Generalized coordinates

The generalized coordinates of a system are a set of variables that describe the position of the system. A specific set of values of these coordinates is called a configuration. In mechanical systems, a configuration is easily interpreted as the literal position in space, but as the name suggests, these coordinates are more general. Instead, they represent position in the space of possible configurations, known as the configuration space.

Generalized coordinates are denoted with $\mathbf{q}\equiv(q_{1},\ldots,q_{N})$ . Their time derivatives, $\dot{\mathbf{q}}\equiv(\dot{q}_{1},\ldots,\dot{q}_{N})$ , are known as generalized velocities. They are simply the total derivative in time of the coordinates:

\dot{\mathbf{q}}=\frac{d\mathbf{q}}{dt}=\sum_{i=1}^{N} \frac{ \partial \mathbf{q} }{ \partial q_{i} } (q_{1}(t),\ldots,q_{N}(t))\ \dot{q}_{i}(t)

Notably, the velocity $\dot{\mathbf{q}}$ may also explicitly depend on time like $\dot{\mathbf{q}}(q_{1}(t),\ldots,q_{N}(t),t)$ as opposed to only implicitly like $\dot{\mathbf{q}}(q_{1}(t),\ldots,q_{N}(t))$ . In such as case, an additional time derivative $\partial \mathbf{q}/ \partial t$ appears on top of the sum above.

If a Lagrangian or Hamiltonian is given, it is also possible to define the linear momenta of these coordinates, which are known as conjugate momenta.

A note on notation

The notations $(q_{1},\ldots,q_{N})$ and especially $(q_{1},\ldots,q_{N},\dot{q}_{1},\ldots,\dot{q}_{N})$ are ubiquitous in analytical mechanics, but are also quite lengthy and verbose. Since they'll come up very often throughout these notes, it pays to have a shorthand notation for these. Namely, the coordinates will be packaged in the vector $\mathbf{q}=(q_{1},\ldots,q_{N})$ and the velocities in the vector $\dot{\mathbf{q}}=(\dot{q}_{1},\ldots,\dot{q}_{N})$ . Together, they'll often show up as $(\mathbf{q},\dot{\mathbf{q}})$ . In many cases, the vector bold font may also be omitted inside of function arguments. For instance, $f(q,\dot{q})\equiv f(q_{1},\ldots,q_{N},\dot{q}_{1},\ldots,\dot{q}_{N})$ .

The vector $\mathbf{q}$ should not in general be confused with the vector that is usually denoted $\mathbf{r}$ , which represents coordinates in "real" space (usually $\mathbb{R}^{3}$ or a similar space) as opposed to configuration space. As such, not only do they represent different positions, they also likely have a completely different dimensionality. $\mathbf{q}$ may have more (as it combines all the coordinates of all the bodies of the system) or less (say, in a single-body system under some constraints). The two are connected to each other via a Coordinate transformation. Often times, the system is solved first for $\mathbf{q}$ and then converted to $\mathbf{r}$ through that transformation. In fact, $\mathbf{r}$ is usually a composite function of $\mathbf{q}$ , as $\mathbf{r}\equiv \mathbf{r}(q_{1}(t),\ldots,q_{n}(t))$ .

It is perhaps more fruitful to explain by example. Consider a singular free particle. Its coordinates in three dimensions are described by the vector $\mathbf{r}=(x,y,z)$ . In this simple case, these coordinates are the generalized coordinates. Since there are no constraints, the configuration space, which we'll call $Q$ , is $Q=\mathbb{R}^{3}$ . Say now we have two free particles. The positions of the particles are $\mathbf{r}_{1}=(x_{1},y_{1},z_{1})$ and $\mathbf{r}_{2}=(x_{2},y_{2},z_{2})$ , but the generalized coordinates are $\mathbf{q}=(x_{1},x_{2},y_{1},y_{2},z_{1},z_{2})$ . This is because the generalized coordinates describes the system as a whole. The configuration space is now $Q=\mathbb{R}^{6}$ .

Say now we have a simple pendulum. The point mass is constrained to swing at a fixed distance $R$ from the joint, which means that most of $\mathbb{R}^{3}$ is now inaccessible to it. The coordinates of the point mass are still $\mathbf{r}=(x,y,z)$ , but the configuration space is much smaller: it is the circle of radius $R$ centered at the joint, a two dimensional space. Notice that one constraint reduced the dimension of $Q$ by one: this is the value of constraints. They can delete variables from a problem, making it considerably easier to solve. In this case, despite being in 3D, we only need two variables to solve the system. The coordinates that are necessary to define $Q$ are called free coordinates, whereas any remaining ones are constrained or cyclical coordinates. The number of free coordinates is called the degrees of freedom of the system and is also the dimension of $Q$ .