Ser Uilyam Rouen Hamilton
**Gamilton mexanikasi** 1833 yilda Lagranj mexanikasining islohoti sifatida paydo boʻldi. Sir Uilyam Rouen Hamilton tomonidan kiritilgan. Gamilton mexanikasi (umumlashtirilgan) tezliklarni almashtiradi {\displaystyle {\dot {q}}^{i}} (umumlashtirilgan) *impuls* bilan Lagranj mexanikasida qoʻllaniladi. Ikkala nazariya ham klassik mexanikaning talqinini beradi va bir xil fizik hodisalarni tavsiflaydi.

Gamilton mexanikasi geometriya (ayniqsa, simplektik geometriya va Puasson tuzilmalari) bilan yaqin aloqada boʻlib, klassik va kvant mexanikasi oʻrtasida bogʻlovchi boʻlib xizmat qiladi.

Umumiy koʻrinish
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### Fazo fazosi koordinatalari (p, q) va Gamilton H

Mayli {\displaystyle (M,{\mathcal {L}})} konfiguratsiya maydoni boʻlgan mexanik tizim boʻlishi kerak {\displaystyle M} va silliq Lagrangian {\displaystyle {\mathcal {L}}.} Standart koordinata tizimini tanlaymiz: {\displaystyle ({\boldsymbol {q}},{\boldsymbol {\dot {q}}})}. {\displaystyle \textstyle p\_{i}({\boldsymbol {q}},{\boldsymbol {\dot {q}}},t)~{\stackrel {\text{def}}{=}}~{\partial {\mathcal {L}}}/{\partial {\dot {q}}^{i}}} miqdorlar *impuls* deyiladi. (Shuningdek, *umumlashtirilgan impuls*, qoʻshma *impuls* va *kanonik impulslar*). Bir zumda {\displaystyle t,} Legendre transformatsiyasi {\displaystyle {\mathcal {L}}}ning xaritasi sifatida belgilanadi {\displaystyle ({\boldsymbol {q}},{\boldsymbol {\dot {q}}})\to \left({\boldsymbol {p}},{\boldsymbol {q}}\right)} silliq teskari boʻlishi taxmin qilinadi {\displaystyle ({\boldsymbol {p}},{\boldsymbol {q}})\to ({\boldsymbol {q}},{\boldsymbol {\dot {q}}}).} bilan tizim uchun {\displaystyle n} erkinlik darajasi, Lagranj mexanikasi *energiya funksiyasini* belgilaydi:


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{\displaystyle E\_{\mathcal {L}}({\boldsymbol {q}},{\boldsymbol {\dot {q}}},t)\,{\stackrel {\text{def}}{=}}\,\sum \_{i=1}^{n}{\dot {q}}^{i}{\frac {\partial {\mathcal {L}}}{\partial {\dot {q}}^{i}}}-{\mathcal {L}}.}

{\displaystyle E_{\mathcal {L}}({\boldsymbol {q}},{\boldsymbol {\dot {q}}},t)\,{\stackrel {\text{def}}{=}}\,\sum _{i=1}^{n}{\dot {q}}^{i}{\frac {\partial {\mathcal {L}}}{\partial {\dot {q}}^{i}}}-{\mathcal {L}}.}
Legendre ning oʻzgarishi {\displaystyle {\mathcal {L}}} aylanadi {\displaystyle E\_{\mathcal {L}}} funksiyaga aylanadi {\displaystyle {\mathcal {H}}({\boldsymbol {p}},{\boldsymbol {q}},t)} *Hamiltonian* sifatida tanilgan. Gamiltonianlik qanoatlantiradi



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{\displaystyle {\mathcal {H}}\left({\frac {\partial {\mathcal {L}}}{\partial {\boldsymbol {\dot {q}}}}},{\boldsymbol {q}},t\right)=E\_{\mathcal {L}}({\boldsymbol {q}},{\boldsymbol {\dot {q}}},t)}

{\displaystyle {\mathcal {H}}\left({\frac {\partial {\mathcal {L}}}{\partial {\boldsymbol {\dot {q}}}}},{\boldsymbol {q}},t\right)=E_{\mathcal {L}}({\boldsymbol {q}},{\boldsymbol {\dot {q}}},t)}
bu shuni anglatadi



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{\displaystyle {\mathcal {H}}({\boldsymbol {p}},{\boldsymbol {q}},t)=\sum \_{i=1}^{n}p\_{i}{\dot {q}}^{i}-{\mathcal {L}}({\boldsymbol {q}},{\boldsymbol {\dot {q}}},t),}

{\displaystyle {\mathcal {H}}({\boldsymbol {p}},{\boldsymbol {q}},t)=\sum _{i=1}^{n}p_{i}{\dot {q}}^{i}-{\mathcal {L}}({\boldsymbol {q}},{\boldsymbol {\dot {q}}},t),}
tezliklar bu yerda {\displaystyle {\boldsymbol {\dot {q}}}=({\dot {q}}^{1},\ldots ,{\dot {q}}^{n})} dan topiladi ({\displaystyle n} -oʻlchovli) tenglama {\displaystyle \textstyle {\boldsymbol {p}}={\partial {\mathcal {L}}}/{\partial {\boldsymbol {\dot {q}}}}} Bu, taxminga koʻra, yagona echilishi mumkin {\displaystyle {\boldsymbol {\dot {q}}}.} ({\displaystyle 2n} -oʻlchovli) juftlik {\displaystyle ({\boldsymbol {p}},{\boldsymbol {q}})} *fazo fazosi koordinatalari* deb ataladi. (Shuningdek, *kanonik koordinatalar*).

