Note: This post is an English adaptation of my original Chinese article (URL). Some parts have been modified for clarity, cultural relevance, or to better fit the English-speaking audience.
When you first started learning analytical mechanics, have you ever been confused about what generalized coordinates really are? Are they just “generalized” versions of Cartesian coordinates? How general can they be?
In essence, generalized coordinates are a set of parameterized, irreducible, and independent variables that can fully describe every possible state of a mechanical system subject to various constraints. (Here, “irreducible” basically means “minimal,” implying that the number of these coordinates is exactly what is needed to completely describe the system under its constraints. If you have more coordinates than that, the set contains dependent coordinates; if you have fewer, you cannot fully describe the system.)
So, what’s the difference compared to Cartesian coordinates? It may seem that generalized coordinates only add “parameterization” and “irreducibility” as properties.
But let’s clarify one point first: both generalized coordinates and Cartesian coordinates can “fully describe the system’s motion under constraints.” Why then do we even need generalized coordinates in analytical mechanics? Is it because Cartesian coordinates are not sufficient? Or are those extra properties — “parameterization” and “irreducibility” — really that crucial? Let’s take an example to answer these questions.
Assume we have a classical mechanical system with $N$ particles in $D$ dimensions, subject to $M$ constraint equations. (We assume the system is “well-behaved,” meaning all constraints are integrable, independent, etc. We’ll use Newtonian mechanics for our discussion to highlight the difference between generalized and Cartesian coordinates.)
- Using Cartesian coordinates: we first write down all the force-component equations for each particle, then add these $M$ constraint equations, leading to a total of $N \times D + M$ equations to solve.
- Using generalized coordinates: we would first apply some parameterization methods, incorporating the known constraint equations to determine a set of generalized coordinates. After that, we only need to write down the force-component equations in terms of these generalized coordinates, which leaves us with $N \times D – M$ equations to solve.
Combined with some past experience of equation-solving, we know that the generalized-coordinate approach is usually better for two obvious reasons:
- The total number of equations to solve is reduced.
- The constraint equations are effectively “built into” the coordinate variables themselves, which often makes the resulting equations easier to handle (e.g., avoiding complicated coupled equations).
Essentially, this is because a set of generalized coordinates reveals the degrees of freedom of the system. However, be careful: in cases involving nonintegrable constraints, the number of generalized coordinates does not necessarily equal the system’s degrees of freedom. (For details, here is an excellent article on Zhihu: Link)
Now, let’s talk about how to determine those irreducible generalized coordinates through parameterization:
The simplest way is to just see it directly. For instance, with a 2D pendulum constraint, it’s quite straightforward to imagine using a single generalized coordinate $\theta$ to parameterize $x$ and $y$. But this only works for very simple problems in which the choice of generalized coordinates is obvious.
When the constraints are more complicated (but still integrable), there is a more systematic and mathematical method: using the Implicit Function Theorem. (For its proof, see this article on Zhihu) Once its conditions are satisfied—which we’ve assumed in our well-behaved system—you can directly conclude the necessary number of irreducible generalized coordinates, i.e., the system’s degrees of freedom. (Again, be reminded: in systems with nonintegrable constraints, the number of generalized coordinates may not equal the degrees of freedom, but we’re excluding such complications here.)
One thing to note is that the result you get from the Implicit Function Theorem is local, not global. Its validity holds in a neighborhood where the system is still well-behaved. For instance, with a single pendulum, applying the theorem at one point is typically enough, since the situation is pretty much the same for the other points — unless the pendulum swings overhead or something similar, in which case we must check if the conditions still hold there.
Once you’ve determined the number of generalized coordinates $X$ (under your assumed well-behaved conditions), you can go ahead and choose coordinates that best fit the system. Of course, you could naively pick $X$ coordinates out of your Cartesian set and call it a day, but that might not be the most efficient approach. Observe the constraints carefully and try to pick the most convenient generalized coordinates possible!
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