Lagrangian to Equations of Motion

A tiny package for symbolically deriving a system's equations of motion in the form $$\mathbf{M}(\mathbf{q})\mathbf{\ddot{q}} + \mathbf{c}(\mathbf{q},\mathbf{\dot{q}}) - \mathbf{\tau}_p(\mathbf{q}) = \mathbf{\tau} + \sum_j \mathbf{W}_j(\mathbf{r}_j,\mathbf{q}) \mathbf{F}_j$$ for a vector of generalized coordinates $\mathbf{q}$ from the system's kinetic energy $T(\mathbf{q},\mathbf{\dot{q}})$ and potential energy $V(\mathbf{q})$, that together form the Lagrangian $L = T-V$.

Additionally, given $\mathbf{r}_j(\mathbf{q})$, i.e. vectors to points where external forces $\mathbf{F}_j$ are applied to the system, wrench matrices can also be derived.

Individual terms

$$ \begin{aligned} \mathbf{M} &= \frac{\partial^2 T}{\partial \mathbf{\dot{q}}^2} \\ \mathbf{c} &= \frac{\partial^2 T}{\partial \mathbf{\dot{q}} \partial \mathbf{q}} \mathbf{\dot{q}} - \frac{\partial T}{\partial \mathbf{q}} \\ \mathbf{\tau}_p &= - \frac{\partial V}{\partial \mathbf{q}} \\ \mathbf{W}_j &= \left(\frac{\partial \mathbf{r}_j}{\partial \mathbf{q}}\right)^\top \end{aligned} $$

Installation

lagrangian2eom can be installed using pip directly from github

pip install lagrangian2eom@git+https://github.com/lieskjur/lagrangian2eom

or by first cloning the repository and then providing the path

pip install <path to lagrangian2eom>

Examples

• Cart-Pole
• Acrobot