Abstract
A unified formulation of rigid body dynamics based on Gauss principle is proposed. The Lagrange, Kirchhoff and Newton–Euler equations are seen to arise from different choices of the quasi-coordinates in the velocity space. The group-theoretical aspects of the method are discussed.
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Notes
According to Eq. (10a), the vector \(\underline{\xi }\) represents the velocity of the point of the body \(\mathfrak B\) instantly located at the space origin of the observer’s frame. As such, it may look a rather factitious object. A better understanding of the symmetry hidden in the representation (11) is gained interpreting the vectors \(-\underline{\xi }\), \(-\underline{\omega }\) respectively as the linear and angular velocity of the frame \(\mathfrak F\) relative to \(\mathfrak F'\), and the vector \(-\underline{v}_P\) as the velocity, relative to \(\mathfrak F'\), of a point \(P\) at rest in \(\mathfrak F\). In this way, Eq. (11) is on the same footing as Eq. (8), namely it describes, up to a sign, the rigid motion of \(\mathfrak F\) relative to \(\mathfrak F'\). Interchanging left and right invariance is therefore equivalent to interchanging the roles of the frames \(\mathfrak F\) and \(\mathfrak F'\), i.e. to replacing each transformation by the corresponding inverse.
Although conceptually preferable, for holonomic systems Gauss’ principle is not strictly necessary in order to establish Eq. (14): one may equally well start with d’Alembert’s principle, make use of the identity \(\frac{\partial {P_i}}{\partial {q^\alpha }}=\frac{\partial {\underline{v}_i}}{\partial {\dot{q}^\alpha }}\), and replace the resulting equations by the linear combinations
$$\begin{aligned} 0=\sum _{i=1}^N\big (m_i\underline{a}_i-\underline{F}_i\big )\cdot \frac{\partial {\underline{v}_i}}{\partial {\dot{q}^\beta }}\,\frac{\partial {\dot{q}^\beta }}{\partial {z^\alpha }}\,=\, \sum _{i=1}^N\big (m_i\underline{a}_i-\underline{F}_i\big )\cdot \frac{\partial {\underline{v}_i}}{\partial {z^\alpha }} \end{aligned}$$
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Massa, E., Vignolo, S. Newton–Euler, Lagrange and Kirchhoff formulations of rigid body dynamics: a unified approach. Meccanica 51, 2019–2023 (2016). https://doi.org/10.1007/s11012-015-0333-7
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DOI: https://doi.org/10.1007/s11012-015-0333-7