I began this category of notes after reading [[cite_manchesterQuaternionVariationalIntegrators2016a]] and asking for Steven's help unpacking this work a bit more quickly. This can be seen here [[@ Steven Derivation Request]]. ## Introduction The problem of optimisation concerns finding the best way to do something. The inherent assumption, at the outset, is that an optimal path exists and thus our immediate concern is to seek conditions that lead to finding this optimal solution. So, bearing in mind this assumption, we define a family of curves around this optimal "way" (or path or trajectory) represented by $x(t, \epsilon)$, where $\epsilon$ is a set of infinitesmally small values. Note that, the curve represents the optimal path when $\epsilon$ is set to zero. ## The Varied Quantity and The Variation The literature also refers to $x(t, \epsilon)$ as the **varied quantity or varied path**, which can be expanded using Taylor's series: $ \begin{align} x(t, \epsilon) &= x(t, 0) + \frac{\partial x}{\partial \epsilon}\Bigr|_{\epsilon=0} (\epsilon - 0) + \mathcal{O}(\epsilon^2)\\ &= x(t, 0) + \frac{\partial x}{\partial \epsilon}\Bigr|_{\epsilon=0} \epsilon \\ \\ &= x(t, 0) + \delta x (t). \\ \end{align} $ Having dropped the second (and higher) order terms above, we then get to define the second term on the RHS as "the variation of $x(t)quot;, which is denoted by $\delta x (t)$. More correctly, this is called the first variation of $x(t)$ . According to [[cite_longuskiOptimalControlAerospace2014]], Lagrange used another form of the first variation: $ \delta x (t) = \epsilon \eta(t). $ ## Next note [[3a Examples of varied quantities]]