# A useful invariant for elastic rods

Florence Bertails-Descoubes, Sébastien Neukirch

► To cite this version:

Florence Bertails-Descoubes, Sébastien Neukirch. A useful invariant for elastic rods. ESMC 2025 - 12th European Solid Mechanics Conference, Jul 2025, Lyon, France. hal-05162593

HAL Id: hal-05162593

https://hal.science/hal-05162593v1

Submitted on 15 Jul 2025

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
Florence Bertails-Descoubes (INRIA Grenoble, France)

Sébastien Neukirch (CNRS & Sorbonne University, Paris, France)

<figure><img src="image_2.png" /></figure>

<figure><img src="image_3.png" /></figure>

<figure><img src="image_4.png" /></figure>
# CONSERVATION OF MECHANICAL ENERGY

## Dynamics of the nonlinear pendulum

<figure><img src="image_5.png" /></figure>

$$ \ddot{\theta}(t) = -\frac{g}{l} \sin \theta(t) $$

$\theta(t)$ using Jacobi Elliptic functions

<figure><img src="image_6.png" /></figure>

<figure><img src="image_7.png" /></figure>

A

B

$$ H = \frac{1}{2} \dot{\theta}^2 - \frac{g}{l} \cos \theta $$

(conserved) mechanical energy, i.e. $\dot{H} = 0$

$$ H_A = H_B \Rightarrow \cos \theta_{max} = 1 - \frac{l}{2g} \dot{\theta}_0^2 $$
KIRCHHOFF ANALOGY

nonlinear pendulum

<figure><img src="image_8.png" /></figure>

dynamics

$$ \ddot{\theta}(t) = - \frac{g}{l} \sin \theta(t) $$

mechanical energy $H = \frac{1}{2} \dot{\theta}^2 - \frac{g}{l} \cos \theta$

planar elastica

<figure><img src="image_9.png" /></figure>

$$ \theta''(s) = - \frac{P}{EI} \sin \theta(s) $$

statics

$$ H = \frac{1}{2} EI \theta'^2 - P \cos \theta $$

Maddocks invariant: $H' = 0$

curvature energy

tension

A.E.H Love 1944 (first occurrence?)

Kehrbaum & Maddocks 1997

O'Reilly 2007

Jung 2011 (discrete version)

Singh 2021 (pseudo-momentum)
MAKING USE OF THE INVARIANT

pinned-pinned planar elastica

<figure><img src="image_10.png" /></figure>

We impose $P$, we obtain the deflection $\theta_0$

Q: values of $y_M$ and $\kappa_M$ ?

$$H_0 = -P \cos \theta_0$$

$$H_M = \frac{1}{2} EI \kappa_M^2 - P$$

$$H_0 = H_M \Rightarrow \begin{cases} \kappa_M = \sqrt{\frac{2P}{EI}(1 - \cos \theta_0)} \\ y_M = \sqrt{\frac{2EI}{P}(1 - \cos \theta_0)} \end{cases}$$

=> could be used for validation of numerics
HOW UNIVERSAL IS IT?

frictionless contact

<figure><img src="image_11.png" /></figure>

curvature
energy

$$H = \frac{1}{2} EI \theta'^2 + \text{tension}$$

* the curvature does not jump

* the tension does not jump

$$H_1 = H_2$$

=> works in presence of contact

see also

Clauvelin 2009 (knot)

O'Reilly 2017 (jump cond.)

Singh 2022 (capstan)
## Euler buckling in cavity

Lubinski 1962, Domokos 1996, Roman 1999, Deboeuf 2024, etc.

<figure><img src="image_12.png" /></figure>

<figure><img src="image_13.png" /></figure>

<figure><img src="image_14.png" /></figure>

<figure><img src="image_15.png" /></figure>

**Question: coexistence?**
# MAKING USE OF THE INVARIANT (CONTACT)

## Euler buckling in cavity

Lubinski 1962, Domokos 1996, Roman 1999, Deboeuf 2024, etc.

<figure><img src="image_16.png" /></figure>

<figure><img src="image_17.png" /></figure>

<figure><img src="image_18.png" /></figure>

<figure><img src="image_19.png" /></figure>

**Question: coexistence?**

**Answer:**

no friction $\Rightarrow P$ uniform

$$H_A = 0 - P$$

$$H_B = \frac{1}{2} EI \kappa_B^2 - P \quad \text{with } \kappa_B > 0$$

$$\Rightarrow H_A = H_B \text{ impossible}$$

$$\Rightarrow \text{coexistence is impossible}$$
MAKING USE OF THE INVARIANT (3)

rigid sliding sleeve

<figure><img src="image_20.png" /></figure>

$$ 
\left\{ 
\begin{array}{l} 
H_{\text{out}} = H_{\text{A}} = H_{\text{B}} = \frac{1}{2} EI \kappa_{\text{B}}^2 - P' \\ 
H_{\text{in}} = H_{\text{C}} = 0 - P 
\end{array} 
\right. 
$$

$$ H_{\text{out}} = H_{\text{in}} \Rightarrow P' = P + \frac{1}{2} EI \kappa_{\text{B}}^2 $$
CONSERVATION OF THE INVARIANT AT
THE EXIT OF A SLIDING SLEEVE

