allows in fact to concretely close and open the music stand and it then means that singular configurations are not always bad configurations that should be avoid as far as possible!!
The quadratic equation given in (10) no longer defines a vector space but only a cone $C_{2,q_0}V$. Calculating then the successive derivatives of $f_3, f_4, \dots$ of $f_1$ involving multi Lie brackets allows us to define a decreasing sequence $(C_{k,q_0}V)_k$ of cones and thus to compute explicitly the intersection of all these cones denoted $C_{q_0}V$ which is the tangent cone at the singularity $q_0$. The decreasing sequence is necessary stationary for some integer $n_0$ that depends on the mechanism. Very often, $n_0 = 2$ and $C_{q_0}V = C_{2,q_0}V$ like for the music stand. Explicit expressions of the successive derivative and the corresponding equations of the cones $C_{k,q_0}V$ can be found partially in [9] and completely in [11] and in [4] where singular open chains are also tackled.

## 1.3 Beam theory

### 1.3.1 General partial differential equation

The third example chosen for illustrating applications of Lie group calculations to mechanics is beam theory namely one dimensional continuous mechanical system theory we call it also curvilinear system theory. It is one of the more investigated topic in mechanics since one can already find some elements of it in Leonardo Da Vinci works. As for any structural mechanic point of view, the idea is to use some privileged direction or dimension in the physical system $\Sigma$ to propose an alternative description to the 3-dimensional usual one. For beam theory, it means that one dimension is privileged that will be described by a parameter $\sigma$ belonging to an interval $[0, \ell_0]$ of $\mathbb{R}$. The choice of this parameter means that we choose a reference configuration $r_\Sigma$ for $\Sigma$. A lot of models suppose that each material section $S_\sigma$ attached to the value $\sigma$ along the privileged direction remains rigid during the possible transformation of $\Sigma$. We adopt this model here and we discuss this hypothesis in section 1.4. For this application to beam theory, first references are [10] and [12] but the main and more complete one is [8].

The configuration space $S_\Sigma$ of $\Sigma$ is then equal $\prod_{\sigma \in [0, \ell_0]} S_\sigma$. Since the configuration $r_\Sigma = (r(\sigma))_{\sigma \in [0, \ell_0]}$ with $r(\sigma) \in S_\sigma$ for each $\sigma \in [0, \ell_0]$ is given, any motion $t \in \mathbb{R} \mapsto (s_\sigma(t))_{\sigma \in [0, \ell_0]} \in S_\Sigma$ is equivalent to a map $(\sigma, t) \in [0, \ell_0] \times \mathbb{R} \mapsto D(\sigma, t) \in \mathbb{D}$ such that $s_\sigma(t) = D(\sigma, t) \cdot r(\sigma)$ for any $(\sigma, t) \in [0, \ell_0] \times \mathbb{R}$. Such a curvilinear system is then a general Cosserat 1D system.

Applying the mechanic principles allows us to obtain the single partial differential equation in the Lie algebra $D(\mathcal{E})$ governing the dynamic evolution. This unique equation is equivalent to a set of 6 scalar partial differential equations for such a curvilinear system. Again, these completely general equations cannot be obtained by hand without any additional hypothesis. This intrinsic dynamic equation reads in a pure Lagrangian form:

$$ \rho_0 \mathcal{T}_0(\sigma, t) + \frac{\partial \Theta_0}{\partial \sigma}(\sigma, t) + [\mathbf{e}_0(\sigma, t), \Theta_0(\sigma, t)] = \rho_0 \left( H_0\left(\frac{\partial V_0(\sigma, t)}{\partial t}\right) + [V_0(\sigma, t), H_0(V_0(\sigma, t))] \right) \quad (11) $$

where $V_0 = \theta_\ell(\frac{\partial D}{\partial t})$ and $\mathbf{e}_0 = \theta_\ell(\frac{\partial D}{\partial \sigma})$ are functions with values in the Lie algebra $D(\mathcal{E})$ and describes velocity and deformation fields of $\Sigma$, $\mathcal{T}_0$ is the mass density of external actions, $\Theta_0$ the field of internal actions and $H_0$ the mass density of inertia operators. These quantities are purely Lagrangian and refer to $r_\Sigma$. This reference configuration is any and can be freely chosen according to the concrete cases. Constitutive laws provide additional relations between $\Theta_0$ and $\mathbf{e}_0$. The usual boundaries conditions involve $\Theta_0, \mathbf{e}_0$ and $D$ for $\sigma = 0$ or $\ell_0$ so that we get a well-founded partial differential problem in $D(\mathcal{E})$. We now illustrate how further developments can be led while remaining coordinate-free.