### Eyler-Lagranj tenglamasidan Gamilton tenglamalariga

Fazoviy fazoda koordinatalar {\displaystyle ({\boldsymbol {p}},{\boldsymbol {q}}),} ({\displaystyle n} -oʻlchovli) Eyler-Lagranj tenglamasi



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{\displaystyle {\frac {\partial {\mathcal {L}}}{\partial {\boldsymbol {q}}}}-{\frac {d}{dt}}{\frac {\partial {\mathcal {L}}}{\partial {\boldsymbol {\dot {q}}}}}=0}

{\displaystyle {\frac {\partial {\mathcal {L}}}{\partial {\boldsymbol {q}}}}-{\frac {d}{dt}}{\frac {\partial {\mathcal {L}}}{\partial {\boldsymbol {\dot {q}}}}}=0}
dagi *Gamilton tenglamalariga* aylanadi {\displaystyle 2n} oʻlchamlari

{\displaystyle {\frac {\mathrm {d} {\boldsymbol {q}}}{\mathrm {d} t}}={\frac {\partial {\mathcal {H}}}{\partial {\boldsymbol {p}}}},\quad {\frac {\mathrm {d} {\boldsymbol {p}}}{\mathrm {d} t}}=-{\frac {\partial {\mathcal {H}}}{\partial {\boldsymbol {q}}}}.}

### Statsionar harakat tamoyilidan Gamilton tenglamalarigacha

Mayli {\displaystyle {\mathcal {P}}(a,b,{\boldsymbol {x}}\_{a},{\boldsymbol {x}}\_{b})} silliq yoʻllar toʻplami boʻlsin {\displaystyle {\boldsymbol {q}}:[a,b]\to M} buning uchun {\displaystyle {\boldsymbol {q}}(a)={\boldsymbol {x}}\_{a}} va {\displaystyle {\boldsymbol {q}}(b)={\boldsymbol {x}}\_{b}.} Funksional harakat {\displaystyle {\mathcal {S}}:{\mathcal {P}}(a,b,{\boldsymbol {x}}\_{a},{\boldsymbol {x}}\_{b})\to \mathbb {R} } orqali aniqlanadi



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{\displaystyle {\mathcal {S}}[{\boldsymbol {q}}]=\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,}

{\displaystyle {\mathcal {S}}[{\boldsymbol {q}}]=\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,}
bu yerda {\displaystyle {\boldsymbol {q}}={\boldsymbol {q}}(t),} va {\displaystyle {\boldsymbol {p}}=\partial {\mathcal {L}}/\partial {\boldsymbol {\dot {q}}}} (yuqoriga qarang). Yoʻl {\displaystyle {\boldsymbol {q}}\in {\mathcal {P}}(a,b,{\boldsymbol {x}}\_{a},{\boldsymbol {x}}\_{b})} ning statsionar nuqtasi hisoblanadi {\displaystyle {\mathcal {S}}} (va demak, harakat tenglamasi) agar va faqat yoʻl boʻlsa {\displaystyle ({\boldsymbol {p}}(t),{\boldsymbol {q}}(t))} fazo fazosida koordinatalar Gamilton tenglamalariga boʻysunadi.

### Asosiy fizikaviy talqin

Gamilton mexanikasining oddiy talqini uni m massali bitta relyativistik boʻlmagan zarrachadan tashkil topgan bir oʻlchovli tizimda qoʻllashdan kelib chiqadi. Qiymat {\displaystyle H(p,q)} Gamiltonian — tizimning umumiy energiyasi, bu holda kinetik va potentsial energiya yigʻindisi, anʼanaviy ravishda mos ravishda T va V bilan belgilanadi. Bu yerda p — impuls mv, q — fazo koordinatasi. Keyin



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{\displaystyle {\mathcal {H}}=T+V,\qquad T={\frac {p^{2}}{2m}},\qquad V=V(q)}

{\displaystyle {\mathcal {H}}=T+V,\qquad T={\frac {p^{2}}{2m}},\qquad V=V(q)}
T faqat p ning funksiyasi, V esa faqat q funksiyasi (yaʼni, T va V skleronomik).

Bu misolda q ning vaqt hosilasi tezlikdir va shuning uchun birinchi Gamilton tenglamasi zarraning tezligi uning impulsga nisbatan kinetik energiyasining hosilasiga teng ekanligini bildiradi. Impuls momentining vaqt hosilasi p *Nyuton kuchiga* teng va shuning uchun ikkinchi Gamilton tenglamasi kuch potensial energiyaning salbiy gradientiga teng ekanligini anglatadi.

Misol
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Sferik mayatnik shar yuzasida ishqalanishsiz harakatlanuvchi *m* massadan iborat boʻlsin. Massaga taʼsir qiluvchi yagona kuchlar sfera va tortishish reaksiyasidir. Sferik koordinatalar massaning holatini (*r*, *θ*, *φ*) koʻrinishida tasvirlash uchun ishlatiladi, bu yerda *r* oʻzgarmas, *r* = *ℓ* .

 Sferik mayatnik : burchaklar va tezliklar.
Ushbu tizim uchun Lagrangian :



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{\displaystyle L={\frac {1}{2}}m\ell ^{2}\left({\dot {\theta }}^{2}+\sin ^{2}\theta \ {\dot {\varphi }}^{2}\right)+mg\ell \cos \theta .}

{\displaystyle L={\frac {1}{2}}m\ell ^{2}\left({\dot {\theta }}^{2}+\sin ^{2}\theta \ {\dot {\varphi }}^{2}\right)+mg\ell \cos \theta .}
Gamiltonian shunday



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{\displaystyle H=P\_{\theta }{\dot {\theta }}+P\_{\varphi }{\dot {\varphi }}-L}

{\displaystyle H=P_{\theta }{\dot {\theta }}+P_{\varphi }{\dot {\varphi }}-L}
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{\displaystyle P\_{\theta }={\frac {\partial L}{\partial {\dot {\theta }}}}=m\ell ^{2}{\dot {\theta }}}