<figure><img src="image_21.png" /></figure>

$$ \text{only normal contact forces} \Rightarrow H_{\text{out}} = H_{\text{in}} $$
# MAKING USE OF THE INVARIANT (4)

## The Elastica Arm Scale

Bosi et al. 2014

<figure><img src="image_22.png" /></figure>

$$H_A = 0 + 0$$

$$H_B = \frac{1}{2} EI \kappa^2 + \text{tension} = 0$$

=> the beam is under compression at point B

$$H_C = \frac{1}{2} EI \kappa_C^2 - Mg \mathbf{e}_y \cdot \text{tangente}_C = 0$$

$$\text{bound. cond.} \Rightarrow \kappa_C = 0$$

$$\Rightarrow \text{tangente}_C \perp \mathbf{e}_y$$
# THE INVARIANT: WHERE DOES IT COME FROM?

<figure><img src="image_23.png" /></figure>

Under the kinematic constraints between $x(s)$, $y(s)$, $\theta(s)$, $\kappa(s)$, $e(s)$

$$x(0) = \dots \qquad x(L) = \dots$$

$$y(0) = \dots \qquad y(L) = \dots$$

$$\theta(0) = \dots \qquad \theta(L) = \dots$$

$\mathcal{L} = W + \text{constraints}$

$$\mathcal{L} = \frac{1}{2} EI (\kappa(s) - \hat{\kappa})^2 + \frac{1}{2} EA (e(s) - 1)^2 + \rho A g y(s) + \lambda_1 (x' - e \cos \theta) + \lambda_2 (y' - e \sin \theta) + \lambda_3 (\theta' - \kappa)$$

study $\mathcal{L}(q, q')$ with $q = \{x, y, e, \theta, \kappa\} \Rightarrow$ Euler-Lagrange: $\frac{\partial \mathcal{L}}{\partial q} = \frac{d}{ds} \frac{\partial \mathcal{L}}{\partial q'}$
# THE INVARIANT: WHERE DOES IT COME FROM?

force field: $-\nabla V$

<figure><img src="image_24.png" /></figure>

<figure><img src="image_25.png" /></figure>

bound. cond.

$$x(0) = \dots \quad x(L) = \dots$$

$$y(0) = \dots \quad y(L) = \dots$$

$$\theta(0) = \dots \quad \theta(L) = \dots$$

force and moment
equilibrium
# THE INVARIANT: WHERE DOES IT COME FROM?

force field: $-\nabla V$

$$ \frac{\partial \mathcal{L}}{\partial \kappa} - \frac{d}{ds} \frac{\partial \mathcal{L}}{\partial \kappa'} = 0 \Rightarrow \lambda_3 = EI(\kappa(s) - \hat{\kappa}) $$

$$ \frac{\partial \mathcal{L}}{\partial e} - \frac{d}{ds} \frac{\partial \mathcal{L}}{\partial e'} = 0 \Rightarrow \underbrace{\lambda_1 \cos \theta + \lambda_2 \sin \theta}_{\text{tension (axial force)}} = EA(e(s) - 1) $$

constitutive relations

<figure><img src="image_26.png" /></figure>

bound. cond.

$$ x(0) = \dots \quad x(L) = \dots $$

$$ y(0) = \dots \quad y(L) = \dots $$

$$ \theta(0) = \dots \quad \theta(L) = \dots $$

$$ \frac{\partial \mathcal{L}}{\partial x} - \frac{d}{ds} \frac{\partial \mathcal{L}}{\partial x'} = 0 \Rightarrow \lambda_1'(s) = - \frac{\partial V(x, y)}{\partial x} $$

$$ \frac{\partial \mathcal{L}}{\partial y} - \frac{d}{ds} \frac{\partial \mathcal{L}}{\partial y'} = 0 \Rightarrow \lambda_2'(s) = - \frac{\partial V(x, y)}{\partial y} $$

$$ \frac{\partial \mathcal{L}}{\partial \theta} - \frac{d}{ds} \frac{\partial \mathcal{L}}{\partial \theta'} = 0 \Rightarrow \lambda_3'(s) = \lambda_1 \sin \theta - \lambda_2 \cos \theta $$

$EI \theta''(s) = \text{shear force}$

force and moment
equilibrium
# THE INVARIANT: WHERE DOES IT COME FROM?

force field: $-\nabla V$

in the case where $EI(s)$ and $EA(s)$ and $\hat{k}(s)$ (no explicit $s$ dependence)

