7-1-32. H. Poincaré. Calculations

[Ca. end of October, 1909]

[deux figures]

$$\begin{array}[]{cc}d\varphi&Z\texttt{ fixe}\\ d\theta&\Omega\\ -d\phi&z\texttt{ mobile}\\ p=\theta^{\prime}&O\Omega\\ q=\varphi^{\prime}\sin\theta&OA\\ r=\varphi^{\prime}\cos\theta-\psi^{\prime}&Oz\end{array}$$

$$2T=A[\theta^{\prime 2}+\varphi^{\prime 2}\sin^{2}\theta]+C(\varphi^{\prime}\cos\theta-\psi^{\prime})^{2}$$

$$2T=A\theta^{\prime 2}+\frac{A+C}{2}\varphi^{\prime 2}+\frac{C-A}{2}\varphi^{\prime 2}\cos 2\theta-2C\varphi^{\prime}\psi^{\prime}\cos\theta+C\psi^{\prime 2}$$

$$\begin{array}[]{cc}\Theta=\frac{dT}{d\theta^{\prime}}=A\theta^{\prime}\texttt{;}&\Psi=\frac{dT}{d\psi^{\prime}}=C\psi^{\prime}-C\varphi^{\prime}\cos\theta=-Cr\end{array}$$

$$\phi=\frac{dT}{d\varphi^{\prime}}=\frac{A+C}{2}\varphi^{\prime}+\frac{C-A}{2}\varphi^{\prime}\cos 2\theta-C\psi^{\prime}\cos\theta$$

$$\begin{array}[]{cc}2T=\alpha x^{2}+2\beta xy+\gamma y^{2}=xX+yY\texttt{;}&\mu=\beta^{2}-\alpha\gamma\\ \mu x=-\gamma X+\beta Y\texttt{;}&\mu y=\beta X-\alpha Y\\ X=\alpha x+\beta y\texttt{;}&Y=\beta x+\gamma y\end{array}$$

$$2T=\frac{-\gamma X^{2}+2\beta XY-\alpha Y^{2}}{\mu}$$

$$\mu=C^{2}\cos^{2}\theta-C[\frac{A+C}{2}+\frac{C-A}{2}\cos 2\theta]$$

$$\mu=\frac{C}{2}[(C-(A+C)+cos2\theta(C-(C-A)]=\frac{AC}{2}(cos2\theta-1)$$

$$T=\frac{\Theta^{2}}{2A}-\frac{C\phi^{2}+2C\cos\theta\phi\psi+\psi^{2}[\frac{A+C}{2}+\frac{C-A}{2}\cos 2\theta]}{2\mu}$$

$$\begin{gathered}\Phi=G;\qquad Cr=G\cos\theta;\qquad\Theta=0\text{ pour}\\ T=G^{2}\frac{2C\cos^{2}\theta+\cos^{2}\theta(A\sin^{2}\theta+C\cos^{2}\theta}{AC(1-\cos^{2}\theta)}\\ T=G^{2}\frac{C(1-\cos^{2}\theta)^{2}+A\cos^{2}\theta\sin^{2}\theta}{2AC\sin^{2}\theta}=G^{2}\frac{C\sin^{2}\theta+A\cos^{2}\theta}{2AC}\\ 2T=Ah^{2}+Cr^{2};\qquad G^{2}=A^{2}h^{2}+C^{2}r^{2}\\ Ah=G\sin\theta;\qquad Cr=G\cos\theta;\qquad 2T=G^{2}\left[\frac{sin^{2}\theta}{A}+\frac{\cos^{2}\theta}{C}\right]\\ \frac{dS}{d\theta}=f(\theta,\phi,\psi,T):\varphi+\int\frac{df}{d\phi}d\theta=\text{ const. }|\psi+\int\frac{df}{d\psi}d\theta=\text{const.}\end{gathered}$$

$$\int\frac{df}{dT}d\theta=t+const.$$

$$f=\sqrt{2A(T+F)}$$

$$\begin{array}[]{cc}\int\frac{Ad\theta}{f}=t+const.\texttt{;}&\varphi+\int\frac{A\frac{dF}{d\phi}d\theta}{f}=const.\end{array}$$

$$\begin{array}[]{ccc}\psi+\int\frac{A\frac{dF}{d\psi}d\theta}{f};\qquad\theta=\text{ const. };\qquad\varphi+\frac{dF}{d\phi}t=\text{ const.}\end{array}$$

$$\psi+\frac{dF}{d\psi}t=const.$$

$$Cr=G\cos\theta_{0}$$

$$\frac{T}{G^{2}}=\frac{C\sin^{2}\theta_{0}+A\cos^{2}\theta_{0}}{2AC};\quad\frac{\Theta^{2}}{2AG^{2}}=\frac{C\sin^{2}\theta_{0}+A\cos^{2}\theta_{0}}{2AC}-\frac{C(1-\cos\theta\cos\theta_{0})^{2}+A\cos^{2}\theta_{0}\sin^{2}\theta}{2AC\sin^{2}\theta}$$

