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

Material acceleration field

The acceleration of a material particle \(\u{X}\) executing its motion in \(\mathbb{E}\) at the instant \(\u{\tau}\) is \(\begin{equation} \u{x}_{\u{X}}''(\u{\tau})=:\mt[\u{\tau}]{\u{a}}(\u{X}), \end{equation}\)

where \(\u{x}_{\u{X}}''(\u{\tau})\) is the value of the Fréchet derivative of \(\u{x}_{\u{X}}'\) at the time instant $\u{\tau}$. To be more precise, the acceleration of the material particle $\u{X}$ at the instant \(\u{\tau}\) equals the value of the Fréchet derivative of the map \(\mathbb{T}\ni \u{\tau} \mapsto \u{x}_{\u{X}}'(\u{\tau})\in \mathbb{V}\), where \(\mathbb{A}:=\mathcal{L}\pr{\mathbb{T},\mathbb{V}}\). The space \(\mathbb{A}\) is called the physical acceleration space and it can be shown that the set \(\pr{\u{a}_{i}}_{i\in\mathcal{I}}\), where \(\u{a}_i\u{\tau}:=\tau \u{v}_i\), provides an orthonormal basis for \(\mathbb{A}\). The mapping \(\mt[\u{\tau}]{\u{v}}:\u{\kappa}_{\rm R}(\mathcal{B})\to \mathbb{A}\) is called the material acceleration field. The material acceleration field answers the question: what is the the acceleration of the material partial \(\u{X}\) at the time instance \(\u{\tau}\).

When \(\u{x}_{\u{\tau}}(\u{X})=x_{i}(X_i(\u{X}),\tau)\u{e}_i\) it follows that \(\begin{align} \mt[\u{\tau}]{\u{a}}(\u{X})=x_{i,\tau\tau}(X_i(\u{X}),\tau)\u{a}_{i} \end{align}\)