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  "body": "\n## Statement\n\nLet $M$ be a connected [[smooth manifold|F45VQ47GPV]] and let $X_1, \\ldots, X_k$ be smooth [[vector fields|0299VCW1TH]] on $M$.\nAssume the bracket-generating condition.\nFor every $p \\in M$, the values at $p$ of the $X_i$ and of their iterated [[Lie brackets|68FGQEBD6J]] $[X_i, X_j]$, $[X_i, [X_j, X_l]]$, and so on, span $T_p M$.\nConsider the driftless control system\n$$x' = \\sum_{i=1}^{k} u_i X_i(x), \\qquad u = (u_1, \\ldots, u_k) \\in \\R^k.$$\nThis system is [[controllable|A9WFM77Z4P]]: any two points of $M$ are joined by a trajectory of the system for a piecewise constant control.\nEquivalently, any two points of $M$ are joined by a piecewise smooth curve whose velocity lies at each point in the span of $X_1, \\ldots, X_k$.\n\n## Remarks\n\nThe set of points reachable from $p$ is the set of endpoints of finite compositions of [[flows|06738GGWWK]] of the fields $\\pm X_i$.\nSince the system is symmetric under $u \\mapsto -u$, reachability is an equivalence relation, and the equivalence classes partition $M$.\nFlowing back and forth along two fields moves a point, to second order, in the direction of their bracket.\nWith the bracket condition, this shows that for suitable $Y_j$ among the $\\pm X_i$, the map $(t_1, \\ldots, t_N) \\mapsto e^{t_N Y_N} \\circ \\cdots \\circ e^{t_1 Y_1}(p)$ has an image with nonempty interior.\nSo each class is open, and connectedness of $M$ leaves only one class.\nExample: on $\\R^3$, the fields $X_1 = \\partial_x - \\frac{y}{2} \\partial_z$ and $X_2 = \\partial_y + \\frac{x}{2} \\partial_z$ satisfy $[X_1, X_2] = \\partial_z$, so $X_1, X_2, [X_1, X_2]$ span $\\R^3$ at each point and every point is reachable, although only two directions are controlled.\nThe bracket condition cannot be dropped: for $X_1 = \\partial_x$ and $X_2 = \\partial_y$ on $\\R^3$ all brackets vanish, and the trajectories from $p$ stay in the plane through $p$ where $z$ is constant.\n\n## Explanations\n\n### L1 human\n\nA car can move forward, backward and turn, but it cannot slide sideways.\nStill, by a sequence of small forward and turning motions, it can park in a gap beside it.\nThe theorem explains why: alternating two allowed motions produces a motion in a new direction, given by their Lie bracket.\nIf the allowed motions and their brackets point in all directions, then every position can be reached.\n\n### L2 human\n\nThe theorem is the basic controllability result of geometric control theory for systems without drift.\nIn [[sub-Riemannian geometry|97NYHNBZ2D]] it shows that the distance defined by curves tangent to a bracket-generating distribution is finite, and it is the starting point of that theory.\nFor systems with drift $x' = X_0(x) + \\sum u_i X_i(x)$ the bracket condition only gives that the reachable sets have nonempty interior, not controllability.\n",
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