Abstract
The theory of rigid bodies is a particularly important part of general mechanics. Firstly, next to the spherically symmetric mass distributions that we studied in Sect. 1.30, the top is the simplest example of a body with finite extension. Secondly, its dynamics is a particularly beautiful model case to which one can apply the general principles of canonical mechanics and where one can study the consequences of the various space symmetries in an especially transparent manner.
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Notes
- 1.
In general, \(J_{\mu \nu }\) depends on time, whenever the \({\varvec{x}}_{i}\) refer to a fixed reference frame in space and when the body rotates with respect to that frame (see Sect. 3.11).
- 2.
This does not necessarily mean that the rigid body has a spherical shape.
- 3.
St. Ebenfeld, F. Scheck: Ann. Phys. (New York) 243, 195 (1995). Note that the vector \(\varvec{\sigma }\) equals \(-\varvec{a}\) in this reference and that the choice of convention for the rotation \({\mathbf {R}}(t)\), while consistent with earlier sections of this chapter, is the inverse of the one employed there.
- 4.
In order to become familiar with this notation and calculus the reader should verify that \(\bigl \langle \hat{\varvec{e}}_{\bar{i}}\vert {{\mathbf {\mathsf{{J}}}}}\vert \hat{\varvec{e}}_{\bar{k}}\bigr \rangle = \text {diag}(I_1,I_1,I_3)\), and that \({{\mathbf {\mathsf{{J}}}}}^{-1}\) is indeed the inverse of \({{\mathbf {\mathsf{{J}}}}}\).
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Appendix: Practical Examples
Appendix: Practical Examples
1. Symmetric Top in a Gravitational Field. Study quantitatively the motion of a symmetric spinning top in the earth’s gravitational field (a qualitative description is given in Sect. 3.16).
Solution. It is convenient to introduce dimensionless variables as follows. For the energy \(E^{\prime }\) (3.100) take
Instead of the projections \(L_{3}\) and \({\bar{L}}_{3}\) introduce
The function f(u) on the right-hand side of (3.104) is replaced with the dimensionless function
As one may easily verify, the ratio \(I^{\prime }_{1} /Mgl\) has dimension (time)\(^{2}\). Thus,\(\omega \,{\buildrel {\scriptstyle \mathrm{def}}\over =}\, \sqrt{Mgl/I^{\prime }_{1}} \) is a frequency. Finally, using the dimensionless time variable \(\tau \,{\buildrel {\scriptstyle \mathrm{def}}\over =}\, \omega t,\) (3.104) becomes
Vertical rotation is stable if \({\bar{L}}^{2} _{3} >4Mgl I^{\prime }_{1},\) i.e. if \({\bar{\lambda }} >2.\) The top is vertical if \(\lambda ={\bar{\lambda }}.\) With \(u\rightarrow 1,\) the critical energy, with regard to stability, is then \(\varepsilon _{\mathrm{crit.}} (\lambda ={\bar{\lambda }}) =0. \) For the suspended top we have \(\lambda =-{\bar{\lambda }} ,\) \(u\rightarrow 1,\) and the critical energy is
The equations of motion now read
and
Curve A of Fig. 3.25 corresponds to the case of a suspended top, i.e. \(\lambda =-{\bar{\lambda }} \) and \(u_{0} =-1.\) We have chosen \(\varepsilon =0,\) \(\lambda =3.0.\) Curve C corresponds to the vertical top, and we have chosen \(\varepsilon =2,\) \(\lambda ={\bar{\lambda }}=5.\) Curve B, finally, describes an intermediate situation. Here we have taken \(\varepsilon =2,\) \(\lambda =4,\) \({\bar{\lambda }}=6.\)
The differential equations (A.5) and (A.6) can be integrated numerically, e.g. by means of the Runge–Kutta procedure described in Practical Example 2.2. For this purpose let
and read (A.11) and (2.12) of the Appendix to Chap. 2 as equations with two components each. This allows us to represent the motion of the axis of symmetry in terms of angular coordinates \((\theta ,\varPhi )\) in the strip between the two parallels defined by \(u_{1}\) and \(u_{2}\). It requires a little more effort to transcribe the results onto the unit sphere and to represent them, by a suitable projection, as in Fig. 3.26.
2. The Tippe Top. Under the assumption that the coefficient of gliding friction is proportional to \(g_\mathrm{n}\), the coefficient that appears in the normal force, numerically integrate the equations of motion (3.120a–3.120c) with the three possible choices for the moments of inertia.
Solution. The assumption is that \(g_\mathrm{fr}=\mu g_\mathrm{n}\). Let \(v=\Vert \varvec{v}\Vert \) be the modulus of the velocity. In order to avoid the discontinuity at \(\varvec{v}=0\) on the right-hand side of (3.120c) one can replace \(\hat{\varvec{v}}\) by
where M is a large positive number. Indeed, the factor \(\tanh (M\Vert \varvec{v}\Vert )\) vanishes at zero and tends quickly, yet in a continuous fashion, to 1. It is useful to introduce appropriate units of length, mass, and time such that \(R=1\), \(m=1\) and \(g=1\). Furthermore, the coefficient of friction \(\mu \) should be chosen sufficiently large, say \(\mu =0.75\), so that the numerical solutions quickly reach the asymptotic state(s). Compare your results with the examples given by Ebenfeld and Scheck (1995) footnote 3.
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Scheck, F. (2018). The Mechanics of Rigid Bodies. In: Mechanics. Graduate Texts in Physics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-55490-6_3
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