Are displacements, velocities, forces etc. in the same vector space?

Question

When we label a diagram with vector arrows for displacement, velocity, acceleration, force etc., are these quantities all in the same vector space, or are they in separate, interacting vector spaces? Do forces perhaps 'act' on bodies in the sense of group action, as well as in the physical sense?

Answer 1

Technically they should be in different vector spaces. Even forces at different points don't have to belong to the same vector space. Velocities at a point belong to the tangent space at that point, and accelerations to the double tangent space.

Answer 2

They are not in the same space. For instance:

• velocities are vectors in the tangent space $T_qQ$(which is a manifold) of the configuration manifold $Q$;
• momenta are in the cotangent bundle $T_{(p,q)}^*Q$;
• forces are proportional to accelerations, and hence are vectors in the tangent space of the tangent bundle $T_q(TQ)$.

If you are working on $Q=\mathbb{R}^n$ since all these different spaces can be identified with $\mathbb{R}^n$ itself, it looks like they belong to the same space. But in general for an arbitrary smooth manifold $Q$ this is not the case.