stylistic:style_transfer, information:classification, linguistic:paraphrasing, linguistic:translation, linguistic:voice_change
From stationary action principle to Hamilton's equations

Let P (a, b, x a, x b) {\displaystyle {\mathcal {P}}(a,b,{\boldsymbol {x}}_{a},{\boldsymbol {x}}_{b})} be the set of smooth paths q: → M {\displaystyle {\boldsymbol {q}}:\to M} for which q (a) = x a {\displaystyle {\boldsymbol {q}}(a)={\boldsymbol {x}}_{a}} and q (b) = x b. {\displaystyle {\boldsymbol {q}}(b)={\boldsymbol {x}}_{b}.} The action functional S: P (a, b, x a, x b) → R {\displaystyle {\mathcal {S}}:{\mathcal {P}}(a,b,{\boldsymbol {x}}_{a},{\boldsymbol {x}}_{b})\to \mathbb {R} } is defined via S = ∫ a b L (t, q (t), q ˙ (t)) d t = ∫ a b (∑ i = 1 n p i q ˙ i − H (p, q, t)) d t, {\displaystyle {\mathcal {S}}=\int _{a}^{b}{\mathcal {L}}(t,{\boldsymbol {q}}(t),{\dot {\boldsymbol {q}}}(t))\,dt=\int _{a}^{b}\left(\sum _{i=1}^{n}p_{i}{\dot {q}}^{i}-{\mathcal {H}}({\boldsymbol {p}},{\boldsymbol {q}},t)\right)\,dt,} where q = q (t) {\displaystyle {\boldsymbol {q}}={\boldsymbol {q}}(t)}, and p = ∂ L / ∂ q ˙ {\displaystyle {\boldsymbol {p}}=\partial {\mathcal {L}}/\partial {\boldsymbol {\dot {q}}}} (see above). A path q ∈ P (a, b, x a, x b) {\displaystyle {\boldsymbol {q}}\in {\mathcal {P}}(a,b,{\boldsymbol {x}}_{a},{\boldsymbol {x}}_{b})} is a stationary point of S {\displaystyle {\mathcal {S}}} (and hence is an equation of motion) if and only if the path (p (t), q (t)) {\displaystyle ({\boldsymbol {p}}(t),{\boldsymbol {q}}(t))} in phase space coordinates obeys the Hamilton's equations.