**Noether's theorem:** $H = \frac{\partial \mathcal{L}}{\partial \mathbf{q}'} \cdot \mathbf{q}' - \mathcal{L}$ with $H'(s) \equiv 0$ at equilibrium $\mathbf{q} = \{x, y, e, \theta, \kappa\}$

<figure><img src="image_27.png" /></figure>

Noether, 'Invariante Variationsprobleme'
Gottinger Nachrichten of 1918, pp. 235-257

bound. cond.

$$H = \frac{\partial \mathcal{L}}{\partial x'} x' + \frac{\partial \mathcal{L}}{\partial y'} y' + \frac{\partial \mathcal{L}}{\partial e'} e' + \frac{\partial \mathcal{L}}{\partial \theta'} \theta' + \frac{\partial \mathcal{L}}{\partial \kappa'} \kappa' - \mathcal{L}$$

$$x(0) = \dots \quad x(L) = \dots$$

$$y(0) = \dots \quad y(L) = \dots$$

$$\theta(0) = \dots \quad \theta(L) = \dots$$

$$H = \frac{1}{2} EI [\kappa^2(s) - \hat{\kappa}^2] + \frac{1}{2} EA [e^2(s) - 1] - V(x, y)$$

<figure><img src="image_28.png" /></figure>

not strain energies!

<figure><img src="image_29.png" /></figure>
# CONSERVATIVE EXTERNAL POTENTIAL

<figure><img src="image_30.png" /></figure>

$$H'(s) = 0 \quad \forall s$$

$$h'(s) \neq 0$$

## soft-wall (or barrier) potential

$$V(y) = \frac{b}{a - y(s)}$$

<figure><img src="image_31.png" /></figure>

<figure><img src="image_32.png" /></figure>
# CONSERVATIVE EXTERNAL POTENTIAL

<figure><img src="image_33.png" /></figure>

$$H = \frac{1}{2} EI \kappa^2(s) + \text{tension} - V[y(s)]$$

$$H'(s) = 0 \quad \forall s$$

$$h'(s) \neq 0$$

## soft-wall (or barrier) potential

$$V(y) = \frac{b}{a - y(s)}$$

<figure><img src="image_34.png" /></figure>

<figure><img src="image_35.png" /></figure>
# CONSERVATIVE EXTERNAL POTENTIAL

<figure><img src="image_36.png" /></figure>

$$H = \frac{1}{2} EI \kappa^2(s) + \text{tension} - V[y(s)]$$

$$H'(s) = 0 \quad \forall s$$

$$h'(s) \neq 0$$

## soft-wall (or barrier) potential

$$V(y) = \frac{b}{a - y(s)}$$

<figure><img src="image_37.png" /></figure>

<figure><img src="image_38.png" /></figure>
# CONSERVATIVE EXTERNAL POTENTIAL

<figure><img src="image_39.png" /></figure>

$$H = \frac{1}{2} EI \kappa^2(s) + \text{tension} - V[y(s)]$$

$$H'(s) = 0 \quad \forall s$$

$$h'(s) \neq 0$$

## soft-wall (or barrier) potential

$$V(y) = \frac{b}{a - y(s)}$$

<figure><img src="image_40.png" /></figure>

<figure><img src="image_41.png" /></figure>
CONSERVATIVE EXTERNAL POTENTIAL

<figure><img src="image_42.png" /></figure>

we use

$EI=1$

$L=1$

$a=0.1$

$\Delta=0.026$

$$H = \frac{1}{2} EI \kappa^2(s) + \text{tension} - 0$$

$$H'(s) = 0 \quad \forall s$$

$$h'(s) = 0 \quad \forall s$$

$$\text{potential} \qquad \int_0^L V(y) = \frac{b}{a - y(s)} ds \xrightarrow[b \to 0]{} 0$$

$$\text{force} \qquad \int_0^L \frac{\partial V}{\partial y} = \frac{b}{(a - y(s))^2} ds \xrightarrow[b \to 0]{} F_{\text{contact}}$$

<figure><img src="image_43.png" /></figure>
THE INVARIANT: WHEN DOES IT FAIL?

<figure><img src="image_44.png" /></figure>

Example: tapered beam

<figure><img src="image_45.png" /></figure>

$$ \mathcal{L} = \frac{1}{2} EI(s) \kappa^2(s) + \lambda_1(x' - \cos \theta) + \lambda_2(y' - \sin \theta) + \lambda_3(\theta' - \kappa) $$

$$ \text{Equilibrium: } [EI(s) \theta'(s)]' = -P \sin \theta(s) $$

$$ \text{Invariant? } H(s) = \frac{1}{2} EI(s) \theta'(s)^2 - P \cos \theta(s) $$

$$ H'(s) = \dots = -\frac{1}{2} EI'(s) \theta'(s)^2 \neq 0 $$

reminder
$\kappa(s) \equiv \theta'(s)$
THE INVARIANT: WHEN DOES IT FAIL?