### 1.3.2 Free-Vibrations of beams

In a first time we show how a complete and general coordinate-free linearization around any equilibrium is possible. Suppose then first that there is an equilibrium position of $\Sigma$ when it is
subjected to a distributed load $\mathcal{T}_{e,0}(\sigma)$. Equilibrium is then defined by the two fields $\sigma \mapsto D_{e,0}(\sigma)$ (leading to the field $\sigma \mapsto \mathbf{e}_{e,0}(\sigma)$) and $\sigma \mapsto \Theta_{e,0}(\sigma)$. The equilibrium equation reads

$$ \rho_0 \mathcal{T}_{e,0}(\sigma) + \frac{\partial \Theta_{e,0}}{\partial \sigma}(\sigma) + [\mathbf{e}_{e,0}(\sigma), \Theta_{e,0}(\sigma)] = 0 \quad (12) $$

As usual we suppose that the displacement field $(\sigma, t) \mapsto D(\sigma, t)$ is the superposition of the equilibrium displacement field $\sigma \mapsto D_{e,0}(\sigma)$ and a time depending field. Since the last field is supposed to be small we express it thanks to the exponential map of the Lie group so that $D(\sigma, t) = \exp X(\sigma, t) \circ D_{e,0}(\sigma)$ where the kinematic unknown is now a field $(\sigma, t) \mapsto X(\sigma, t)$ with values in the Lie algebra. The power of the Lie group approach arises by leading to the first order expression of (11) with respect to $X(\sigma, t)$ and its derivatives. By duality, we also suppose that the magnitude of $\chi_0(\sigma, t) = \Theta(\sigma, t) - \Theta_{e,0}(\sigma)$ is of same order than the one of $X(\sigma, t)$. Using (12) in a first order approximation of (11) leads to the following equation

$$ \rho_0 \left( H_0(\mathrm{Ad} D_0^{-1}(\sigma) \frac{\partial^2 X(\sigma, t)}{\partial t^2}) \right) = \frac{\partial \chi_0}{\partial \sigma}(\sigma, t) + [\mathrm{Ad} D_0^{-1}(\sigma) \left( \frac{\partial X(\sigma, t)}{\partial \sigma} \right), \Theta_{e,0}(\sigma)] + [\mathbf{e}_{e,0}(\sigma), \chi_0 \sigma, t] \quad (13) $$

which is the most general vibration equation of a Cosserat beam (no constitute law has yet been chosen). The two unknown fields are $(\sigma, t) \mapsto X(\sigma, t)$ and $(\sigma, t) \mapsto \chi_0(\sigma, t)$. Obviously a constitutive law linking these two fields must be added to have a closed formulation. To lead to a usual form with only the kinematic field $(\sigma, t) \mapsto X(\sigma, t)$ as unknown, we now suppose that vibrations are elastic.

### 1.3.3 Free-Vibrations of elastic beams

Linear elastic vibrations means here that there is an operator $E_0$ (we suppose the material homogeneous so that $E_0$ does not depend on $\sigma$) such that $\chi_0(\sigma, t) = E_0(\mathrm{Ad} D_{e,0}^{-1}(\sigma) \frac{\partial X(\sigma, t)}{\partial \sigma})$. Using the equilibrium configuration as a new reference configuration consists in applying $\mathrm{Ad} D_{e,0}(\sigma)$ to (13) by the left. The final expression is then

$$ \rho_0 \tilde{H}_0 \left( \frac{\partial^2 X(\sigma, t)}{\partial t^2} \right) = \tilde{E}_0 \left( \frac{\partial^2 X(\sigma, t)}{\partial \sigma^2} \right) - \tilde{E}_0 \left( [\tilde{\mathbf{e}}_0(\sigma), \frac{\partial X(\sigma, t)}{\partial \sigma}] \right) + \left[ \frac{\partial X(\sigma, t)}{\partial \sigma}, \tilde{\Theta}_{e,0}(\sigma) \right] + \left[ \tilde{\mathbf{e}}_0(\sigma), \tilde{E}_0 \left( \frac{\partial X(S_0, t)}{\partial S_0} \right) \right] \quad (14) $$

