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  "body": "\n## Summary\n\nThe Lie group method studies a differential equation through its symmetries: continuous groups of changes of variables that map solutions to solutions.\nSophus Lie showed that such symmetries can be found by solving a linear system of determining equations for their infinitesimal generators.\nOnce found, they produce new solutions from old ones and reduce the equation to one with fewer variables.\nFor ordinary differential equations it is the [[Lie symmetry method|X3S9XHX83H]].\n[[Similarity solutions|579J5K9K5Z]] and [[traveling waves|M37M411KP7]] are the invariant solutions of scaling and translation symmetries.\nSee the [[definition|SYXNSN5JX0]].\n\n## Explanations\n\n### L1 human\n\nMoving a solution of the heat equation to the right, or starting it later, again gives a solution.\nStretching space by $2$ and time by $4$ also gives a solution.\nThese are symmetries of the equation, and each comes in a family with a continuous parameter.\nLie's method finds all such families by a mechanical computation.\nA solution that is unchanged by one family depends on fewer variables, so it solves a simpler equation.\n",
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