<figure><img src="image_46.png" /></figure>

Example: curved beam

<figure><img src="image_47.png" /></figure>

$$ \mathcal{L} = \frac{1}{2} EI (\kappa(s) - \hat{\kappa}(s))^2 + \lambda_1 (x' - \cos \theta) + \lambda_2 (y' - \sin \theta) + \lambda_3 (\theta' - \kappa) $$

$$ \text{Equilibrium: } EI [\theta'(s) - \hat{\kappa}(s)]' = -P \sin \theta(s) $$

$$ \text{Invariant? } H(s) = \frac{1}{2} EI (\theta'(s)^2 - \hat{\kappa}^2(s)) - P \cos \theta(s) $$

$$ H'(s) = \dots = -\frac{1}{2} EI \hat{\kappa}'(s) (\theta' - \hat{\kappa}) \neq 0 $$

reminder
$\kappa(s) \equiv \theta'(s)$
BACK TO CONTACT

<figure><img src="image_48.png" /></figure>
BACK TO CONTACT

<figure><img src="image_49.png" /></figure>
BACK TO CONTACT

New term in the Lagrangian: $\mathcal{L}_c = -\mu(s) g(\mathbf{r}(s))$

<figure><img src="image_50.png" /></figure>
BACK TO CONTACT

New term in the Lagrangian: $\mathcal{L}_c = -\mu(s) g(\mathbf{r}(s))$

<figure><img src="image_51.png" /></figure>

Does it break Noether's invariant ?
BACK TO CONTACT

New term in the Lagrangian: $\mathcal{L}_c = -\mu(s) g(\mathbf{r}(s))$

<figure><img src="image_52.png" /></figure>

Does it break Noether's invariant ?

-> No!

**Signorini law for frictionless contact**

$$0 \le g(\mathbf{r}(s)) \perp \mu(s) \ge 0$$
BACK TO CONTACT

New term in the Lagrangian: $\mathcal{L}_c = -\mu(s) g(\mathbf{r}(s))$

Does it break Noether's invariant ?

-> No!

**Signorini law for frictionless contact**

$$0 \le g(\mathbf{r}(s)) \perp \mu(s) \ge 0$$

<figure><img src="image_53.png" /></figure>
New term in the Lagrangian: $\mathcal{L}_c = -\mu(s) g(\mathbf{r}(s))$

<figure><img src="image_54.png" /></figure>

Does it break Noether's invariant ?

-> No!

**Signorini law for frictionless contact**

$$0 \le g(\mathbf{r}(s)) \perp \mu(s) \ge 0$$
New term in the Lagrangian: $\mathcal{L}_c = -\mu(s) g(\mathbf{r}(s))$

<figure><img src="image_55.png" /></figure>

Does it break Noether's invariant ?

-> No!

**Signorini law for frictionless contact**

$$0 \le g(\mathbf{r}(s)) \perp \mu(s) \ge 0$$

Moreover, *H*'s formula is unchanged !
3D RODS: KINEMATICS

<figure><img src="image_56.png" /></figure>

Centreline **r(s)**
Material frame $\mathcal{R}(s) = \{\mathbf{d}_1(s), \mathbf{d}_2(s), \mathbf{d}_3(s)\}$
# 3D RODS: KINEMATICS

<figure><img src="image_57.png" /></figure>

Centreline $r(s)$
Material frame $\mathcal{R}(s) = \{d_1(s), d_2(s), d_3(s)\}$

## Two types of constraints:

1/ Coupling between $r$ and $\mathcal{R}$
e.g.,

$r' = d_3$ (Kirchhoff)

or shear $\quad 1+extension$

$r' = \nu_1 d_1 + \nu_2 d_2 + \nu_3 d_3$ (Reissner)

→ General case handled here
# 3D RODS: KINEMATICS

<figure><img src="image_58.png" /></figure>

<table><tr><td>Centreline</td><td>r(s)</td></tr><tr><td>Material frame</td><td>R(s) = {d<sub>1</sub>(s), d<sub>2</sub>(s), d<sub>3</sub>(s)}</td></tr></table>

## Two types of constraints:

2/ $\mathcal{R} \in SO(3)$ Rotation group

→ Existence of a vector $\mathbf{u} = \{\mathbf{u}_1, \mathbf{u}_2, \mathbf{u}_3\}_{d_j}$ such that

$$ \left\{ \begin{array}{l} \mathbf{d}'_1 = \mathbf{u} \times \mathbf{d}_1 \\ \mathbf{d}'_2 = \mathbf{u} \times \mathbf{d}_2 \\ \mathbf{d}'_3 = \mathbf{u} \times \mathbf{d}_3 \end{array} \right. $$

