nLab
canonical momentum

Idea

The archetypical example of a mechanical system is a particle propagating on a manifold $\Sigma$. The phase space of this particular system happens to be canonically identified with the cotangent bundle $X≔T^{*}\Sigma$ of $\Sigma$. Here the covector $(x,p)$ in $X$ over a point $x\in \Sigma$ is physically interpretd as describing a state of the system where the particle is at position $x\in \Sigma$ and has momentum (essentially: speed) as given by $p$.

Therefore locally for coordinate patch $\varphi :{ℝ}^{2n}\simeq {ℝ}^n\times {ℝ}^n\to T^{*}\Sigma$ the $2n$-canonical coordinates of the Cartesian space ${ℝ}^n$ are naturally thought of as decomposed into $n$ “canonical coordinates” on the first $n$ factors and a set of “canonical momenta”, being the canonical coordinates on the second ${ℝ}^n$-factor.

Notice that “canonical” here refers (at best) to the canonical coordinates of the Cartesian space ${ℝ}^n$ once $\varphi$ has been chosen. The choice of $\varphi$ however is arbitrary. Hence, despite the (standard) term, there is nothing much canonical about these “canonical coordinates” and “canonical momenta”.

In general, the phase space of a physical system is a symplectic manifold which need not be a cotangent bundle as for the particle sigma-model.

But locally over a coordinate patch every symplectic manifold looks like ${ℝ}^{2n}\simeq {ℝ}^n\times {ℝ}^n$ such that under this identification the symplectic form reads ${\sum }_{i=1}^{n}d{q}_i\wedge d{p}^i$, for $\{q_i\}$ the canonical coordinates on one ${ℝ}^n$ and $\{p^i\}$ for the other.

Therefore generally, in the context of mechanics, with such a local identification one calls $p^i$ the canonical momentum of the coordinate (or sometimes “canonical coordinate”) $q_i$.

Globally the notion of canonical momenta may not exist at all. The notion that does exist globally is that of a polarization of a symplectic manifold. See there for more details.

Related concepts

• momentum

• canonical coordinates

• phase space

• Hamiltonian mechanics

• geometric quantization