{\displaystyle P_{\theta }={\frac {\partial L}{\partial {\dot {\theta }}}}=m\ell ^{2}{\dot {\theta }}}
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{\displaystyle P\_{\varphi }={\frac {\partial L}{\partial {\dot {\varphi }}}}=m\ell ^{2}\sin ^{2}\!\theta \,{\dot {\varphi }}.}

{\displaystyle P_{\varphi }={\frac {\partial L}{\partial {\dot {\varphi }}}}=m\ell ^{2}\sin ^{2}\!\theta \,{\dot {\varphi }}.}
Koordinatalar va impulslar nuqtai nazaridan, Gamiltonian buni oʻqiydi



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{\displaystyle H=\underbrace {\left[{\frac {1}{2}}m\ell ^{2}{\dot {\theta }}^{2}+{\frac {1}{2}}m\ell ^{2}\sin ^{2}\!\theta \,{\dot {\varphi }}^{2}\right]} \_{T}+\underbrace {{\Big [}-mg\ell \cos \theta {\Big ]}} \_{V}={\frac {P\_{\theta }^{2}}{2m\ell ^{2}}}+{\frac {P\_{\varphi }^{2}}{2m\ell ^{2}\sin ^{2}\theta }}-mg\ell \cos \theta }

{\displaystyle H=\underbrace {\left[{\frac {1}{2}}m\ell ^{2}{\dot {\theta }}^{2}+{\frac {1}{2}}m\ell ^{2}\sin ^{2}\!\theta \,{\dot {\varphi }}^{2}\right]} _{T}+\underbrace {{\Big [}-mg\ell \cos \theta {\Big ]}} _{V}={\frac {P_{\theta }^{2}}{2m\ell ^{2}}}+{\frac {P_{\varphi }^{2}}{2m\ell ^{2}\sin ^{2}\theta }}-mg\ell \cos \theta }
Gamilton tenglamalari toʻrtta birinchi tartibli differensial tenglamalarda koordinatalar va qoʻshma impulslarning vaqt evolyutsiyasini beradi,





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{\displaystyle {\begin{aligned}{\dot {\theta }}&={P\_{\theta } \over m\ell ^{2}}\\[6pt]{\dot {\varphi }}&={P\_{\varphi } \over m\ell ^{2}\sin ^{2}\theta }\\[6pt]{\dot {P\_{\theta }}}&={P\_{\varphi }^{2} \over m\ell ^{2}\sin ^{3}\theta }\cos \theta -mg\ell \sin \theta \\[6pt]{\dot {P\_{\varphi }}}&=0.\end{aligned}}}

{\displaystyle {\begin{aligned}{\dot {\theta }}&={P_{\theta } \over m\ell ^{2}}\\[6pt]{\dot {\varphi }}&={P_{\varphi } \over m\ell ^{2}\sin ^{2}\theta }\\[6pt]{\dot {P_{\theta }}}&={P_{\varphi }^{2} \over m\ell ^{2}\sin ^{3}\theta }\cos \theta -mg\ell \sin \theta \\[6pt]{\dot {P_{\varphi }}}&=0.\end{aligned}}}
impulsi, bu burchak momentumining vertikal komponentiga mos keladi {\displaystyle L\_{z}=\ell \sin \theta \times m\ell \sin \theta \,{\dot {\varphi }}}, harakat doimiysi. Bu vertikal oʻq atrofida tizimning aylanish simmetriyasining natijasidir. Gamiltonian, azimutdan yoʻq boʻlish {\displaystyle \varphi } siklik koordinata boʻlib, uning qoʻshma impulsining saqlanishini nazarda tutadi.

Gamilton tenglamalarini chiqarish
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Gamilton tenglamalarini Lagrangian bilan hisoblash yoʻli bilan olish mumkin {\displaystyle {\mathcal {L}}}, umumlashtirilgan pozitsiyalar qi va umumlashtirilgan tezliklar q̇i, bu yerda {\displaystyle i=1,\ldots ,n}. Bu yerda biz qobiqdan tashqari ishlaymiz, yaʼni {\displaystyle q^{i},{\dot {q}}^{i},t} Har qanday harakat tenglamalariga rioya qilish uchun cheklanmagan faza fazosidagi mustaqil koordinatalar (xususan, {\displaystyle {\dot {q}}^{i}} ning hosilasi emas {\displaystyle q^{i}}). Lagrangianning umumiy farqi :


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{\displaystyle \mathrm {d} {\mathcal {L}}=\sum \_{i}\left({\frac {\partial {\mathcal {L}}}{\partial q^{i}}}\mathrm {d} q^{i}+{\frac {\partial {\mathcal {L}}}{\partial {\dot {q}}^{i}}}\,\mathrm {d} {\dot {q}}^{i}\right)+{\frac {\partial {\mathcal {L}}}{\partial t}}\,\mathrm {d} t\ .}

{\displaystyle \mathrm {d} {\mathcal {L}}=\sum _{i}\left({\frac {\partial {\mathcal {L}}}{\partial q^{i}}}\mathrm {d} q^{i}+{\frac {\partial {\mathcal {L}}}{\partial {\dot {q}}^{i}}}\,\mathrm {d} {\dot {q}}^{i}\right)+{\frac {\partial {\mathcal {L}}}{\partial t}}\,\mathrm {d} t\ .}
Umumlashtirilgan impuls koordinatalari quyidagicha aniqlandi:{\displaystyle p\_{i}=\partial {\mathcal {L}}/\partial {\dot {q}}^{i}}, shuning uchun tenglamani quyidagicha qayta yozishimiz mumkin:



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{\displaystyle {\begin{aligned}\mathrm {d} {\mathcal {L}}=&\sum \_{i}\left({\frac {\partial {\mathcal {L}}}{\partial q^{i}}}\,\mathrm {d} q^{i}+p\_{i}\mathrm {d} {\dot {q}}^{i}\right)+{\frac {\partial {\mathcal {L}}}{\partial t}}\mathrm {d} t\\=&\sum \_{i}\left({\frac {\partial {\mathcal {L}}}{\partial q^{i}}}\,\mathrm {d} q^{i}+\mathrm {d} (p\_{i}{\dot {q}}^{i})-{\dot {q}}^{i}\,\mathrm {d} p\_{i}\right)+{\frac {\partial {\mathcal {L}}}{\partial t}}\,\mathrm {d} t\,.\end{aligned}}}

