## Quaternions This subsection is mostly from [[cite_manchesterQuaternionVariationalIntegrators2016a]] where he says "the exponential of a quaternion having zero scalar part is always a unit quaternion, a varied unit quaternion can be defined as: $ {^\epsilon}q = qe^{\epsilon \hat{\boldsymbol{\eta}}} \tag{3a.1} $ " In the above equation, ${^\epsilon}q$ denotes a varied quaternion. > [!seealso] hat notation > The $\hat{}$ operator was first seen in Equation ($4$) in [[3b Quaternions#Rotations with Unit Quaternions]]. > One can also compute the time derivative of this varied quaternion as: $ {^\epsilon}\dot{q} = \dot{q}e^{\epsilon \hat{\boldsymbol{\eta}}} + \epsilon q \dot{\hat{\boldsymbol{\eta}}} e^{\epsilon \hat{\boldsymbol{\eta}}} \tag{3a.2} $ ### First Variation of quaternion The first variation of the quaternion is given by (see [[3 What is a varied quantity in variational calculus#The Varied Quantity and The Variation|immediate parent to this note]]): $ \delta q (t) = \frac{\partial {{^\epsilon}q}}{\partial \epsilon}\Bigr|_{\epsilon=0} \epsilon \tag{3a.3} $ which ought to simplify to: $ \delta q (t) = qe^{\epsilon \hat{\boldsymbol{\eta}}}\hat{\boldsymbol{\eta}}\Bigr|_{\epsilon=0} \epsilon \tag{3a.4} $ which, upon setting $\epsilon=0$, results in: $ \delta q (t) = \epsilon q\hat{\boldsymbol{\eta}} \tag{3a.5} $ ## Rotation Matrices This subsection is mostly from [[cite_leeLieGroupVariational2005]], where they say "the configuration space is the rotation group $SO(3)$, the variation should be consistent with the rotation group". The varied rotation matrix is $R^\epsilon \in \mathrm{SO}(3)$ and obtained by: $ R{^\epsilon} = Re^{\epsilon \eta} \tag{3a.6} $ where: - $R \in \mathrm{SO}(3)$ is the rotation matrix from the body-fixed frame to the inertial frame; - $\epsilon \in \mathbb{R}$, $\eta \in \mathfrak{so}(3)$ denotes a variation in the Lie algebra of skew symmetric matrices vanishing at $t_0$ and $t_f $. One can also compute the time derivative of this varied rotation matrix as: $ \dot{R}{^\epsilon} = \dot{R}e^{\epsilon {\eta}} + \epsilon R e^{\epsilon {\eta}} \dot{\eta} \tag{3a.7} $ ### First Variation of Rotation Matrices