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- About the definition of moment of inertia tensor (inertia tensor) from a differential geometry point of view
Hello,
I'd like to better understand the definition of inertia tensor from a mathematical viewpoint.
As discussed here, one defines the (0,2)-rank system's moment of inertia tensor (inertia tensor) ##\mathbf I ## w.r.t. the system's CoM. Of course such a tensor ##\mathbf I## depends on the system's orientation in space, i.e. it depends on the "point" representing it in configuration space.
Just to fix ideas, consider a rigid dumbbell. To describe its orientation in 3D space 2 parameters are required (e.g. spherical coordinates ##\theta, \varphi## w.r.t. its CoM), hence the dumbbell configuration space is two-dimensional. Now, at each point in such configuration space, one can assign the (0,2)-rank inertial tensor.
My question: does this assignment actually define a tensor field? AFAIK, a tensor field requires the underlying notion of "tensor bundle" built from a differentiable manifold ##M##. Here ##M## is the system's configuration space, however the dimensions do not match.
Namely ##M## as configuration space is two-dimensional (like the dimension of the tangent space at each point) while the inertia tensor "eats" three-dimensional vectors (i.e. angular velocities).
What is actually going on? Thanks.
I'd like to better understand the definition of inertia tensor from a mathematical viewpoint.
As discussed here, one defines the (0,2)-rank system's moment of inertia tensor (inertia tensor) ##\mathbf I ## w.r.t. the system's CoM. Of course such a tensor ##\mathbf I## depends on the system's orientation in space, i.e. it depends on the "point" representing it in configuration space.
Just to fix ideas, consider a rigid dumbbell. To describe its orientation in 3D space 2 parameters are required (e.g. spherical coordinates ##\theta, \varphi## w.r.t. its CoM), hence the dumbbell configuration space is two-dimensional. Now, at each point in such configuration space, one can assign the (0,2)-rank inertial tensor.
My question: does this assignment actually define a tensor field? AFAIK, a tensor field requires the underlying notion of "tensor bundle" built from a differentiable manifold ##M##. Here ##M## is the system's configuration space, however the dimensions do not match.
Namely ##M## as configuration space is two-dimensional (like the dimension of the tangent space at each point) while the inertia tensor "eats" three-dimensional vectors (i.e. angular velocities).
What is actually going on? Thanks.