with $\tilde{H}_0 = \tilde{H}_0(\sigma) = \mathrm{Ad} D_{e,0}(\sigma) \circ H_0 \circ \mathrm{Ad} D_{e,0}^{-1}(\sigma)$, $\tilde{E}_0 = \tilde{E}_0(\sigma) = \mathrm{Ad} D_{e,0}(\sigma) \circ E_0 \circ \mathrm{Ad} D_{e,0}^{-1}(\sigma)$ and $\tilde{\mathbf{e}}_0(\sigma) = \mathrm{Ad} D_{e,0}(\sigma)(\mathbf{e}_0(\sigma))$ are the corresponding quantities pushed forward from the reference configuration to the equilibrium one. One recognizes the right-hand side term of the equivariant property of inertia operators.

This equation explicitly contains all the coupling terms between the vibrations and the equilibrium configuration and supposes linear elasticity only for the small motions. The calculations of the equilibrium may involve non linear elasticity or even inelastic transformations.

### 1.3.4 Variational formulation for spectral analysis

The last illustration of Lie group calculations deals with spectral analysis and variational formulation. Looking for time harmonic solution $X(\sigma, t) = \sin \omega t U(\sigma)$ leads to find a function $\sigma \mapsto U(\sigma)$ with some convenient properties of regularity (derivative could be understood in the sense of distributions) and verifying some boundary conditions we do not now specify. The map $\sigma \mapsto U(\sigma)$ is then solution of the differential equation

$$ \tilde{E}_0(U''(\sigma)) - \tilde{E}_0([\tilde{\mathbf{e}}_0(\sigma), U'(\sigma)]) + [U'(\sigma), \tilde{\Theta}_{e,0}(\sigma)] + [\tilde{\mathbf{e}}_0(\sigma), \tilde{E}_0(U'(\sigma))] + \rho_0 \omega^2 \tilde{H}_0(U(\sigma)) = 0 \quad (15) $$
Using the Klein form [., .] allows us to define a scalar product on the space of functions defined on [0, ℓ₀] (axioms of scalar product can be checked due to properties of inertia operators of rigid bodies) denoted [[., .]] and associated with the kinetic energy of Σ by

$$ [[U | V]] = \int_{0}^{\ell_0} \rho_0[\tilde{H}_0(U(\sigma)) | V(\sigma)]d\sigma \quad (16) $$

The spectral analysis is then brought back to the analysis of bilinear form (U, V) ↦ C(U, V) = ∫₀^ℓ₀ [E₀(U''(σ)) - E₀([e₀(σ), U'(σ)])] + [U'(σ), Θ_{e,0}(σ)] + [e₀(σ), E₀(U'(σ))] | V(σ)]dσ and more precisely to the operator A such that C(U, V) = [[A(U) | V]]. The symmetry or not of C is especially an important step in stability analysis and can be further led when boundary conditions are specified.

## 1.4 Some research avenues

Numerous authors use today Lie group approaches (often partially by fixing a point in the space and describing then the group D(Σ) by a semi-direct product of translations and rotations about this point) in robotics. As developed in section 1.3 this approach has also been extended to investigate continuous systems like Cosserat beams providing a unique partial differential equation in the Lie algebra D(Σ) without any coordinate system and allowing to make all the derivative calculations for example for a vibration analysis in a complete intrinsic approach. In the same context, a similar approach has also been used for the control of an artificial model of eel locomotion by F. Boyer (see for example [2]) leading finally to very performing prototypes. Strangely, the control theory does not systematically use this approach and investigates the controllability of each mechanical system from the scalar equations derived from Lagrange general equations leading to equations like (1) that are purely formal and do not translate the specifics of each system. As already mentioned, it is quasi impossible to generate these specific equations by hand for a little bit complicated mechanical system and it is the reason why tools of control theory are often used for mechanical systems with a very small number of degrees of freedom. We think that it can be a large field of investigations to generalize the coupling of the control theory tools with the intrinsic approach by Lie group theory.

