## Lagrangian ![[Fig1-eps-converted-to.pdf]] The Lagrangian of the above GVMS is given by $ \mathcal{L} = \frac{1}{2}\int _{m} {^N\mathbf{v}^P} \cdot {^N\mathbf{v}^P} \, dm \tag{2a1.1} $ > [!NOTE] At this point, we can investigate a simple pendulum with varying mass, which is explored in another note in "Next notes" below. In the parent note, we have shown that the velocity of $P$ can be transferred to the mass centre $S^*$. This allows $\mathcal{L}$ to be expressed as: $ \begin{align} \mathcal{L} &&= \frac{1}{2}\int _{m} \bigg( {^N\mathbf{v}^*} + ({^N\boldsymbol{\omega}^B \times \mathbf{p}) + ( {^B\mathbf{v}^P} - {^B\mathbf{v}^*})} \bigg)\\ \cdot &&\bigg( {^N\mathbf{v}^*} + ({^N\boldsymbol{\omega}^B \times \mathbf{p}) + ( {^B\mathbf{v}^P} - {^B\mathbf{v}^*})} \bigg) \, dm\\ \end{align} \tag{2a1.2} $ which can be simplified to: $ \tag{2a1.3} \begin{align} \mathcal{L} = \frac{1}{2}\int _{m} {^N\mathbf{v}^*} &\cdot {^N\mathbf{v}^*}\, dm\\ + \frac{1}{2} {^N\boldsymbol{\omega}^B} &\cdot \Bigg[\int _{m} \mathbf{p} \times({^N\boldsymbol{\omega}^B \times \mathbf{p})\, dm\\ + 2\int _{m} \mathbf{p} \times {^B\mathbf{v}^P}}\, dm\Bigg] \end{align} $ or $ \mathcal{L} = \frac{1}{2}{m} {^N\mathbf{v}^*} \cdot {^N\mathbf{v}^*} + \frac{1}{2} {^N\boldsymbol{\omega}^B} \cdot \Bigg[\int _{m} \mathbf{p} \times({^N\boldsymbol{\omega}^B \times \mathbf{p})\, dm\\ + 2\int _{m} \mathbf{p} \times {^B\mathbf{v}^P}}\, dm\Bigg] \tag{2a1.3} $ which, I will write as two terms; translational and rotational terms of the Lagrangian $\mathcal{L}_{t}$ and $\mathcal{L}_{rot}$, respectively. They are defined as: $ \mathcal{L}_{t} = \frac{1}{2}{m} {^N\mathbf{v}^*} \cdot {^N\mathbf{v}^*} \tag{2a1.4} $ and $ \mathcal{L}_{rot} = \frac{1}{2} {^N\boldsymbol{\omega}^B} \cdot \Bigg[\int _{m} \mathbf{p} \times({^N\boldsymbol{\omega}^B \times \mathbf{p})\, dm\\ + 2\int _{m} \mathbf{p} \times {^B\mathbf{v}^P}}\, dm\Bigg] $, which is further written in short hand as: $ \mathcal{L}_{rot} = (\mathcal{L}_{rot})_{1} + (\mathcal{L}_{rot})_{2} \tag{2a1.5} $ where $ (\mathcal{L}_{rot})_{1} = \frac{1}{2} {^N\boldsymbol{\omega}^B} \cdot \int _{m} \mathbf{p} \times({^N\boldsymbol{\omega}^B} \times \mathbf{p})\, dm \tag{2a1.6} $ and $ (\mathcal{L}_{rot})_{2} = {^N\boldsymbol{\omega}^B} \cdot \int _{m} \mathbf{p} \times {^B\mathbf{v}^P}\, dm \tag{2a1.7} $ We can make use of a result from [[cite_kaneDynamicsTheoryApplications1985]] (Equation ($31$), page 70) to rewrite Equation ($2a1.6$) as $ (\mathcal{L}_{rot})_{1} = \frac{1}{2} {^N\boldsymbol{\omega}^B} \cdot \Bigg[\int _{m} \mathbf{p}^2 \mathbf{U} - \mathbf{p} \mathbf{p} \, dm \Bigg] \cdot {^N\boldsymbol{\omega}^B} \tag{2a1.8} $ In Equation ($2a1.8$), $\mathbf{U}$ is a special dyadic called the *unit dyadic* (see [[cite_kaneDynamicsTheoryApplications1985]], Equation ($12$), page 67)- it is defined as : $ \mathbf{U} = \hat{\mathbf{b}}_{1}\hat{\mathbf{b}}_{1} + \hat{\mathbf{b}}_{2} \hat{\mathbf{b}}_{2} + \hat{\mathbf{b}}_{3}\hat{\mathbf{b}}_{3} \tag{2a1.9} $ where $\hat{\mathbf{b}}_{i}$ $(i = 1,2,3)$ are mutually perpendicular unit vectors. Now, if the system above was of constant mass, then according to another result from [[cite_kaneDynamicsTheoryApplications1985]] (Equation ($16$), page 68) the integral in Equation ($2a1.6$) evaluates to the central inertia dyadic. We will abstain from that for now because we know we will do some other manipulations in deriving the equations of motion of the variable mass system. ## Next notes: There are two pick-off points here, according to me: - [[2a1a Action Integral of GVMS]], which can be used to derive the continuous equations of motion using the principle of least action. - [[2a1b Lagrangian of Variable Mass Simple Pendulum]] similar to that done by Lee et al.