<figure><img src="image_59.png" /></figure>

(Darboux equations)
THE INVARIANT IN THE 3D CASE: TWO METHODS

**Method $\mathbb{R}^n$ ('flattened') :**

$\mathcal{L} = \mathcal{L}[\mathbf{q}(s), \mathbf{q}'(s)]$ with $\mathbf{q} = (\mathbf{r}, \mathbf{d}_1, \mathbf{d}_2, \mathbf{d}_3, u_1, u_2, u_3, v_1, v_2, v_3)$ and $\mathbf{q}' = (\mathbf{r}', \mathbf{d}'_1, \mathbf{d}'_2, \mathbf{d}'_3, \dots)$

<figure><img src="image_60.png" /></figure>

→ Add constraints **1** and **2** in $\mathcal{L}$ with multipliers

$\frac{\partial \mathcal{L}}{\partial \mathbf{q}'}$

Classical Noether invariant: $H = \frac{\partial \mathcal{L}}{\partial \mathbf{q}'} \cdot \mathbf{q}' - \mathcal{L}$
# THE INVARIANT IN THE 3D CASE: TWO METHODS

**Method $\mathbb{R}^n$ ('flattened') :**

$\mathcal{L} = \mathcal{L}[\mathbf{q}(s), \mathbf{q}'(s)]$ with $\mathbf{q} = (\mathbf{r}, \mathbf{d}_1, \mathbf{d}_2, \mathbf{d}_3, u_1, u_2, u_3, v_1, v_2, v_3)$ and $\mathbf{q}' = (\mathbf{r}', \mathbf{d}'_1, \mathbf{d}'_2, \mathbf{d}'_3, \dots)$

<figure><img src="image_61.png" /></figure>

→ Add constraints **1** and **2** in $\mathcal{L}$ with multipliers

Classical Noether invariant: $H = \frac{\partial \mathcal{L}}{\partial \mathbf{q}'} \cdot \mathbf{q}' - \mathcal{L}$

**Method $\mathbb{R}^n \times SO(3)$ ('compact') :**

$\mathcal{L} = \mathcal{L}[\mathbf{q}(s), \tilde{\mathbf{q}}(s)]$ with $\mathbf{q} = (\mathbf{r}, \mathbf{d}_1, \mathbf{d}_2, \mathbf{d}_3, v_1, v_2, v_3)$ and $\tilde{\mathbf{q}} = (\mathbf{r}', \mathbf{u}_1, \mathbf{u}_2, \mathbf{u}_3, \dots) \neq \mathbf{q}'$

<figure><img src="image_62.png" /></figure>

→ Add only constraint **1** in $\mathcal{L}$ with multipliers

New Noether invariant, valid on $\mathbb{R}^n \times SO(3)$: $H = \frac{\partial \mathcal{L}}{\partial \mathbf{r}'} \cdot \mathbf{r}' + \frac{\partial \mathcal{L}}{\partial \bar{\mathbf{u}}} \cdot \bar{\mathbf{u}} - \mathcal{L}$
# THE INVARIANT IN THE 3D CASE: TWO METHODS

**Method $\mathbb{R}^n$ ('flattened') :**

$$ \mathcal{L} = \mathcal{L}[\mathbf{q}(s), \mathbf{q}'(s)] \text{ with } \mathbf{q} = (\mathbf{r}, \mathbf{d}_1, \mathbf{d}_2, \mathbf{d}_3, u_1, u_2, u_3, v_1, v_2, v_3) \text{ and } \mathbf{q}' = (\mathbf{r}', \mathbf{d}'_1, \mathbf{d}'_2, \mathbf{d}'_3, \dots) $$

<figure><img src="image_63.png" /></figure>

→ Add constraints **1** and **2** in $\mathcal{L}$ with multipliers

$$ \text{Classical Noether invariant: } H = \frac{\partial \mathcal{L}}{\partial \mathbf{q}'} \cdot \mathbf{q}' - \mathcal{L} $$

**Method $\mathbb{R}^n \times SO(3)$ ('compact') :**

$$ \mathcal{L} = \mathcal{L}[\mathbf{q}(s), \tilde{\mathbf{q}}(s)] \text{ with } \mathbf{q} = (\mathbf{r}, \mathbf{d}_1, \mathbf{d}_2, \mathbf{d}_3, v_1, v_2, v_3) \text{ and } \tilde{\mathbf{q}} = (\mathbf{r}', \mathbf{u}_1, \mathbf{u}_2, \mathbf{u}_3, \dots) \neq \mathbf{q}' $$

<figure><img src="image_64.png" /></figure>

→ Add only constraint **1** in $\mathcal{L}$ with multipliers

$$ \text{New Noether invariant, valid on } \mathbb{R}^n \times SO(3): H = \frac{\partial \mathcal{L}}{\partial \mathbf{r}'} \cdot \mathbf{r}' + \frac{\partial \mathcal{L}}{\partial \bar{\mathbf{u}}} \cdot \bar{\mathbf{u}} - \mathcal{L} \quad \begin{cases} \bar{\mathbf{u}} = \{\mathbf{u}_1, \mathbf{u}_2, \mathbf{u}_3\} \\ \text{s.t. } \mathbf{u} = \mathcal{R} \bar{\mathbf{u}} \end{cases} $$
# THE INVARIANT IN THE 3D CASE: TWO METHODS