Another very important aspect for a large number of mechanical systems targeted by so-called mobile robotics is the intuitive discrimination of "global rigid motion" of the system and "deformation" of the system. For example for a rocket, the global rigid motion is controlled and the deformation (often vibrations) is passively subjected whereas for the artificial model of the eel mentioned above, the control is made on the "internal degrees of freedom" (namely the deformation) in order to produce automatically a "global rigid motion" (viewed here as the motion of the head of the eel). A third situation between these two extreme dual situations is for example illustrated by artificial satellites where the global motion (around the earth for example) and the deformation are both controlled. In fact, this aspect of the dynamics falls under the field of the dynamics in a principal fiber bundle (see for example [6], [1] where the Lagrangian or Hamiltonian points of view are privileged). Roughly speaking, principal fiber bundle is a structure equipping a set on which a Lie group acts non transitively. It is the completely general case of a configuration space of a rigid-body system. The quotient space by this action (some technical hypotheses must be done) is the space of shapes of the system and called the base of the fiber bundle and each fiber over the base is the set of configurations frozen with the corresponding shape, the motion along this fiber over this shape being a rigid motion. The concept of connection in such a fiber space with their associated concepts of curvature and torsion is then central to geometrically describe the dynamics in such a space. As noted above and according to the targeted system, the control can be done along the fibers, on the base or in a coupled way. This
geometric view can be a deep help for control theory and we are convinced that a larger use
of this geometric approach should be known, shared and used in the community of mechanics
since it decouples as far as possible the two sides of the general motion. An optimal numerical
approach should also then start from these intrinsic equation set. In the next section 2, in a
complete different framework, we will also meet this structure.

Regarding intrinsic beam theory, we could also point out another interesting line of research. It should consist in considering the group acting on each section of the curvilinear system to be the full affine group $Ga(\mathcal{E})$ and not the displacement group $D(\mathcal{E})$ which is a subgroup of $Ga(\mathcal{E})$. A general investigation for a single body can be found in [3]. It should allow to take into account a part of the deformations of the sections which necessarily occur in the real life.

## 2 Granular mechanics

The second topic that we now consider comes from recent researches regarding granular me-
chanics. In a certain sense, it is an opposite framework to the one dealt in the previous section.
Indeed, in this context, the inertia of the elementary objects (the grains) will be in a first time
neglected (incremental quasi-static evolutions) and we essentially focus on the incremental law
of the evolution of the system. It could be surprising that the most used numerical method to
deal with such systems (DEM) mainly involves the dynamic Newton laws for each grain. It is
in fact a way found to deal with potential discontinuities like shear band ( [7]) and continual
re-arrangement of the grain system. For us, the following investigations were the continuity of
works dealing with stability in geo-mechanics and especially in granular mechanics. The initial
and main purpose of these investigations was to reconcile an instability criterion dealing with
the tangent stiffness matrix $K$ (the usual Euler criterion $\det K = 0$) and another instability
criterion involving the symmetric part $K_s$ of $K$ (the second order work criterion or Hill crite-
rion $\det K_s = 0$). A complete view of this question and of its solution can be found in [14] or
in [15]. This solution implements the new concept of Kinematic Structural Stability (KiSS) and
its dual object of geometric degree of non conservativity (GDNC) which are really significant
for nonconservative systems leading to non symmetric stiffness matrices (hypo-elastic systems,
non-associate plastic systems, follower external actions,...).

A large part of the solution is in fact algebraic. In the geometric language, it means that the
calculations are done "over a point". To deal with complete evolutions means to pass from the
punctual point of view to (at least!) a local one. To remain intrinsic, this passage must not
depend on a choice of coordinate system. The way to pass from the punctual to the local or
event global aspect leads us to the language of vector bundles. An additional difficulty, already
met in the last paragraph 1.4, is the necessity to distinguish in the framework of discrete systems
the deformations (change of shape) and the displacements namely the principal bundle structure
of the configuration space. Main references for this part are [13] and [16]. Remark that the
Polish school is very active in the topic of geometric mechanics dealing with these structures
and another more general and more complicated like Dirac structures, groupoïds, .... But our
goal here, as for Lie group approach, is more modest and is to show how a convenient geometric
language highlights outdated concepts and proposes performing and understanding calculation
process.

2.1 A geometric approach

The proposed geometric approach borrows from Lagrange’s formalism the basic concepts.