{\displaystyle {\begin{aligned}\mathrm {d} {\mathcal {L}}=&\sum _{i}\left({\frac {\partial {\mathcal {L}}}{\partial q^{i}}}\,\mathrm {d} q^{i}+p_{i}\mathrm {d} {\dot {q}}^{i}\right)+{\frac {\partial {\mathcal {L}}}{\partial t}}\mathrm {d} t\\=&\sum _{i}\left({\frac {\partial {\mathcal {L}}}{\partial q^{i}}}\,\mathrm {d} q^{i}+\mathrm {d} (p_{i}{\dot {q}}^{i})-{\dot {q}}^{i}\,\mathrm {d} p_{i}\right)+{\frac {\partial {\mathcal {L}}}{\partial t}}\,\mathrm {d} t\,.\end{aligned}}}
Qayta tartibga solishdan keyin quyidagilar olinadi:


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{\displaystyle \mathrm {d} \!\left(\sum \_{i}p\_{i}{\dot {q}}^{i}-{\mathcal {L}}\right)=\sum \_{i}\left(-{\frac {\partial {\mathcal {L}}}{\partial q^{i}}}\,\mathrm {d} q^{i}+{\dot {q}}^{i}\mathrm {d} p\_{i}\right)-{\frac {\partial {\mathcal {L}}}{\partial t}}\,\mathrm {d} t\ .}

{\displaystyle \mathrm {d} \!\left(\sum _{i}p_{i}{\dot {q}}^{i}-{\mathcal {L}}\right)=\sum _{i}\left(-{\frac {\partial {\mathcal {L}}}{\partial q^{i}}}\,\mathrm {d} q^{i}+{\dot {q}}^{i}\mathrm {d} p_{i}\right)-{\frac {\partial {\mathcal {L}}}{\partial t}}\,\mathrm {d} t\ .}
Chap tarafdagi qavs ichidagi atama faqat Gamiltoniandir

{\textstyle {\mathcal {H}}=\sum p\_{i}{\dot {q}}^{i}-{\mathcal {L}}}

ilgari belgilangan, shuning uchun:


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{\displaystyle \mathrm {d} {\mathcal {H}}=\sum \_{i}\left(-{\frac {\partial {\mathcal {L}}}{\partial q^{i}}}\,\mathrm {d} q^{i}+{\dot {q}}^{i}\,\mathrm {d} p\_{i}\right)-{\frac {\partial {\mathcal {L}}}{\partial t}}\,\mathrm {d} t\ .}

{\displaystyle \mathrm {d} {\mathcal {H}}=\sum _{i}\left(-{\frac {\partial {\mathcal {L}}}{\partial q^{i}}}\,\mathrm {d} q^{i}+{\dot {q}}^{i}\,\mathrm {d} p_{i}\right)-{\frac {\partial {\mathcal {L}}}{\partial t}}\,\mathrm {d} t\ .}
Gamiltonianning umumiy differensialini ham hisoblash mumkin {\displaystyle {\mathcal {H}}} koordinatalariga nisbatan {\displaystyle q^{i},p\_{i},t} oʻrniga {\displaystyle q^{i},{\dot {q}}^{i},t}, hosil beradi:


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{\displaystyle \mathrm {d} {\mathcal {H}}=\sum \_{i}\left({\frac {\partial {\mathcal {H}}}{\partial q^{i}}}\mathrm {d} q^{i}+{\frac {\partial {\mathcal {H}}}{\partial p\_{i}}}\mathrm {d} p\_{i}\right)+{\frac {\partial {\mathcal {H}}}{\partial t}}\,\mathrm {d} t\ .}

{\displaystyle \mathrm {d} {\mathcal {H}}=\sum _{i}\left({\frac {\partial {\mathcal {H}}}{\partial q^{i}}}\mathrm {d} q^{i}+{\frac {\partial {\mathcal {H}}}{\partial p_{i}}}\mathrm {d} p_{i}\right)+{\frac {\partial {\mathcal {H}}}{\partial t}}\,\mathrm {d} t\ .}
Endi bu ikki ifodani tenglashtirish mumkin {\displaystyle d{\mathcal {H}}}, jihatidan biri {\displaystyle {\mathcal {L}}}, ikkinchisi jihatidan {\displaystyle {\mathcal {H}}} :


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{\displaystyle \sum \_{i}\left(-{\frac {\partial {\mathcal {L}}}{\partial q^{i}}}\mathrm {d} q^{i}+{\dot {q}}^{i}\mathrm {d} p\_{i}\right)-{\frac {\partial {\mathcal {L}}}{\partial t}}\,\mathrm {d} t\ =\ \sum \_{i}\left({\frac {\partial {\mathcal {H}}}{\partial q^{i}}}\mathrm {d} q^{i}+{\frac {\partial {\mathcal {H}}}{\partial p\_{i}}}\mathrm {d} p\_{i}\right)+{\frac {\partial {\mathcal {H}}}{\partial t}}\,\mathrm {d} t\ .}

{\displaystyle \sum _{i}\left(-{\frac {\partial {\mathcal {L}}}{\partial q^{i}}}\mathrm {d} q^{i}+{\dot {q}}^{i}\mathrm {d} p_{i}\right)-{\frac {\partial {\mathcal {L}}}{\partial t}}\,\mathrm {d} t\ =\ \sum _{i}\left({\frac {\partial {\mathcal {H}}}{\partial q^{i}}}\mathrm {d} q^{i}+{\frac {\partial {\mathcal {H}}}{\partial p_{i}}}\mathrm {d} p_{i}\right)+{\frac {\partial {\mathcal {H}}}{\partial t}}\,\mathrm {d} t\ .}
Ushbu hisob-kitoblar qobiqdan tashqari boʻlganligi sababli, tegishli koeffitsientlarni tenglashtirish mumkin {\displaystyle \mathrm {d} q^{i},\mathrm {d} p\_{i},\mathrm {d} t} ikki tomondan:



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{\displaystyle {\frac {\partial {\mathcal {H}}}{\partial q^{i}}}=-{\frac {\partial {\mathcal {L}}}{\partial q^{i}}}\quad ,\quad {\frac {\partial {\mathcal {H}}}{\partial p\_{i}}}={\dot {q}}^{i}\quad ,\quad {\frac {\partial {\mathcal {H}}}{\partial t}}=-{\partial {\mathcal {L}} \over \partial t}\ .}

{\displaystyle {\frac {\partial {\mathcal {H}}}{\partial q^{i}}}=-{\frac {\partial {\mathcal {L}}}{\partial q^{i}}}\quad ,\quad {\frac {\partial {\mathcal {H}}}{\partial p_{i}}}={\dot {q}}^{i}\quad ,\quad {\frac {\partial {\mathcal {H}}}{\partial t}}=-{\partial {\mathcal {L}} \over \partial t}\ .}
Bunda, biri parametrik funksiyalarni almashtiradi {\displaystyle q^{i}=q^{i}(t)} tezliklar bilan faza fazosida trayektoriyani belgilaydi {\textstyle {\dot {q}}^{i}={\tfrac {d}{dt}}q^{i}(t)}, Lagrange tenglamalariga boʻysunish:



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{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial {\mathcal {L}}}{\partial {\dot {q}}^{i}}}-{\frac {\partial {\mathcal {L}}}{\partial q^{i}}}=0\ .}

{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial {\mathcal {L}}}{\partial {\dot {q}}^{i}}}-{\frac {\partial {\mathcal {L}}}{\partial q^{i}}}=0\ .}
Qobiqdagi nuqtai nazardan qayta tartibga solish va yozish {\displaystyle p\_{i}=p\_{i}(t)} beradi:



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{\displaystyle {\frac {\partial {\mathcal {L}}}{\partial q^{i}}}={\dot {p}}\_{i}\ .}

{\displaystyle {\frac {\partial {\mathcal {L}}}{\partial q^{i}}}={\dot {p}}_{i}\ .}
Shunday qilib, Lagranj tenglamalari Gamilton tenglamalariga ekvivalentdir:



∂

H

∂

q

i


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{\displaystyle {\frac {\partial {\mathcal {H}}}{\partial q^{i}}}=-{\dot {p}}\_{i}\quad ,\quad {\frac {\partial {\mathcal {H}}}{\partial p\_{i}}}={\dot {q}}^{i}\quad ,\quad {\frac {\partial {\mathcal {H}}}{\partial t}}=-{\frac {\partial {\mathcal {L}}}{\partial t}}\,.}

{\displaystyle {\frac {\partial {\mathcal {H}}}{\partial q^{i}}}=-{\dot {p}}_{i}\quad ,\quad {\frac {\partial {\mathcal {H}}}{\partial p_{i}}}={\dot {q}}^{i}\quad ,\quad {\frac {\partial {\mathcal {H}}}{\partial t}}=-{\frac {\partial {\mathcal {L}}}{\partial t}}\,.}
Vaqtdan mustaqil boʻlgan holda {\displaystyle {\mathcal {H}}} va {\displaystyle {\mathcal {L}}}, yaʼni {\displaystyle \partial {\mathcal {H}}/\partial t=-\partial {\mathcal {L}}/\partial t=0} , Gamilton tenglamalari 2*n* birinchi tartibli differensial tenglamalardan, Lagranj tenglamalari esa n ikkinchi tartibli tenglamalardan iborat. Gamilton tenglamalari odatda aniq yechimlarni topish qiyinligini kamaytirmaydi, lekin ulardan muhim nazariy natijalarni olish mumkin, chunki koordinatalar va momentlar deyarli simmetrik rollarga ega boʻlgan mustaqil oʻzgaruvchilardir.
Gamilton tenglamalarining Lagranj tenglamalaridan yana bir afzalligi bor: agar tizim simmetriyaga ega boʻlsa, baʼzi koordinatalar {\displaystyle q\_{i}} Gamiltonianda (yaʼni *tsiklik koordinatada*) yuzaga kelmaydi, mos keladigan impuls koordinatasi. {\displaystyle p\_{i}} har bir traektoriya boʻylab saqlanadi va bu koordinatani toʻplamning boshqa tenglamalarida doimiyga kamaytirish mumkin. Bu muammoni n koordinatadan (*n* − 1) koordinatagacha samarali qisqartiradi: bu geometriyadagi simplektik qisqarishning asosidir. Lagranj tizimida impulsning saqlanishi ham darhol sodir boʻladi, ammo barcha umumiy tezliklar {\displaystyle {\dot {q}}\_{i}} hali ham Lagranjiyada uchraydi va n koordinatali tenglamalar sistemasi haligacha yechilishi kerak.
Lagrangian va Gamilton yondashuvlari klassik mexanikada chuqurroq natijalar uchun zamin yaratadi va kvant mexanikasida oʻxshash formulalarni taklif qiladi: yoʻl integral formulasi va Shredinger tenglamasi .