**Method $\mathbb{R}^n$ ('flattened') :**

$\mathcal{L} = \mathcal{L}[\mathbf{q}(s), \mathbf{q}'(s)]$ with $\mathbf{q} = (\mathbf{r}, \mathbf{d}_1, \mathbf{d}_2, \mathbf{d}_3, u_1, u_2, u_3, v_1, v_2, v_3)$ and $\mathbf{q}' = (\mathbf{r}', \mathbf{d}'_1, \mathbf{d}'_2, \mathbf{d}'_3, \dots)$

<figure><img src="image_65.png" /></figure>

→ Add constraints **1** and **2** in $\mathcal{L}$ with multipliers

Classical Noether invariant: $H = \frac{\partial \mathcal{L}}{\partial \mathbf{q}'} \cdot \mathbf{q}' - \mathcal{L}$

**Method $\mathbb{R}^n \times SO(3)$ ('compact') :**

$\mathcal{L} = \mathcal{L}[\mathbf{q}(s), \tilde{\mathbf{q}}(s)]$ with $\mathbf{q} = (\mathbf{r}, \mathbf{d}_1, \mathbf{d}_2, \mathbf{d}_3, v_1, v_2, v_3)$ and $\tilde{\mathbf{q}} = (\mathbf{r}', \mathbf{u}_1, \mathbf{u}_2, \mathbf{u}_3, \dots) \neq \mathbf{q}'$

<figure><img src="image_66.png" /></figure>

→ Add only constraint **1** in $\mathcal{L}$ with multipliers

New Noether invariant, valid on $\mathbb{R}^n \times SO(3)$: $H = \underbrace{\frac{\partial \mathcal{L}}{\partial \mathbf{r}'} \cdot \mathbf{r}'}_{\text{translation}} + \underbrace{\frac{\partial \mathcal{L}}{\partial \bar{\mathbf{u}}} \cdot \bar{\mathbf{u}}}_{\text{rotation}} - \mathcal{L} \quad \begin{cases} \bar{\mathbf{u}} = \{\mathbf{u}_1, \mathbf{u}_2, \mathbf{u}_3\} \\ \text{s.t.} \quad \mathbf{u} = \mathcal{R} \bar{\mathbf{u}} \end{cases}$
THE INVARIANT IN THE 3D CASE

$$
\left.
\begin{aligned}
W_{\text{bend}} &= \frac{1}{2}B_1(u_1(s) - \hat{u}_1)^2 + \frac{1}{2}B_2(u_2(s) - \hat{u}_2)^2 + \frac{1}{2}B_3(u_3(s) - \hat{u}_3)^2 &\text{Reissner model} \\
W_{\text{shear}} &= \frac{1}{2}A_1(v_1(s) - \hat{v}_1)^2 + \frac{1}{2}A_2(v_2(s) - \hat{v}_2)^2 + \frac{1}{2}A_3(v_3(s) - \hat{v}_3)^2
\end{aligned}
\right\}
$$
THE INVARIANT IN THE 3D CASE

<figure><img src="image_67.png" /></figure>
THE INVARIANT IN THE 3D CASE

<figure><img src="image_68.png" /></figure>

where $\mathcal{L} = \mathcal{L}[\mathbf{q}(s), \mathbf{q}'(s)]$ with $\mathbf{q} = (\mathbf{r}, \mathbf{d}_1, \mathbf{d}_2, \mathbf{d}_3, u_1, u_2, u_3, v_1, v_2, v_3)$ and $\mathbf{q}' = (\mathbf{r}', \mathbf{d}'_1, \mathbf{d}'_2, \mathbf{d}'_3, \dots)$

Method $\mathbb{R}^n$

$$H = \frac{\partial \mathcal{L}}{\partial \mathbf{q}'} \cdot \mathbf{q}' - \mathcal{L}$$
# THE INVARIANT IN THE 3D CASE

<figure><img src="image_69.png" /></figure>

constraints

where $\mathcal{L} = \mathcal{L}[\mathbf{q}(s), \tilde{\mathbf{q}}(s)]$ with $\mathbf{q} = (\mathbf{r}, \mathbf{d}_1, \mathbf{d}_2, \mathbf{d}_3, v_1, v_2, v_3)$ and $\tilde{\mathbf{q}} = (\mathbf{r}', u_1, u_2, u_3, \dots) \neq \mathbf{q}'$

$$ \text{Method} \quad \mathbb{R}^n \times \text{SO}(3) \qquad H = \frac{\partial \mathcal{L}}{\partial \mathbf{r}'} \cdot \mathbf{r}' + \frac{\partial \mathcal{L}}{\partial \bar{\mathbf{u}}} \cdot \bar{\mathbf{u}} - \mathcal{L} $$
THE INVARIANT IN THE 3D CASE

