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  "body": "\n## Summary\n\nThe translation equation is the [[functional equation|SRAN4EKWS0]] $F(F(x, s), t) = F(x, s + t)$ for an unknown map $F \\colon X \\times \\R \\to X$.\nIts solutions with $F(x, 0) = x$ are the flows on the set $X$, that is, actions of the additive group $\\R$ on $X$.\nSo it is the basic equation of continuous-time [[dynamical systems|BBZP1R872N]] and of iteration with non-integer exponents.\nWith the real line replaced by a group, it becomes the equation of a [[group action|0WCY42EV9V]].\n\n## Explanations\n\n### L1 human\n\nThink of $F(x, t)$ as the position at time $t$ of a particle that starts at $x$.\nMoving for time $s$ and then for time $t$ should give the same position as moving for time $s + t$.\nFor example, $F(x, t) = x + t$ moves every point at unit speed along the line.\nThe map $F(x, t) = x e^t$ is another solution, in which points move away from $0$ exponentially fast.\n",
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  "name": "Translation equation",
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  "source": "https://github.com/tamnd/math2/blob/main/nodes/CPE/CPE1ANDDJ0.md",
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