Gamiltonianning xossalari
-------------------------

* Gamiltonning qiymati {\displaystyle {\mathcal {H}}} tizimning umumiy energiyasi, agar energiya funksiyasi boʻlsa {\displaystyle E\_{\mathcal {L}}} bir xil xususiyatga ega. (Tanrifiga qarang {\displaystyle {\mathcal {H}}).}
* {\displaystyle {\frac {d{\mathcal {H}}}{dt}}={\frac {\partial {\mathcal {H}}}{\partial t}}} qachonki {\displaystyle \mathbf {p} (t),\mathbf {q} (t)} Gamilton tenglamalarining yechimini hosil qilamiz. Haqiqatdan ham, {\textstyle {\frac {d{\mathcal {H}}}{dt}}={\frac {\partial {\mathcal {H}}}{\partial {\boldsymbol {p}}}}\cdot {\dot {\boldsymbol {p}}}+{\frac {\partial {\mathcal {H}}}{\partial {\boldsymbol {q}}}}\cdot {\dot {\boldsymbol {q}}}+{\frac {\partial {\mathcal {H}}}{\partial t}},} va oxirgi muddatdan tashqari hamma narsa bekor qilinadi.
* {\displaystyle {\mathcal {H}}} *nuqta oʻzgarishlarida*, yaʼni silliq oʻzgarishlarda oʻzgarmaydi {\displaystyle {\boldsymbol {q}}\leftrightarrow {\boldsymbol {q'}}} kosmik koordinatalar. (Energetika funksiyasining oʻzgarmasligidan kelib chiqadi {\displaystyle E\_{\mathcal {L}}} nuqta oʻzgarishlari ostida. {\displaystyle E\_{\mathcal {L}}} ning oʻzgarmasligi bevosita oʻrnatilishi mumkin).
* {\displaystyle {\frac {\partial {\mathcal {H}}}{\partial t}}=-{\frac {\partial {\mathcal {L}}}{\partial t}}.} (Qarang: Gamilton tenglamalarini chiqarish).
* {\displaystyle -{\frac {\partial {\mathcal {H}}}{\partial q^{i}}}={\dot {p}}\_{i}={\frac {\partial {\mathcal {L}}}{\partial q^{i}}}.} (Gemilton va Eyler-Lagranj tenglamalarini solishtiring yoki Gamilton tenglamalarini chiqarishga qarang).
* {\displaystyle {\frac {\partial {\mathcal {H}}}{\partial q^{i}}}=0} agar va faqat agar {\displaystyle {\frac {\partial {\mathcal {L}}}{\partial q^{i}}}=0.} Oxirgi tenglama bajariladigan koordinata *tsiklik* (yoki *inkor etilmaydigan*) deyiladi. Har bir siklik koordinata {\displaystyle q^{i}} tomonidan erkinlik darajalari sonini kamaytiradi {\displaystyle 1,} mos keladigan impulsni keltirib chiqaradi {\displaystyle p\_{i}} saqlanishi kerak va Gamilton tenglamalarini yechishni osonlashtiradi .

Elektromagnit maydondagi zaryadlangan zarrachaning Gamiltoniani
---------------------------------------------------------------

Gamilton mexanikasining yetarlicha tasviri elektromagnit maydondagi zaryadlangan zarrachaning Gamiltonian tomonidan berilgan. Dekart koordinatalarida elektromagnit maydondagi relyativistik boʻlmagan klassik zarrachaning lagranjiyani (SI birliklarida):



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{\displaystyle {\mathcal {L}}=\sum \_{i}{\tfrac {1}{2}}m{\dot {x}}\_{i}^{2}+\sum \_{i}q{\dot {x}}\_{i}A\_{i}-q\varphi }

{\displaystyle {\mathcal {L}}=\sum _{i}{\tfrac {1}{2}}m{\dot {x}}_{i}^{2}+\sum _{i}q{\dot {x}}_{i}A_{i}-q\varphi }
Bu yerda q — zarrachaning elektr zaryadi, φ — elektr skayar potentsiali va Ai — magnit vektor potentsialining tarkibiy qismlari, ularning barchasi aniq bogʻliq boʻlishi mumkin. {\displaystyle x\_{i}} va {\displaystyle t} .

Ushbu Lagranj Eyler-Lagranj tenglamasi bilan birgalikda Lorents kuch qonunini hosil qiladi.

  


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{\displaystyle m{\ddot {\mathbf {x} }}=q\mathbf {E} +q{\dot {\mathbf {x} }}\times \mathbf {B} \,,}

{\displaystyle m{\ddot {\mathbf {x} }}=q\mathbf {E} +q{\dot {\mathbf {x} }}\times \mathbf {B} \,,}va minimal ulanish deb ataladi.

Kanonik impulslar quyidagicha ifodalanadi:



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{\displaystyle p\_{i}={\frac {\partial {\mathcal {L}}}{\partial {\dot {x}}\_{i}}}=m{\dot {x}}\_{i}+qA\_{i}}

{\displaystyle p_{i}={\frac {\partial {\mathcal {L}}}{\partial {\dot {x}}_{i}}}=m{\dot {x}}_{i}+qA_{i}}
Gamiltonian, Lagrangianning Legendre oʻzgarishi sifatida, shuning uchun:



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{\displaystyle {\mathcal {H}}=\sum \_{i}{\dot {x}}\_{i}p\_{i}-{\mathcal {L}}=\sum \_{i}{\frac {\left(p\_{i}-qA\_{i}\right)^{2}}{2m}}+q\varphi }

{\displaystyle {\mathcal {H}}=\sum _{i}{\dot {x}}_{i}p_{i}-{\mathcal {L}}=\sum _{i}{\frac {\left(p_{i}-qA_{i}\right)^{2}}{2m}}+q\varphi }
Bu tenglama kvant mexanikasida tez-tez ishlatiladi.

