Physlib.ClassicalMechanics.Force

Given a reference frame `frame` and a type of `Object`, this instance allows an element $F$ of type `frame.Force Object` to be treated as a function from time to vectors. Specifically, for any $t \in \text{Time}$, $F(t)$ returns the vector value of the force at that time within the specified reference frame.

For a given reference frame and object type, an internal force $F$ can be treated as a function that maps a time $t \in \text{Time}$ to a vector $V \in \text{Vector}$. This instance allows the use of the notation $F(t)$ to represent the vector value of the internal force at that specific time.

Given a reference frame and a type of objects, an internal force (which consists of a source object, a target object, and a time-dependent vector) can be coerced into a general force (which consists only of a target object and a time-dependent vector). This is achieved by the function `InternalForce.toForce`, which retains the target and the vector value while disregarding the source object.

For an internal force $F$ in a given reference frame, characterized by a source object $S$, a target object $T$, and a time-dependent vector value $\mathbf{v}(t)$, the function `reverse` defines the corresponding reaction force. This resulting internal force has $T$ as its source and $S$ as its target, with its vector value at any time $t$ being the additive inverse $-\mathbf{v}(t)$ of the original force. This corresponds to the "equal and opposite" force described by Newton's Third Law.

In a given reference frame, the net force on a specific object $o$ at time $t$ is the vector sum of all forces acting on that object. Given a multiset of internal forces $\mathcal{I}$ and a multiset of external forces $\mathcal{E}$, the net force is computed by: 1. Converting all internal forces in $\mathcal{I}$ to general forces (retaining their target and vector value). 2. Combining these with the external forces $\mathcal{E}$ to form a single collection of forces $\mathcal{F}$. 3. Summing the vector values $\vec{F}(t)$ for all forces $\vec{F} \in \mathcal{F}$ whose target is the object $o$. Mathematically, this is expressed as: $$\text{netForce}(o, \mathcal{I}, \mathcal{E}, t) = \sum_{\{\vec{F} \in (\mathcal{I} + \mathcal{E}) \mid \text{target}(\vec{F}) = o\}} \vec{F}(t)$$ where the sum is taken over the multiset of forces whose target property matches $o$.