What are the mathematical models for force, acceleration and velocity?

Question

1. In mechanics, the space can be described as a Riemann manifold. Forces, then, can be defined as vector fields of this manifold. Accelerations are linear functions of forces, so they are covector fields. But what about velocities and many other kinds of vectors?

2. Of course velocities are not forces, so I don't think it is right to reuse vector fields of this manifold. But does this mean that this manifold has many different tangent spaces at each point?

3. This sounds very strange to me. I think the problem is that math models have no physical units, maybe somehow we can create a many-sorted manifold to accommodate units?

Answer 1

Velocities and Spatial Accelerations are twists and Forces and Momenta are wrenches. Both are screws (two-vectors) with one vector free and the other a spatial field. All of them transform with the same laws and their interactions have many dual properties.

_{NOTE: See "A treatise on the theory of screws", Stawell R Ball, https://archive.org/details/theoryscrews00ballrich}

The proportionality tensor transforming twists to wrenches is the 6×6 spatial mass matrix converting motion into momentum and acceleration into forces.

For example below I am composing a velocity twist and a momentum wrench. Do you spot the similarities?

$$\begin{aligned} {\hat v} &= \begin{pmatrix} {\bf \omega} \\{\bf r} \times {\bf \omega} \end{pmatrix} & {\hat p} &= \begin{pmatrix} {\bf p} \\{\bf r} \times {\bf p} \end{pmatrix}\end{aligned} $$

Answer 2

In classical mechanics a system is described by a Lagrangian $\mathscr{L}\colon TQ\to \mathbb{R}$, with $Q$ being the configuration space and $TQ$ its tangent bundle, namely the union over $q\in Q$ of all tangent spaces $T_qQ$: $TQ = \cup_q T_qQ$. A local chart on $Q$ looks like $(q_1, \ldots, q_n)$, the $q_k$ being the degrees of freedom of the system. The Lagrangian is then $\mathscr{L}\equiv\mathscr{L}\big(q(t), v(t)\big)$ and the equations of motion are: $$ \frac{d}{dt}\frac{\partial \mathscr{L}}{\partial v^{\mu}} - \frac{\partial \mathscr{L}}{\partial q^{\mu}}=0. $$ The solution is a collection of $\big(q^{\mu}(t), v^{\mu}(t)\big)$ that live on $TQ$; if we make the further requirement that, on those solutions, $v=\dot{q}$, then the path on $TQ$ projects uniquely onto a path on $Q$, whose flow is given by the velocity fields.

To directly answer your questions:

Forces, then, can be defined as vector fields of this manifold. Accelerations are linear functions of forces, so they are covector fields. But what about velocities and many other kinds of vectors?

Wrong. Positions and velocities are coordinates of local charts $\phi$ from the tangent bundle $\phi\colon U\subset TQ\to\mathbb{R}$: as such, they transform contra-variantly. Forces, in the above formalism, are related to the conjugate momenta $p_{\mu}=\partial\mathscr{L}/\partial{v^{\mu}}$ and hence transform co-variantly, with the inverse matrix.

Of course velocities are not forces, so I don't think it is right to reuse vector fields of this manifold. But does this mean that this manifold has many different tangent spaces at each point?

See above. Also, manifolds just have one tangent space at each point, defined as the set of all directional derivatives calculated in that point.

I think the problem is that math models have no physical units, maybe somehow we can create a many-sorted manifold to accommodate units?

That has absolutely nothing to do with units.