Generalized momentum

Given a Lagrangian classical system with $(C, L)$ with coordinates $q_{i}$ , the generalized momentum or conjugate momentum is

p_{i} := \frac{\partial L}{\partial {\dot{q}}_{i}}

Motivation: If we compare the Euler-Lagrange equations

\frac{\partial L}{\partial q} - \frac{d}{d t} \frac{\partial L}{\partial \dot{q}} = 0

with Newton equation

\frac{d}{d t} p = F

we get the feeling that

on the one hand, $F = \frac{\partial L}{\partial q} = \frac{\partial}{\partial q} (T - V) = - \frac{\partial V}{\partial q}$
on the other, $\frac{\partial L}{\partial \dot{q}}$ must play the role of $p$ .

It turns out that generalized momentum is a covector. Roughly speaking, it can be understood in the following way. The Lagrangian $L$ is a modification of the kinetic energy $T$ , which is a "kind of" squared length of the velocity ${\dot{q}}_{i}$ . In a vector space with an inner product $g (-, -)$ , a length is computed in the following way:

we take the vector $v$
we obtain the dual vector $g (v, -)$ .
we compute $L e n g t h = g (v, v)$
Therefore, loosely speaking

\frac{\partial L e n g t h}{\partial v} = \frac{\partial g (v, v)}{\partial v} = g (v, -)

which is a covector.