\[\renewcommand{\u}[1]{\boldsymbol{#1}} \renewcommand{\b}[1]{\mathbb} \newcommand{\set}[2]{\left\{#1\,\big|\,#2\right\}} \newcommand{\pr}[1]{\left(#1\right)} \newcommand{\st}[2][]{#2_{\mathscr{s}#1}} \newcommand{\mt}[2][]{#2_{\mathscr{m}#1}} \newcommand{\lsc}[2][\mathscr{l}]{ { }^{#1}\!#2 } \newcommand{\norm}[1]{\lVert #1 \rVert}\]

Spatial acceleration field

The spatial acceleration field yields the acceleration at the time instance \(\u{\tau}\) of the material particle that is located \(\u{x}\) at the time instance \(\u{\tau}\). The material particle that is located at \(\u{x}\) at time \(\u{\tau}\) as \(\u{\varphi}_{\u{\tau}}^{-1}(\u{x})\). The acceleration of the material particle \(\u{\varphi}_{\u{\tau}}^{-1}(\u{x})\) at the time instance \(\u{\tau}\) is of course \(\u{A}_{\u{\tau}}\pr{\u{\varphi}_{\u{\tau}}^{-1}(\u{x})}\). Thus, we have the spatial acceleration field as \(\begin{align} \u{a}_{\u{\tau}}\pr{\u{x}}&=\mt[\u{\tau}]{\u{a}}(\u{\varphi}_{\u{\tau}}^{-1}(\u{x}))\\ &=x_{i,\tau\tau}(X_i\circ\u{\varphi}_{\u{\tau}}^{-1}(\u{x}),\tau)\u{a}_{i} \end{align}\)

Note \(\begin{equation} \u{a}(\u{x},t)\neq\frac{\partial \u{v}(\u{x},t)}{\partial t} \end{equation}\)