Example

The N-fold 3D pendulum

We describe here the specific problem of a chain of \(N\) connected 3D pendulums, whose dynamics evolves on \((TS^2)^N\). The dynamics of this mechanical system is described in terms of a Lie group \(G\) acting transitively on the phase space \(\mathcal{M}\). The equations of motion are presented in terms of the infinitesimal generator of the transitive action.

Equations of motion

Let us consider a chain of \(N\) pendulums subject to constant gravity \(g\). The system is modeled by \(N\) rigid, massless links serially connected by spherical joints, with the first link connected to a fixed point placed at the origin of the ambient space \(\mathbb{R}^3\). We neglect friction and interactions among the pendulums.

The modeling part comes from (Lee, Leok and McClamroch, (2018)) and we omit details. We denote by \(q_i\in S^2\) the configuration vector of the \(i\) -th mass, \(m_i\), of the chain. Following (Lee, Leok and McClamroch, (2018)), we express the Euler–Lagrange equations for our system in terms of the configuration variables \((q_1,\dots,q_N)\in (S^2)^N\subset\mathbb{R}^{3N}\), and their angular velocities \((\omega_1,...,\omega_N)\in T_{q_1}S^2\times ... \times T_{q_N}S^2\subset\mathbb{R}^{3N}\), defined be the following kinematic equations:

(1)\[\begin{align} \dot{q}_i = \omega_i\times q_i, \quad i=1,\dots,N. \end{align}\]

The Euler–Lagrange equations of the system can be written as

(2)\[\begin{split}\begin{align} R(q)\dot{\omega} = \left[\sum_{\substack{j=1\\ j\neq i}}^N M_{ij}|\omega_j|^2\hat{q}_i q_j - \Big(\sum_{j=i}^N m_j\Big)gL_i \hat{q}_i e_3 \right]_{i=1,...,N} = \begin{bmatrix}r_1\\ \vdots \\ r_N \end{bmatrix}\in\mathbb{R}^{3N}, \end{align}\end{split}\]

where \(R(q)\in\mathbb{R}^{3N\times 3N}\) is a symmetric block matrix defined as

(3)\[\begin{align} R(q)_{ii} = \Big(\sum_{j=i}^Nm_j\Big)L_i^2I_3\in\mathbb{R}^{3\times 3}, \end{align}\]

(4)\[\begin{align} R(q)_{ij} = \Big(\sum_{k=j}^N m_k\Big)L_iL_j\hat{q}_i^T\hat{q}_j\in\mathbb{R}^{3\times 3} = R(q)_{ji}^T,\; i<j, \end{align}\]

and

(5)\[\begin{align} M_{ij} =\Big(\sum_{k={\text{max}}\{i,j\}}^N m_k\Big)L_iL_j I_3\in\mathbb{R}^{3\times 3}. \end{align}\]

Equations (1) and (2) define the dynamics of the N-fold pendulum, and hence a vector field \(F\in\mathfrak{X}((TS^2)^N)\). We now find a function \(f:(TS^2)^N\rightarrow \mathfrak{se}(3)^N\) such that

(6)\[\begin{align} \Psi_*(f(m))\vert_m = F\vert_m,\;\;\forall m\in (TS^2)^N. \end{align}\]

Since \(R(q)\) defines a linear invertible map (see (Celledoni, Çokaj, Leone, Murari and Owren, (2021) International Journal of Computer Mathematics)).

(7)\[\begin{align} A_{q}:T_{q_1}S^2\times ... \times T_{q_N}S^2 \rightarrow T_{q_1}S^2 \times ... \times T_{q_N}S^2,\quad A_q(\omega):=R(q)\omega, \end{align}\]

we can rewrite the ODEs for the angular velocities as follows:

(8)\[\begin{split}\begin{align} \dot{\omega}= A_{q}^{-1}\left(\begin{bmatrix}r_1\\ \vdots \\ r_N \end{bmatrix}\right) =\begin{bmatrix} h_1(q,\omega) \\ \vdots \\ h_N(q,\omega)\end{bmatrix} = \begin{bmatrix} a_1(q,\omega)\times q_1 \\ \vdots \\ a_N(q,\omega)\times q_N \end{bmatrix}. \end{align}\end{split}\]

In equation (8) the \(r_i\) -s for \(i = 1, ..., N\) are defined as in (2) , and \(a_1,...,a_N:(TS^2)^N\rightarrow \mathbb{R}^3\) can be defined as \(a_i(q,\omega):=q_i\times h_i(q,\omega)\), \(i = 1, ..., N\). Thus, the map \(f\) is given by

(9)\[\begin{split}\begin{align} f(q,\omega) = \begin{bmatrix} \omega_1 \\ q_1\times h_1(q,\omega) \\ \vdots \\ \omega_N \\ q_N\times h_N(q,\omega) \end{bmatrix}\in\mathfrak{se}(3)^N\simeq \mathbb{R}^{6N}. \end{align}\end{split}\]