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George Keeling

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- I am completely stuck on problem 2.45 of Blennow's book Mathematical Models for Physics and Engineering.

I am completely stuck on problem 2.45 of Blennow's book Mathematical Models for Physics and Engineering. @Orodruin It says

"We just stated that the moment of inertia tensor ##I_{ij}## satisfies the relation$${\dot{I}}_{ij}\omega_j=\varepsilon_{ijk}\omega_jI_{kl}\omega_l$$Show that this relation is true by starting from Eq. 2.146 and using the fact that ##\vec{v}=\vec{\omega}\times\vec{x}##."

##\vec{\omega}## is the angular velocity. The overdot signifies a time derivative. We're in Cartesian coordinates. Eq. 2.146 was a formula for moment of inertia about the origin for a solid object of constant density ##\rho## as a volume integral. The ##x_i## are the coordinates (or components of the position vector ##\vec{x}##) of the volume element ##dV##.$$I_{ij}=\int_{V}^{\ }\rho\left(x_kx_k\delta_{ij}-x_ix_j\right)dV$$I have studied all the examples about angular stuff and learned that angular velocity ##\vec{\omega}##, angular momentum ##\vec{L}## and moment of inertia ##I## are always relative to a point. (I always thought they were relative to an axis).

I tried to differentiate the integral with respect to time, but I can't get it to make sense and I'm not sure it's even possible.

I tried writing the infinitesimal version of the formula as$$dI_{ij}=\rho\left(x_kx_k\delta_{ij}-x_ix_j\right)dV$$then$${\dot{I}}_{ij}=\frac{dI_{ij}}{dt}=\rho\left(x_kx_k\delta_{ij}-x_ix_j\right)\frac{dV}{dt}=\left(x_kx_k\delta_{ij}-x_ix_j\right)\frac{dM}{dt}$$where ##dM## is the mass of the volume element. But ##\frac{dM}{dt}## is zero except when the boundary of the solid is at the coordinates and then it is very large.

I tried fiddling with the RHS of the relation in question and got $$\varepsilon_{ijk}\omega_jI_{kl}\omega_l=\omega_j\rho\int_{V}^{\ }\left[\varepsilon_{ijk}\omega_kx_mx_m-\varepsilon_{ijk}\omega_lx_kx_l\right]dV$$$$=\omega_j\rho\int_{V}^{\ }{\varepsilon_{ijk}\omega_kx_mx_mdV}-\rho\int_{V}^{\ }{\varepsilon_{ijk}\omega_jx_k\omega_lx_ldV}=\omega_j\rho\int_{V}^{\ }{\varepsilon_{ijk}\omega_kx_mx_mdV}-\rho\int_{V}^{\ }{v_i\omega_lx_ldV}$$so I managed to use ##\vec{v}=\vec{\omega}\times\vec{x}## in the last part but am not much wiser.

I tried a few other things and nothing helped.

Is there another way to get going?

"We just stated that the moment of inertia tensor ##I_{ij}## satisfies the relation$${\dot{I}}_{ij}\omega_j=\varepsilon_{ijk}\omega_jI_{kl}\omega_l$$Show that this relation is true by starting from Eq. 2.146 and using the fact that ##\vec{v}=\vec{\omega}\times\vec{x}##."

##\vec{\omega}## is the angular velocity. The overdot signifies a time derivative. We're in Cartesian coordinates. Eq. 2.146 was a formula for moment of inertia about the origin for a solid object of constant density ##\rho## as a volume integral. The ##x_i## are the coordinates (or components of the position vector ##\vec{x}##) of the volume element ##dV##.$$I_{ij}=\int_{V}^{\ }\rho\left(x_kx_k\delta_{ij}-x_ix_j\right)dV$$I have studied all the examples about angular stuff and learned that angular velocity ##\vec{\omega}##, angular momentum ##\vec{L}## and moment of inertia ##I## are always relative to a point. (I always thought they were relative to an axis).

I tried to differentiate the integral with respect to time, but I can't get it to make sense and I'm not sure it's even possible.

I tried writing the infinitesimal version of the formula as$$dI_{ij}=\rho\left(x_kx_k\delta_{ij}-x_ix_j\right)dV$$then$${\dot{I}}_{ij}=\frac{dI_{ij}}{dt}=\rho\left(x_kx_k\delta_{ij}-x_ix_j\right)\frac{dV}{dt}=\left(x_kx_k\delta_{ij}-x_ix_j\right)\frac{dM}{dt}$$where ##dM## is the mass of the volume element. But ##\frac{dM}{dt}## is zero except when the boundary of the solid is at the coordinates and then it is very large.

I tried fiddling with the RHS of the relation in question and got $$\varepsilon_{ijk}\omega_jI_{kl}\omega_l=\omega_j\rho\int_{V}^{\ }\left[\varepsilon_{ijk}\omega_kx_mx_m-\varepsilon_{ijk}\omega_lx_kx_l\right]dV$$$$=\omega_j\rho\int_{V}^{\ }{\varepsilon_{ijk}\omega_kx_mx_mdV}-\rho\int_{V}^{\ }{\varepsilon_{ijk}\omega_jx_k\omega_lx_ldV}=\omega_j\rho\int_{V}^{\ }{\varepsilon_{ijk}\omega_kx_mx_mdV}-\rho\int_{V}^{\ }{v_i\omega_lx_ldV}$$so I managed to use ##\vec{v}=\vec{\omega}\times\vec{x}## in the last part but am not much wiser.

I tried a few other things and nothing helped.

Is there another way to get going?