Oʻlchov transformatsiyasi ostida:



A

→

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{\displaystyle \mathbf {A} \rightarrow \mathbf {A} +\nabla f\,,\quad \varphi \rightarrow \varphi -{\dot {f}}\,,}

{\displaystyle \mathbf {A} \rightarrow \mathbf {A} +\nabla f\,,\quad \varphi \rightarrow \varphi -{\dot {f}}\,,}Bu yerda *f*(**r**, *t*) fazo va vaqtning har qanday skalyar funksiyasi boʻlib, yuqorida aytib oʻtilgan Lagranj, kanonik impuls va Gamilton oʻzgarishiga oʻxshash:



L
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{\displaystyle L\rightarrow L'=L+q{\frac {df}{dt}}\,,\quad \mathbf {p} \rightarrow \mathbf {p'} =\mathbf {p} +q\nabla f\,,\quad H\rightarrow H'=H-q{\frac {\partial f}{\partial t}}\,,}

{\displaystyle L\rightarrow L'=L+q{\frac {df}{dt}}\,,\quad \mathbf {p} \rightarrow \mathbf {p'} =\mathbf {p} +q\nabla f\,,\quad H\rightarrow H'=H-q{\frac {\partial f}{\partial t}}\,,}
Bu hali ham xuddi shu Gamilton tenglamasini ishlab chiqaradi:





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{\displaystyle {\begin{aligned}\left.{\frac {\partial H'}{\partial {x\_{i}}}}\right|\_{p'\_{i}}&=\left.{\frac {\partial }{\partial {x\_{i}}}}\right|\_{p'\_{i}}({\dot {x}}\_{i}p'\_{i}-L')=-\left.{\frac {\partial L'}{\partial {x\_{i}}}}\right|\_{p'\_{i}}\\&=-\left.{\frac {\partial L}{\partial {x\_{i}}}}\right|\_{p'\_{i}}-q\left.{\frac {\partial }{\partial {x\_{i}}}}\right|\_{p'\_{i}}{\frac {df}{dt}}\\&=-{\frac {d}{dt}}\left(\left.{\frac {\partial L}{\partial {{\dot {x}}\_{i}}}}\right|\_{p'\_{i}}+q\left.{\frac {\partial f}{\partial {x\_{i}}}}\right|\_{p'\_{i}}\right)\\&=-{\dot {p}}'\_{i}\end{aligned}}}

{\displaystyle {\begin{aligned}\left.{\frac {\partial H'}{\partial {x_{i}}}}\right|_{p'_{i}}&=\left.{\frac {\partial }{\partial {x_{i}}}}\right|_{p'_{i}}({\dot {x}}_{i}p'_{i}-L')=-\left.{\frac {\partial L'}{\partial {x_{i}}}}\right|_{p'_{i}}\\&=-\left.{\frac {\partial L}{\partial {x_{i}}}}\right|_{p'_{i}}-q\left.{\frac {\partial }{\partial {x_{i}}}}\right|_{p'_{i}}{\frac {df}{dt}}\\&=-{\frac {d}{dt}}\left(\left.{\frac {\partial L}{\partial {{\dot {x}}_{i}}}}\right|_{p'_{i}}+q\left.{\frac {\partial f}{\partial {x_{i}}}}\right|_{p'_{i}}\right)\\&=-{\dot {p}}'_{i}\end{aligned}}}
Kvant mexanikasida toʻlqin funksiyasi ham oʻlchagich transformatsiyasi vaqtida mahalliy U(1) guruh transformatsiyasiga oʻtadi, bu esa mahalliy U(1) transformatsiyalar ostida barcha fizik natijalar oʻzgarmas boʻlishi kerakligini bildiradi.

### Elektromagnit maydondagi relativistik zaryadlangan zarracha

Zarra uchun relativistik lagranj (tinch massa {\displaystyle m} va zaryadlash {\displaystyle q}) tomonidan berilgan:



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{\displaystyle {\mathcal {L}}(t)=-mc^{2}{\sqrt {1-{\frac {{{\dot {\mathbf {x} }}(t)}^{2}}{c^{2}}}}}+q{\dot {\mathbf {x} }}(t)\cdot \mathbf {A} \left(\mathbf {x} (t),t\right)-q\varphi \left(\mathbf {x} (t),t\right)}

{\displaystyle {\mathcal {L}}(t)=-mc^{2}{\sqrt {1-{\frac {{{\dot {\mathbf {x} }}(t)}^{2}}{c^{2}}}}}+q{\dot {\mathbf {x} }}(t)\cdot \mathbf {A} \left(\mathbf {x} (t),t\right)-q\varphi \left(\mathbf {x} (t),t\right)}
Shunday qilib, zarrachaning kanonik impulsi:


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{\displaystyle \mathbf {p} (t)={\frac {\partial {\mathcal {L}}}{\partial {\dot {\mathbf {x} }}}}={\frac {m{\dot {\mathbf {x} }}}{\sqrt {1-{\frac {{\dot {\mathbf {x} }}^{2}}{c^{2}}}}}}+q\mathbf {A} }

{\displaystyle \mathbf {p} (t)={\frac {\partial {\mathcal {L}}}{\partial {\dot {\mathbf {x} }}}}={\frac {m{\dot {\mathbf {x} }}}{\sqrt {1-{\frac {{\dot {\mathbf {x} }}^{2}}{c^{2}}}}}}+q\mathbf {A} }
yaʼni kinetik impuls va potensial impulsning yigʻindisi.