1. The configuration set of the system Σ is supposed to be a finite dimension manifold M.
The natural non-transitive action of a Lie subgroup G (depending on each system Σ) of the
Lie group $\mathcal{D}(\mathcal{E})$ on $M$ (with some technical assumptions) allows to define a principal bundle structure $(M, \mathcal{G}, \mathcal{B}, \varpi)$ on $M$. $\mathcal{B}$ is then the space of shapes of $\Sigma$. If $\epsilon = \varpi(m)$, the fiber $\varpi^{-1}\{\epsilon\}$ is a submanifold diffeomorphic to the Lie group $\mathcal{G}$. If two configurations $m$ and $m'$ are on the same fiber (namely if $\varpi(m) = \varpi(m')$) there is a unique rigid displacement $g_{mm'} \in \mathcal{G}$ such that $m' = g_{mm'} \bullet m$.

2. $TM = \bigcup_{m \in M} T_m M$ is the (total space of the) tangent bundle of $M$. A vector $\delta m \in T_m M$ describes any infinitesimal displacement or a rate of displacement in the configuration $m$. This infinitesimal displacement is a rigid infinitesimal displacement if $\varpi^T(\delta m) = 0$. A rigid infinitesimal displacement is denoted $\delta^r m$. The set of infinitesimal rigid displacement in the configuration $m$ is the vector subspace $\varpi^{T^{-1}}\{(\varpi(m), 0)\} = T_m \varpi^{-1}\{\varpi(m)\}$ of $T_m M$. The tangent bundle is denoted $\tau(M) = (TM, M, \pi_M)$.

3. Generically the elements of $\mathcal{B}$ describing the shapes are denoted $\epsilon$. It means that there is a configuration $m$ such that $\varpi(m) = \epsilon$ and $\epsilon$ is then the shape of $\Sigma$ in the configuration $m$. $TB$ is the tangent bundle of $\mathcal{B}$. We stress that there is no canonical way to associate a vector $\delta m$ of $T_m M$ to a vector $\delta \epsilon$ of $T_\epsilon \mathcal{B}$ when $\epsilon = \varpi(m)$ since if $\varpi^T(\delta m) = \delta \epsilon$ then $\varpi^T(\delta m + \delta^r m) = \delta \epsilon$ for any infinitesimal rigid displacement $\delta^r m$ at $m$. Elements $\delta \epsilon$ are called rates of pure deformation. The tangent bundle is denoted $\tau(\mathcal{B}) = (TB, \mathcal{B}, \pi_\mathcal{B})$.

4. A punctual description of an external force $\phi$ acting on $\Sigma$ in a configuration $m$ is an element $\mathbf{f}$ of $T_m^*M$. A punctual description of an internal action $\phi_\ell$ of $\Sigma$ in the configuration $m$ has two equivalent formulations. It is an element $\mathbf{f}_\ell$ of $T_m^*M$ such that $\mathbf{f}_\ell(\delta^r m) = 0$ for any infinitesimal rigid displacement $\delta^r m$ in the configuration $m$ or it is an element $\tilde{\mathbf{f}}_{def}$ of $T_\epsilon^*\mathcal{B}$ with $\varpi(m) = \epsilon$. The relation between the two representations reads $\tilde{\mathbf{f}}_{def}(\delta\epsilon) = -\mathbf{f}_\ell(\delta m)$ if $\delta\epsilon = \varpi^T(\delta m)$. It means that $\mathbf{f}_\ell = -\varpi^*(\tilde{\mathbf{f}}_{def})$: $\mathbf{f}_\ell$ is the pull-back of $-\tilde{\mathbf{f}}_{def}$ by $\varpi$. The corresponding vector bundles are $\tau^*(M) = (T^*M, M, p_M)$ with $T^*M = \bigcup_{m \in M} T_m^*M$ the total space of the cotangent bundle $\tau^*(M)$ of $M$ and $\tau^*(\mathcal{B}) = (T^*\mathcal{B}, \mathcal{B}, p_\mathcal{B})$ with $T^*\mathcal{B} = \bigcup_{\epsilon \in \mathcal{B}} T_\epsilon^*\mathcal{B}$ the total space of the cotangent bundle $\tau^*(\mathcal{B})$ of $\mathcal{B}$. We often identify the vector bundle and its total space when no confusion is possible.