<figure><img src="image_70.png" /></figure>

where $\mathcal{L} = \mathcal{L}[\mathbf{q}(s), \tilde{\mathbf{q}}(s)]$ with $\mathbf{q} = (\mathbf{r}, \mathbf{d}_1, \mathbf{d}_2, \mathbf{d}_3, v_1, v_2, v_3)$ and $\tilde{\mathbf{q}} = (\mathbf{r}', u_1, u_2, u_3, \dots) \neq \mathbf{q}'$

$$ \text{Method} \quad \mathbb{R}^n \times \text{SO}(3) \qquad H = \boxed{\frac{\partial \mathcal{L}}{\partial \mathbf{r}'}} \cdot \mathbf{r}' + \boxed{\frac{\partial \mathcal{L}}{\partial \bar{\mathbf{u}}}} \cdot \bar{\mathbf{u}} - \mathcal{L} \qquad \begin{array}{c} \text{Internal force} \\ \text{Internal moment} \end{array} $$
# HOW DOES THE TENSION ENTER THE INVARIANT FORMULA (3D)

Noether invariant:

$$H = \mathbf{n} \cdot \mathbf{v} + \mathbf{m} \cdot \mathbf{u} - W_{\text{bend}} - W_{\text{shear}} - V(\mathbf{r})$$

$$= \frac{1}{2} B_i [u_i^2(s) - \hat{u}_i^2] + \frac{1}{2} A_i [v_i^2(s) - \hat{v}_i^2] - V(\mathbf{r})$$

$$= \frac{1}{2} B_i [u_i(s) - \hat{u}_i]^2 + m_i \hat{u}_i + \frac{1}{2} A_i [v_i(s) - \hat{v}_i]^2 + n_i \hat{v}_i - V(\mathbf{r})$$
# HOW DOES THE TENSION ENTER THE INVARIANT FORMULA (3D)

Noether invariant:

$$H = \mathbf{n} \cdot \mathbf{v} + \mathbf{m} \cdot \mathbf{u} - W_{bend} - W_{shear} - V(\mathbf{r})$$

$$= \frac{1}{2} B_i [u_i^2(s) - \hat{u}_i^2] + \frac{1}{2} A_i [v_i^2(s) - \hat{v}_i^2] - V(\mathbf{r})$$

$$= \frac{1}{2} B_i [u_i(s) - \hat{u}_i]^2 + m_i \hat{u}_i + \frac{1}{2} A_i [v_i(s) - \hat{v}_i]^2 + n_i \hat{v}_i - V(\mathbf{r})$$

$$
\left\{
\begin{array}{ll}
\text{general case} & \\
\text{not the energies !} &
\end{array}
\right.
$$
# HOW DOES THE TENSION ENTER THE INVARIANT FORMULA (3D)

Noether invariant:

$$H = \mathbf{n} \cdot \mathbf{v} + \mathbf{m} \cdot \mathbf{u} - W_{\text{bend}} - W_{\text{shear}} - V(\mathbf{r})$$

$$= \frac{1}{2} B_i [u_i^2(s) - \hat{u}_i^2] + \frac{1}{2} A_i [v_i^2(s) - \hat{v}_i^2] - V(\mathbf{r})$$

$$= \frac{1}{2} B_i [u_i(s) - \hat{u}_i]^2 + m_i \hat{u}_i + \frac{1}{2} A_i [v_i(s) - \hat{v}_i]^2 + n_i \hat{v}_i - V(\mathbf{r})$$

$$= \frac{1}{2} B_i u_i^2(s) + \frac{1}{2} A_i v_i^2(s) + n_3 - V(\mathbf{r})$$

$$= W_{\text{bend}} + W_{\text{shear}} + \text{tension} - V(\mathbf{r})$$

general case
not the energies !

if $\hat{u}_i = 0$ and $\hat{v}_{1,2} = 0$ and $\hat{v}_3 = 1$

Maddocks 1994
(Intuited case $V = 0$)

(naturally flat rod, with naturally straight cross-section)
MAKING USE OF THE INVARIANT IN 3D: THE TREFOIL KNOT

<figure><img src="image_71.png" /></figure>

Limit case of vanishing thickness

$$H_{3D} = \text{curvature} + \text{twist} + \text{tension}$$

<figure><img src="image_72.png" /></figure>

A simple way to retrieve quickly a theory
MAKING USE OF THE INVARIANT IN 3D: THE TREFOIL KNOT

