Generalized coordinates

Definition

Consider a system of $N$ point particles with position vector $x \in \R^{3N}$, subject to holonomic constraints that may depend on time $t$. For each $t$, let $Q_t \subseteq \R^{3N}$ be the set of positions allowed at time $t$, the configuration space at that time. Let $V \subseteq \R^n$ be open and let $I$ be an open interval of times. Generalized coordinates on a part of the system's configurations are given by a smooth map $$x \colon V \times I \to \R^{3N}, \qquad (q, t) \mapsto x(q, t),$$ such that:

1. For each $t \in I$, the map $q \mapsto x(q, t)$ is injective and its image is an open subset of $Q_t$.
2. For each $(q, t)$, the $3N \times n$ matrix $\partial x / \partial q$ has rank $n$.

The numbers $q_1, \dots, q_n$ are then called generalized coordinates, and $\dot q_1, \dots, \dot q_n$ are the generalized velocities.