5. When we want to pass from a pure punctual point of view to a local one, the natural way in vector bundles like $\tau^*M$ or $\tau^*\mathcal{B}$ is realized through the concept of cross-section or simply section. It is quite natural to suppose that all external actions may be described by such a section but we will distinguish given fixed actions like gravity (section $F_f$ of $\tau^*M$) and control or loading actions that are the cause of the evolution of the system (section $F_c(\sigma)$ where $\sigma$ is the load parameter). To simplify the presentation we suppose that $F_c(0) = 0_{\tau^*M}$ where $0_{\tau^*M}$ is the nil section of the vector bundle $\tau^*M$. We can add the sections of a vector bundle and $F_g(\sigma) = F_f + F_c(\sigma)$ is the section of the total external and given actions acting on $\Sigma$ along the loading process. $\sigma \mapsto F_g(\sigma)$ is then a given function with values in the modulus $\Gamma(\tau^*M)$ of sections of $\tau^*M$. The way to describe the local representation of internal actions is precisely the rule of the constitutive law of the system $\Sigma$. We deal with this just hereafter. When forces are described by a section we say that it is an integrable situation.

## 2.2 Geometric foundations of usual concepts

We now present some basic concepts for the integrable case in this coordinate-free framework.

### 2.2.1 Elasticity, hyperelasticity, weak hypoelasticity, strong hypoelasticity

We now translate the differentiable constitutive laws (elasticity, hyperelasticity, hypoelasticity) in this vector bundle language. The passage from integrable (or weak) to not integrable (or
strong) hypoelasticity leads to pass from a one order description to a second order one and involves in fact the so-called double vector bundles concept. We will not insist here on this more complicated structure.

1. $\Sigma$ is elastic if there is a section $\tilde{F}_{el}$ of $T^*\mathbb{B}$ such that (at least locally) for all $\epsilon \in \mathbb{B}$ the deformations forces $\tilde{\mathbf{f}}_{def} \in T_{\epsilon}^*\mathbb{B}$ in the system $\Sigma$ in the shape state $\epsilon$ is the value of this section at $\epsilon$: $\tilde{\mathbf{f}}_{def} = \tilde{F}_{el}(\epsilon)$.

2. if the 1-form $\tilde{F}_{el}$ is exact namely if there is a function $U_{el}$ defined (at least locally) on $\mathbb{B}$ such that $\tilde{F}_{el} = dU_{el}$ then $\Sigma$ is called hyperelastic.

3. if the 1-form $\tilde{F}_{el}$ is not closed namely if the exterior derivative $d\tilde{F}_{el}$ is not nil, then $\Sigma$ is called weak-hypoelastic or integrable-hypoelastic.

4. the most general description of hypoelasticity cannot longer be done through a section of $T^*\mathbb{B}$ but through the infinitesimal aspect of sections namely a connection. An incremental or strong hypoelastic or not integrable hypoelastic constitutive law is an (Ehresmann) connection $\mathcal{H}$ on $T^*\mathbb{B}$ namely a differentiable field $\mathcal{H}: \tilde{\mathbf{f}}_{def} \in T^*\mathbb{B} \mapsto \mathcal{H}(\tilde{\mathbf{f}}_{def}) \subset T_{\tilde{\mathbf{f}}_{def}}T^*\mathbb{B}$ of so-called horizontal spaces of $TT^*\mathbb{B}$ (double vector bundle) such that

$$T_{\tilde{\mathbf{f}}_{def}} T^*\mathbb{B} = V(\tilde{\mathbf{f}}_{def}) \oplus \mathcal{H}(\tilde{\mathbf{f}}_{def})$$

where $V(\tilde{\mathbf{f}}_{def})$ is the vertical space at $\tilde{\mathbf{f}}_{def}$ canonically identified (since it is a vector space) to $T_{\tilde{\mathbf{f}}_{def}}^*\mathbb{B}$ itself.

When the curvature of this connection vanishes it becomes integrable and we come back to weak hypoelasticity, the section $F_{def}$ being the integral manifold of the differential system associated with the connection.

### 2.2.2 Equilibrium and T-stability of equilibrium

A configuration $m_e$ is an equilibrium of $\Sigma$ subjected to external given forces described by the section $F_g(\sigma)$ ($\sigma$ is a load parameter as above) when the deformation forces $\tilde{f}_{def} \in T_{\epsilon_e}T^*\mathbb{B}$ developed into $\Sigma$ are such that

$$F_g(\sigma)(m_e) = \varpi^*(\tilde{f}_{def}) \qquad (17)$$

with $\epsilon_e = \varpi(m_e)$.