<figure><img src="image_73.png" /></figure>

Limit case of vanishing thickness

$$H_{3D} = \text{curvature} + \text{twist} + \text{tension}$$

→ A simple way to retrieve quickly a theory

$$H_A = 0 + 0 + T$$

$$H_B = \frac{EI}{2R^2} + 0 + 0$$
MAKING USE OF THE INVARIANT IN 3D: THE TREFOIL KNOT

<figure><img src="image_74.png" /></figure>

Limit case of vanishing thickness

$$H_{3D} = \text{curvature} + \text{twist} + \text{tension}$$

<figure><img src="image_75.png" /></figure>

A simple way to retrieve quickly a theory

<figure><img src="image_76.png" /><figcaption>Arai et al. 1999
actin knots</figcaption></figure>

$$H_A = 0 + 0 + T$$

$$H_B = \frac{1 EI}{2 R^2} + 0 + 0$$

$$\frac{H_A = H_B \Rightarrow T = \frac{1 EI}{2 R^2}}$$
# MAKING USE OF THE INVARIANT IN 3D: THE TREFOIL KNOT

<figure><img src="image_77.png" /></figure>

Super-helix model
+ non-smooth contact solver

Joint work with T. Metivet and V. Romero (Inria)

## Case of finite thickness (simulation)

$$H_{3D} = \text{curvature} + \text{twist} + \text{tension}$$

→ A simple way to **validate** a simulator
# MAKING USE OF THE INVARIANT IN 3D: THE TREFOIL KNOT

Joint work with T. Metivet and V. Romero (Inria)

<figure><img src="image_78.png" /></figure>

Super-helix model
+ non-smooth contact solver

## Case of finite thickness (simulation)

$$H_{3D} = \text{curvature} + \text{twist} + \text{tension}$$

<figure><img src="image_79.png" /></figure>

→ A simple way to **validate** a simulator

<figure><img src="image_80.png" /></figure>

Elastic energy
density

<figure><img src="image_81.png" /></figure>

Tension
# MAKING USE OF THE INVARIANT IN 3D: THE TREFOIL KNOT

Joint work with T. Metivet and V. Romero (Inria)

<figure><img src="image_82.png" /><figcaption>Super-helix model
+ non-smooth contact solver</figcaption></figure>

<figure><img src="image_83.png" /></figure>

<figure><img src="image_84.png" /><figcaption>Elastic energy
density</figcaption></figure>

<figure><img src="image_85.png" /><figcaption>Tension</figcaption></figure>
<figure><img src="image_86.png" /></figure>

$$p = \frac{EI}{R^3} \frac{\sin^4 \theta}{\cos 2\theta}$$

$$\tau = \frac{EI}{R} \frac{2 \sin^3 \theta \cos \theta}{\cos 2\theta}$$

$$T = \frac{EI}{R^2} \frac{\sin^4 \theta}{\cos 2\theta}$$
PLECTONEMES

<figure><img src="image_87.png" /></figure>
PLECTONEMES

<figure><img src="image_88.png" /></figure>
PLECTONEMES

<figure><img src="image_89.png" /></figure>
PLECTONEMES

$$H = \text{curvature} + \text{twist} + \text{tension}$$

<figure><img src="image_90.png" /></figure>

$$H_A = \frac{EI}{2} \sin^4 \theta + \frac{GJ\tau^2}{2} + \frac{EI}{R^2} \frac{\sin^4 \theta}{\cos 2\theta}$$

$$H_B = 0 + \frac{1}{2}GJ\tau^2 + T_{ext}$$

$$\frac{H_A = H_B \Rightarrow T_{ext} = \frac{EI}{2R^2} \sin^4 \theta \left( 1 + \frac{2}{\cos 2\theta} \right)}{\text{approx. helical}}$$
PLECTONEMES

<figure><img src="image_91.png" /></figure>

$$T_{\text{ext}} = \frac{EI}{2R^2} \sin^4 \theta \left( 1 + \frac{2}{\cos 2\theta} \right)$$
CONCLUSION

- invariant for 2D and 3D Kirchhoff / Reissner elastic rods

- deduced from variational approach

- valid in case of (frictionless) contact / sleeves

- useful in numerical code validation

- dynamics: other invariants (statics/dynamics/vibrations)

<figure><img src="image_92.png" /></figure>

preprint

<figure><img src="image_93.png" /></figure>
## CONCLUSION

* invariant for 2D and 3D Kirchhoff / Reissner elastic rods

* deduced from variational approach

* valid in case of (frictionless) contact / sleeves

* useful in numerical code validation

* dynamics: other invariants (statics/dynamics/vibrations)

<figure><img src="image_94.png" /><figcaption>EMMY NOETHER
(1882 - 1935)</figcaption></figure>

* University of Erlangen, Göttingen then USA

* Exceptional mathematical contributions (algebra)

* Strongly discriminated against because was a woman

* Things have (fortunately) changed but... stay careful!

preprint

<figure><img src="image_95.png" /></figure>