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  "body": "\n## Summary\n\nMorse theory on Hilbert manifolds extends classical Morse theory from compact manifolds to functions on infinite-dimensional [[Hilbert manifolds|R8DMMW0E9M]].\nRichard Palais and Stephen Smale replaced compactness of the manifold by a compactness condition on the function, the [[Palais–Smale condition|4724KRNHAT]].\nWith it, sublevel sets change only at critical values, and passing a nondegenerate critical point of finite index $\\lambda$ attaches a $\\lambda$-cell.\nThe main application is the study of geodesics as critical points of the energy on spaces of paths and loops.\n\n## Explanations\n\n### L1 human\n\nClassical Morse theory reads the shape of a compact manifold off the critical points of a function on it.\nMany natural problems live on infinite-dimensional spaces, such as the space of all paths between two points, where the function is the energy of a path.\nSuch spaces are not compact, so a sequence that looks more and more like a critical point might escape to infinity.\nThe Palais–Smale condition rules this out, and then the classical theory goes through.\nFor example, critical points of the energy on paths from $p$ to $q$ are the geodesics from $p$ to $q$, and their indices count conjugate points.\n",
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