For an elastic system in the general sense defined above, this equilibrium equation is equivalent to

$$F_t(\sigma)(m_e) = 0 \qquad (18)$$

with $F_t = F_t(\sigma) = F_g(\sigma) - \varpi^*(\tilde{F}_{el})$ is the section of $T^*\mathbb{M}$ of all the actions. Geometrically, we say that the section $F_t$ intersects or cuts the nil section $0_{T^*\mathbb{M}}$ of the cotangent bundle at $m_e = m_e(\sigma)$.

To deal with stability of an equilibrium $m_e$ we must envisage perturbations about $m_e$ which geometrically means to investigate the nature of the intersection of the two sections $F_t$ and $0_{T^*\mathbb{M}}$ at $m_e$. Thom transversality is the appropriate concept to do it: an equilibrium $m_e = m_e(\sigma)$ is said to be T-stable if these two sections $F_t(\sigma)$ and $0_{T^*\mathbb{M}}$ transversally intersect at $m_e(\sigma)$. Remark that this definition is purely geometric and is not related to any calculation of the stiffness matrix for example. Such an equilibrium will be called T-stable or regular. If $m_e(\sigma)$ is regular, thanks to properties of transversality, equilibriums remain regular for small perturbations of the load parameter $\sigma$.
### 2.2.3 Vertical derivative and stiffness operator

In this paragraph, we deal with analytic features of above purely geometric aspects. As announced, in the last step of our path we will come back to the more usual and accepted coordinate approach. The non integrable framework is more complicated but it can be also treated as completely as the integrable case we now present (see [16]).

The idea is to make the derivative of equation (18) since the surjective map $F_t(\sigma) : \mathbb{M} \to T^*\mathbb{M}$ is differentiable. Recall that, by definition of a cross-section, $p_{\mathbb{M}} \circ F_t(\sigma) = id_{\mathbb{M}}$. We obviously can consider the usual derivative $dF_t(\sigma)$ of $F_t(\sigma)$ but it is a map from $TM$ to $TT^*\mathbb{M}$ that forgets the bundle structure of $T^*\mathbb{M}$. A connection on $T^*\mathbb{M}$ is a way to make derivatives of sections preserving the bundle structure but there is here, namely in the problem of incremental evolutions, no natural connection. The appropriate way here is to use the nil section $0_{T^*\mathbb{M}}$ as an horizontal space which leads to the intrinsic concept of vertical derivative $dF_t(\sigma)^{ver}(m_e)$ of $F_t(\sigma)$ at the equilibrium configuration $m_e$: $dF_t(\sigma)^{ver}(m_e)$ is a linear map from $T_{m_e}\mathbb{M}$ on $T_{(m_e,0)}V_{m_e}(T^*\mathbb{M}) \approx V_{m_e}(T^*\mathbb{M}) = T_{m_e}^*\mathbb{M}$ since the vertical space is a vector space. By definition, using the intrinsic geometric decomposition $T_{(m_e,0)}T^*\mathbb{M} = T_{m_e}\mathbb{M} \oplus T_{m_e}^*\mathbb{M}$ (valid a priori only at an equilibrium configuration), we have for all $\delta m \in T_{m_e}\mathbb{M}$:

$$dF_t(\sigma)(m_e(\sigma))(\delta m) = \delta m + dF_t(\sigma)^{ver}(m_e(\sigma))(\delta m)$$

The element $dF_t(\sigma)^{ver}(m_e) \in \mathcal{L}(T_{m_e}\mathbb{M}, T_{m_e}^*\mathbb{M})$ is called the stiffness operator at $m_e$. The transversality condition for T-stability of $m_e$ is equivalent to the surjectivity of $dF_t(\sigma)^{ver}(m_e)$. For dimension reasons, it is in fact equivalent to the invertibility of $dF_t(\sigma)^{ver}(m_e)$.

Chosen a coordinate system $q = (q_1, \dots, q_n)$ on $\mathbb{M}$ at $m_e$ leads to a basis $(\frac{\partial}{\partial q_1}, \dots, \frac{\partial}{\partial q_n})$ of $T_{m_e}\mathbb{M}$ and to the dual basis $(dq_1, \dots, dq_n)$ of $T_{m_e}^*\mathbb{M}$ canonically associated to the coordinate system $q = (q_1, \dots, q_n)$. The matrix of the stiffness operator $dF_t(\sigma)^{ver}(m_e)$ in these bases is the usual stiffness matrix and the transversality condition for T-stability of $m_e$ is then equivalent to the invertibility of the stiffness matrix namely the usual Euler criterion of Structural Mechanics.

