## Discrete Lagrangian ![[Fig1-eps-converted-to.pdf]] It is key to note that the exact exact discrete Lagrangian, $\mathcal{L}_{d}^{E}$, is different from the discrete Lagrangian, $\mathcal{L}_{d}$. This note builds on that concept. The exact discrete Lagrangian, $\mathcal{L}_{d}^{E}$, relates to the Lagrangian in the following way $ \begin{aligned} \mathcal{L}_{d}^{E} = \int _{{t_{k}}} ^{t_{k+1}} \mathcal{L} \, dt \end{aligned} $ The Lagrangian of the above variable mass system is given by $ \mathcal{L} = \int _{m} {^N\mathbf{v}^P} \cdot {^N\mathbf{v}^P} \, dm $ which can be treated by transferring velocity computation to point $O$ on the rigid body; this is detailed in Nanjangud and Eke: $ \begin{aligned} \mathcal{L} &&= \int _{m} \bigg( {^N\mathbf{v}^O} + ({^N\boldsymbol{\omega}^B \times \mathbf{r}) + ( {^B\mathbf{v}^P} - {^B\mathbf{v}^*})} \bigg)\\ \cdot &&\bigg( {^N\mathbf{v}^O} + ({^N\boldsymbol{\omega}^B \times \mathbf{r}) + ( {^B\mathbf{v}^P} - {^B\mathbf{v}^*})} \bigg) \, dm\\ \end{aligned} $