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  "body": "\n## Summary\n\nThe Hamiltonian of an optimal control problem is a function that combines the dynamics and the running cost with a vector of multipliers, the costate.\nIt has the form $H = p \\cdot f - \\lambda_0 L$, or $H = L + p \\cdot f$ in the sign convention of the minimum principle.\n[[Pontryagin's maximum principle|7RPV2T29CF]] says that an optimal control maximizes the Hamiltonian at almost every time.\nIt plays the role that the Hamiltonian of [[Hamiltonian systems|0EDKGA2FAK]] plays in mechanics.\n\n## Explanations\n\n### L1 human\n\nThe Hamiltonian is a bookkeeping device that weighs the immediate cost of a control against its effect on the future.\nThe costate $p$ acts as a price for each unit of change in the state.\nA control is good at an instant if it gives a high value of \"price times rate of change minus cost\".\nFor example, for $x' = u$ with cost $\\int u^2/2 \\, dt$, the Hamiltonian is $H = p u - u^2/2$, which is largest at $u = p$.\n",
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