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2

$T_\mu^\nu = T_{\mu\sigma}g^{\sigma\nu}$


1

Of course you can specialise the derivation of the three dimensional Euler equations for your special case. However, unless you are expressly asked to do this, I doubt that your teacher means for you to take this path because it is very much an overkill. The yoyo is spinning in a plane about its axis of symmetry. The inertia tensor is now a simple scalar ...


1

Integrating your equations (1) and (2) you get that $(\text i) \ \omega_1 = C_1$ a constant in time, while another constant in time. This equality you can re-write as $(\text {ii}) \ \omega_2^2 + \omega_3^2 = C_2^2$, As to the vector $J_{\Omega}\omega$ you know that its components are $\{J_1\omega_1, J_2\omega_2, J_2\omega_3\}$, therefore $(\text ...


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I will answer the mass moment of inertia tension question. For an infinitesimal clump of mass ${\rm d}m$ located at $\vec{r}$ its effect on the inertia tensor is $${\rm d}{\bf J} = -[\vec{r}\times][\vec{r}\times]{\rm d}m$$ where $[\vec{r}\times]$ is the skew symmetric cross product operator $$ \begin{pmatrix}x\\y\\z \end{pmatrix}\times = ...


0

The important point is that both $t^a$ and $v^b$ are vectoror fields defined (at least) on $C$. The idea is that if $t^a\nabla_av^b=0$ and you take $v^b$ at a particular point $p$ on $C$ then $v^b$ at all other points on $C$ is the parallel transported vector from $p$. Another way of thinking about it is that the covariant derivative in a given direction is ...



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