$$(1-\cos\theta\cos\theta_{0})^{2}-\sin^{2}\theta\sin^{2}\theta_{0}=0$$

$$[1-\cos(\theta+\theta_{0})][1-\cos(\theta-\theta_{0})]$$

$X,Y,Z$ fixes
$x,y,z$ mobiles

$$\begin{array}[]{ccc}\varphi&OZ&(\text{angle de}OM^{\prime}z\text{avec}yOz)\\ \psi&OX&(\text{angle de}OP^{\prime}\text{avec}Oz)\\ \chi&OZ&(\text{angle de}P^{\prime}OM^{\prime}\text{avec}P^{\prime}OZ)\\ \omega&OX&(\text{angle de}OP^{\prime}\text{avec}OZ)\\ \theta&OZ&(\text{angle de}POZ\text{avec}YOZ)\end{array}$$

$OM^{\prime}$ lié aux axes mobiles ; deux systèmes d’axes coïncident.

$$OP^{\prime}=OZ=Oz$$

1${}^{\text{ère}}$ rotation $OP^{\prime}$ reste fixe.

$$Oz=OP^{\prime}=OZ;\qquad ZP^{\prime}OM^{\prime}z\perp OX$$

2${}^{\text{e}}$ rotation $OP^{\prime}$ fixe

$$\begin{array}[]{ccc}OP^{\prime}=OZ\texttt{;}&Oz\texttt{angle}\psi\texttt{avec}OP^{\prime}\texttt{cet angle ne changera plus ;}&ZP^{\prime}OM^{\prime}z\perp OX\end{array}$$

3${}^{\text{e}}$ rotation $OP^{\prime}$ peu importe $P^{\prime}OZ$ futur(?) lié.

$$\begin{array}[]{cc}OP^{\prime}=OZ\texttt{;}&P^{\prime}OZ\texttt{futur}\perp OX\end{array}$$

4${}^{\text{e}}$ rotation OP’ et P’OZ futur liés axes mobiles
OP’ angle $\omega$ avec OZ changera plus ; P’OZ futur [illisible] à poser par OP’ et OZ

5${}^{\text{e}}$ rotation $OP^{\prime}$ et $P^{\prime}OZ$ futur liés
$P^{\prime}OZ$ futur fait angle $\theta$ avec $?OZ$

$d\varphi$ autour de $Oz$ ; $d\psi$ perp. à $P^{\prime}OzM^{\prime}$
$d\chi$ autour de $OP^{\prime}$ ; $d\omega$ perp. à $P^{\prime}OZ$ ; $d\theta$ autour de $OZ$

$$l\frac{d\chi}{dt}=h\frac{d\varphi}{dt}=\frac{A-C}{A}rh$$

$$\begin{array}[]{ccc}lG=(A-C)hr\texttt{;}&A^{2}h^{2}+C^{2}r^{2}=G^{2}\texttt{;}&Ah^{2}+Cr^{2}=2T\end{array}$$

$$(\lambda+i\mu+j\nu+k\rho)=e^{k\frac{\varphi}{2}}(C\psi+is\psi)e^{k\frac{\chi}{2}}(C\omega+is\omega)e^{k\frac{\theta}{2}}$$

$$\begin{array}[]{cc}ik=-ki=-j\texttt{;}&e^{k\frac{\varphi}{2}}i=ie^{-k\frac{\varphi}{2}}\end{array}$$

$$C\psi C\omega e^{\frac{k}{2}(\varphi+\chi+\theta)}+is\psi C\omega e^{\frac{k}{2}(\chi+\theta-\varphi)}+is\omega C\psi e^{\frac{k}{2}(\theta-\varphi-\chi)}-s\omega s\psi e^{\frac{k}{2}(\theta-\varphi-\chi)}$$

$$\lambda=C\psi C\omega\cos\frac{\varphi+\chi+\theta}{2}-s\psi s\omega\cos\frac{\theta+\varphi-\chi}{2}$$

$$\mu=s\psi C\omega\cos\frac{\chi+\theta-\varphi}{2}+C\psi s\omega\cos\frac{\theta-\chi-\varphi}{2}$$

$$\nu=s\psi C\omega\sin\frac{\varphi-\chi-\theta}{2}+C\psi s\omega\sin\frac{\varphi+\chi-\theta}{2}$$

$$\rho=C\psi C\omega\sin\frac{\varphi+\chi+\theta}{2}-s\psi s\omega\sin\frac{\theta+\varphi-\chi}{2}$$

AD 3p. Collection particulière, Paris 75017.

Time-stamp: "22.06.2022 12:00"

Notes

• 1 The three pages of calculations transcribed here are from a one-page letter from Benjamin Baillaud to Poincaré, 22.10.1910 (§ 3-2-3).