Constraint - Aetherwisp

A constraint is a limitation put on a system that prevents it from moving in certain locations in space. Applying a constraint to a system reduces the number of unknowns of that systems, leading to a simplified description of the same phenomenon. A bound system can only move in a subset of the space it would have otherwise had access to. In mathematical terms, each constraint shrinks the configuration space of the system by reducing the degrees of freedom: one constraint removes one degree of freedom.

Most mechanical bindings can be described as forces with Newton's second law:

m\mathbf{a}=\mathbf{F}+\Phi

$m$ is mass, $\mathbf{a}$ is acceleration, $\mathbf{F}$ is an active force, which is known, and $\Phi$ is a constraint reaction, which is not known.

Properties#

A constraint is said to be ideal if the configuration space is "smooth", that is, if the constraint reaction in any configuration $P$ is always orthogonal to $Q$ . Mathematically, this is like saying that the Scalar product of $\Phi$ and any tangent element $\delta \mathbf{r}$ is always zero:

\Phi\cdot \delta\mathbf{r}=0\quad\forall P\in \mathbb{R}^{n}\text{ and }\forall \delta \mathbf{r}\in T_{P}Q\quad\Leftrightarrow\quad \Phi\cdot \frac{ \partial \mathbf{r} }{ \partial q_{i} } \text{ where }i=1\ldots,n

where $T_{P}Q$ is the tangent space of $Q$ in the point $P$ . If the constraint applies to $N$ point masses, only the total reaction needs to be orthogonal:

\sum_{i=1}^{N} \Phi\cdot \delta\mathbf{r}_{i}=0\quad \forall \delta \mathbf{r}_{i}\in T_{P}Q

A constraint is said to be scleronomous (or mobile) if it changes in time, otherwise it is rheonomous (or fixed). In other words, the equation of a scleronomous constraint is explicitly dependent on time, whereas a rheonomous one isn't. An example of scleronomous constraint would be an object attached to an elevator: the object is always constrained in the same way but the elevator (the constraint) moves, pulling the object with it.

A constraint is said to be holonomic if it can be expressed as a function of generalized coordinates and potentially time that is set to zero: $f(q_{1},\ldots,q_{N},t)=0$ . Common examples are forcing a particle to move on a Curve or Surface, in which case the equation of the curve or surface is the constraint.