## A 3D Variable Mass Pendulum [[cite_leeLieGroupVariational2005]] develop a variational integrator for 3D constant mass rigid pendulum. In this note, we derive the Lagrangian for a variable mass simple pendulum. ![[Fig1-eps-converted-to.pdf]] We start with the Lagrangian of the system: $ \mathcal{L} = \frac{1}{2} \int _{m} {^N\mathbf{v}^P} \cdot {^N\mathbf{v}^P} \, dm $ Transferring velocity to point $O$ on the rigid body results in the following $\mathcal{L}$: $ \mathcal{L} = \frac{1}{2} \int _{m} \bigg( {^N\mathbf{v}^O} + ({^N\boldsymbol{\omega}^B \times \mathbf{r}) + {^B\mathbf{v}^P}} \bigg) \cdot \bigg( {^N\mathbf{v}^O} + ({^N\boldsymbol{\omega}^B \times \mathbf{r}) + {^B\mathbf{v}^P}} \bigg) \, dm $ Assuming $O$ to be the pivot point of the pendulum implies, ${^N\mathbf{v}^O} = 0$, which simplifies the Lagrangian to: $ \mathcal{L} = \frac{1}{2} \int _{m} \bigg(({^N\boldsymbol{\omega}^B \times \mathbf{r}) + {^B\mathbf{v}^P}} \bigg) \cdot \bigg(({^N\boldsymbol{\omega}^B \times \mathbf{r}) + {^B\mathbf{v}^P}} \bigg) \, dm. $ Expanding the above $ \begin{align} \mathcal{L} = \frac{1}{2} {^N\boldsymbol{\omega}^B} \cdot \Bigg[\int _{m} \mathbf{r} \times({^N\boldsymbol{\omega}^B \times \mathbf{r})\, dm\\ + 2\int _{m} \mathbf{r} \times {^B\mathbf{v}^P}}\, &dm \Bigg]\\ + \frac{1}{2}\int _{m} {^B\mathbf{v}^P} \cdot {^B\mathbf{v}^P}\, &dm \end{align} $ $ \delta S = \delta \int _{t_{0}}^{t_{f}} \mathcal{L} \, dt = \frac{\partial}{\partial \epsilon}\Bigr|_{\epsilon=0} \int _{t_{0}}^{t_{f}} \mathcal{L} \, dt $