### 2.2.4 Intrinsic equations

Along a loading path, when the loading parameter is increasing from 0, the incremental evolution reads $\sigma \mapsto m_e(\sigma)$ and is ensured as long as $m_e(\sigma)$ is regular. The curve $\sigma \mapsto m_e(\sigma)$ is obtained incrementally in the following way. At the step $\sigma$ of the loading path, the equilibrium described by $m_e(\sigma)$ solution of (18) and $F_t(\sigma)$ is disturbed by an infinitesimal increasing $\delta\sigma$. This leads to a perturbation $\frac{\partial F_c(\sigma)}{\partial \sigma} \delta\sigma$ of control external forces that are purely vertical. In order that the evolution remains on the nil section of equilibriums, the full contribution of all the vertical components must vanish. The incremental equation then reads

$$\frac{\partial F_c(\sigma)}{\partial \sigma} \delta\sigma + dF_t(\sigma)^{ver}(m_e(\sigma))(\delta m) = 0 \qquad (19)$$

and we can deduce the infinitesimal variation $\delta m$ as long as $dF_t(\sigma)^{ver}(m_e)$ is invertible namely as long as the T-stability condition holds:

$$m_e(\sigma) = \int_0^\sigma (dF_t(u)^{ver}(m_e(u)))^{-1} \left(\frac{\partial F_c(u)}{\partial u}\right) du$$

Remark that the two vectors of the sum in (19) have not the same origine in $TT^*\mathbb{M}$. It means that their sum has no sense in general but they can be added here only because they are vectors tangent to the same fiber $T_{m_e(\sigma)}\mathbb{M}$ (the vertical space over $m_e$ which is a vector space). This geometric property of control forces are never underlined.
## 2.3 Some research avenues

In order not to be too much technical, we have dealt here only with the integrable case when all actions are described by cross-sections of the cotangent bundle. In this case, we remark that the incremental equation (19) finally reads in $T^*M$. Dealing with the non integrable case imposes to write the incremental process fully in $TT^*M$ dealing with a full second order process but leading however again to a coordinate-free formulation thanks to some appropriate projections in $TT^*M$ (see [16] for a complete presentation). The power of the approach is undoubtedly the most evident in this non integrable case.

Since all the operations leading to the equations are explicit and are finally reduced to algebraic equations and operations, it is possible to program all the process before dealing with any particular system. The SageManifolds developed at the LUX could be for example a good environment to do it. In [16], an example of a not integrable hypoelastic system illustrates the automatic process for building the incremental evolution equations. It shows how, once the geometric framework is well identified, this complex question is reduced even for non integrable systems, to only terms of building linear projections and of solving linear systems.

Regarding possible mechanic research developments, we may mention two ways. First, an extension to dynamics for hypoelastic discrete systems and an interesting presence of at least two geometric connections: the first one describing the constitutive law (the above Ehresman connection) and the Riemanian one linked to the kinetic energy of the system. It could be an original way to deal with instability in granular systems when, as it is observed in practice, they involve bursts of kinetic energy as they lose their stability. A second research avenue deals obviously with the direct extension of constitutive laws from hypoelasticity to plasticity. The deeply non differentiable nature of plasticity is a real challenge for any geometric method involving differential geometry.

## Conclusion

In this paper, we aimed to quickly present two geometric tools deeply present in mechanics: Lie groups and vector bundles. We aimed to illustrate the power of these geometric approaches targeting very various topics of mechanics: Non linear Dynamics, Robotics, Mechanisms Theory, Beam Theory, Granular materials and Constitutive laws etc. Under the mutiplicity and the variety of applications, a constant message is present: the geometric language is not a language for insiders but is in fact often the good language for really understand the deep meaning of the concepts and also the way to solve open questions since it the closest to the reality as far as human intelligence can grasp it. But what it is less often underlined is that this proximity with the "real world" does not distance these methods from the computational aspects and are in fact a good approach for a fruitful and optimal dialogue with computational mechanics. We began with a quote of René Thom and we will end with one from Jean-Marie Souriau: "Mathematics is like a pair of shoes for the hiker: it allows him to reach new summits and to observe the beauty of landscapes that would otherwise remain hidden".

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