## Discrete Action Sum With this note's parent describing what the continuous Lagrangian and Action sums are, we can now define the discrete version of the Action Integral i. e. the discrete action sum by [[cite_leeLieGroupVariational2005]]. ![[Fig1-eps-converted-to.pdf]] $ \begin{aligned} \mathcal{S}_{d} = \sum_{k=0}^{N-1} \int _{{t_{k}}} ^{t_{k+1}} \mathcal{L} \, dt \end{aligned} $ where, $\mathcal{S}_{d}$ is the discrete action sum. Note that on the right hand side of the above equation, the integral of the Lagrangian is taken over a single time step $t_{k}$ to ${t_{k+1}}$. This is known as the exact discrete Lagrangian \cite{[[cite_marsdenDiscreteMechanicsVariational2001]]} denoted by $\mathcal{L}_{d}^{E}$: $ \begin{aligned} \mathcal{L}_{d}^{E} = \int _{{t_{k}}} ^{t_{k+1}} \mathcal{L} \, dt \end{aligned} $ [[2a1 Lagrangian of GVMS|We already know]] that the continuous Lagrangian is given by from : $ \mathcal{L} = \int _{m} {^N\mathbf{v}^P} \cdot {^N\mathbf{v}^P} \, dm $ ## Next notes This creates two lines of work: - [[2a1a1a Discrete Lagrangian of GVMS]] - [[2a1b Lagrangian of Variable Mass Simple Pendulum]] that builds on Lee et al's work on constant mass pendulums.