Are Generalized Coordinates Necessary for Simplifying Complex Systems?

[Tac-Tics]

Are generalized coordinates, as used in Legrangian mechanics, just a different name for coordinates on a chart in a manifold? The idea of generalized coordinates never quite "clicked" with me, but after reading a paper today, it seems that they are just an implicit way of working with manifolds.

So, say you're working with a double-pendulumn system. You have two objects (the ends of the pendulumns) in 3D space, for a total of 6 degrees of freedom, so the entire system can be modeled in R^6. But, to simplify the math, we can use our knowledge of the system's invariants to reduce this down to a 2D submanifold, parametrized by the angle of each pendulumn as each swings in a plane.

Am I correct in coming to this conclusion?

 

[Ben Niehoff]

Yes, the coordinates are on a manifold referred to as "configuration space". The system trajectory is a curve in configuration space parametrized by the variable 't' (time).

In Hamiltonian mechanics, you have not just N coordinates, but also N momenta. These 2N components label points in "phase space". Phase space has some nice properties that come about due to the invariance of Poisson brackets with respect to different canonical sets of variables. A canonical transformation is a particular kind of coordinate transformation on phase space, called a "symplectomorphism". The Hamiltonian itself can be considered the generator (via Poisson brackets) of a one-parameter continuous family of symplectomorphisms (where the parameter is again time) that describes the system evolution. Also some other neat things come about, like Liouville's theorem, which is important for statistical mechanics.

 

[Tac-Tics]

I'm glad to see I'm on track with my intuition.

You used a lot of big words I don't know there. What is a Poisson bracket? Wikipedia goes on to talk about symplectic manifolds, which I am also unfamiliar with. (Wikipedia is a big believer in defining big words in terms of even bigger words in math and physics articles).

 

[Ben Niehoff]

If the coordinates are $q_i$ and their corresponding canonical momenta are $p_i$, then the Poisson bracket of two functions f(q,p), g(q,p) is given by

$$\{f, g\}_{PB} = \sum_i \frac{\partial f}{\partial q_i} \frac{\partial g}{\partial p_i} - \sum_i \frac{\partial f}{\partial p_i} \frac{\partial g}{\partial q_i}$$

Then, the time evolution of any given function $\phi(q, p, t)$ is given by

$$\frac{d\phi}{dt} = \frac{\partial \phi}{\partial t} + \{\phi, H\}_{PB}$$

where H is the Hamiltonian.

It will be easiest to understand if you apply it to a simple one-dimensional system that you know already, like a harmonic oscillator or something. Just walk through the steps and see what happens.

 

[bpet]

So, say you're working with a double-pendulumn system. You have two objects (the ends of the pendulumns) in 3D space, for a total of 6 degrees of freedom, so the entire system can be modeled in R^6. But, to simplify the math, we can use our knowledge of the system's invariants to reduce this down to a 2D submanifold, parametrized by the angle of each pendulumn as each swings in a plane.

To be precise, we use our knowledge of the system's invariants/constraints to show that coordinates lie on a 2D manifold that can be parameterised with two independent coordinates. The 6 coordinates could also be used as generalized coordinates if the corresponding constraint terms are included in the Lagrangian.

Many systems cannot be described fully by independent coordinates, e.g. rolling sphere.