Tezlikni yechib, biz olamiz




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{\displaystyle {\dot {\mathbf {x} }}(t)={\frac {\mathbf {p} -q\mathbf {A} }{\sqrt {m^{2}+{\frac {1}{c^{2}}}{\left(\mathbf {p} -q\mathbf {A} \right)}^{2}}}}}

{\displaystyle {\dot {\mathbf {x} }}(t)={\frac {\mathbf {p} -q\mathbf {A} }{\sqrt {m^{2}+{\frac {1}{c^{2}}}{\left(\mathbf {p} -q\mathbf {A} \right)}^{2}}}}}
Gamiltonian ham shunday



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{\displaystyle {\mathcal {H}}(t)={\dot {\mathbf {x} }}\cdot \mathbf {p} -{\mathcal {L}}=c{\sqrt {m^{2}c^{2}+{\left(\mathbf {p} -q\mathbf {A} \right)}^{2}}}+q\varphi }

{\displaystyle {\mathcal {H}}(t)={\dot {\mathbf {x} }}\cdot \mathbf {p} -{\mathcal {L}}=c{\sqrt {m^{2}c^{2}+{\left(\mathbf {p} -q\mathbf {A} \right)}^{2}}}+q\varphi }
Natijada kuch tenglamasi (Eyler-Lagranj tenglamasiga teng)




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{\displaystyle {\dot {\mathbf {p} }}=-{\frac {\partial {\mathcal {H}}}{\partial \mathbf {x} }}=q{\dot {\mathbf {x} }}\cdot ({\boldsymbol {\nabla }}\mathbf {A} )-q{\boldsymbol {\nabla }}\varphi =q{\boldsymbol {\nabla }}({\dot {\mathbf {x} }}\cdot \mathbf {A} )-q{\boldsymbol {\nabla }}\varphi }

{\displaystyle {\dot {\mathbf {p} }}=-{\frac {\partial {\mathcal {H}}}{\partial \mathbf {x} }}=q{\dot {\mathbf {x} }}\cdot ({\boldsymbol {\nabla }}\mathbf {A} )-q{\boldsymbol {\nabla }}\varphi =q{\boldsymbol {\nabla }}({\dot {\mathbf {x} }}\cdot \mathbf {A} )-q{\boldsymbol {\nabla }}\varphi }
undan olish mumkin





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{\displaystyle {\begin{aligned}{\frac {\mathrm {d} }{\mathrm {d} t}}\left({\frac {m{\dot {\mathbf {x} }}}{\sqrt {1-{\frac {{\dot {\mathbf {x} }}^{2}}{c^{2}}}}}}\right)&={\frac {\mathrm {d} }{\mathrm {d} t}}(\mathbf {p} -q\mathbf {A} )={\dot {\mathbf {p} }}-q{\frac {\partial \mathbf {A} }{\partial t}}-q({\dot {\mathbf {x} }}\cdot \nabla )\mathbf {A} \\&=q{\boldsymbol {\nabla }}({\dot {\mathbf {x} }}\cdot \mathbf {A} )-q{\boldsymbol {\nabla }}\varphi -q{\frac {\partial \mathbf {A} }{\partial t}}-q({\dot {\mathbf {x} }}\cdot \nabla )\mathbf {A} \\&=q\mathbf {E} +q{\dot {\mathbf {x} }}\times \mathbf {B} \end{aligned}}}

{\displaystyle {\begin{aligned}{\frac {\mathrm {d} }{\mathrm {d} t}}\left({\frac {m{\dot {\mathbf {x} }}}{\sqrt {1-{\frac {{\dot {\mathbf {x} }}^{2}}{c^{2}}}}}}\right)&={\frac {\mathrm {d} }{\mathrm {d} t}}(\mathbf {p} -q\mathbf {A} )={\dot {\mathbf {p} }}-q{\frac {\partial \mathbf {A} }{\partial t}}-q({\dot {\mathbf {x} }}\cdot \nabla )\mathbf {A} \\&=q{\boldsymbol {\nabla }}({\dot {\mathbf {x} }}\cdot \mathbf {A} )-q{\boldsymbol {\nabla }}\varphi -q{\frac {\partial \mathbf {A} }{\partial t}}-q({\dot {\mathbf {x} }}\cdot \nabla )\mathbf {A} \\&=q\mathbf {E} +q{\dot {\mathbf {x} }}\times \mathbf {B} \end{aligned}}}

Yuqoridagi hosila vektor hisob identifikatoridan foydalanadi:



1
2

∇

(

A

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A

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{\displaystyle {\tfrac {1}{2}}\nabla \left(\mathbf {A} \cdot \mathbf {A} \right)=\mathbf {A} \cdot \mathbf {J} \_{\mathbf {A} }=\mathbf {A} \cdot (\nabla \mathbf {A} )=(\mathbf {A} \cdot \nabla )\mathbf {A} +\mathbf {A} \times (\nabla \times \mathbf {A} ).}

{\displaystyle {\tfrac {1}{2}}\nabla \left(\mathbf {A} \cdot \mathbf {A} \right)=\mathbf {A} \cdot \mathbf {J} _{\mathbf {A} }=\mathbf {A} \cdot (\nabla \mathbf {A} )=(\mathbf {A} \cdot \nabla )\mathbf {A} +\mathbf {A} \times (\nabla \times \mathbf {A} ).}Relyativistik (kinetik) impulsning funksiyasi sifatida Gamiltonian uchun ekvivalent ifoda, {\displaystyle \mathbf {P} =\gamma m{\dot {\mathbf {x} }}(t)=\mathbf {p} -q\mathbf {A} }, hisoblanadi



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{\displaystyle {\mathcal {H}}(t)={\dot {\mathbf {x} }}(t)\cdot \mathbf {P} (t)+{\frac {mc^{2}}{\gamma }}+q\varphi (\mathbf {x} (t),t)=\gamma mc^{2}+q\varphi (\mathbf {x} (t),t)=E+V}

{\displaystyle {\mathcal {H}}(t)={\dot {\mathbf {x} }}(t)\cdot \mathbf {P} (t)+{\frac {mc^{2}}{\gamma }}+q\varphi (\mathbf {x} (t),t)=\gamma mc^{2}+q\varphi (\mathbf {x} (t),t)=E+V}
Bu kinetik impulsning afzalligi {\displaystyle \mathbf {P} } kanonik impulsni eksperimental ravishda oʻlchash mumkin, {\displaystyle \mathbf {p} }da mumkin emas. Eʼtibor bering, Gamiltonian (toʻliq energiya) nisbiy energiya (kinetik + dam) yigʻindisi sifatida koʻrib chiqilishi mumkin, {\displaystyle E=\gamma mc^{2}}, ortiqcha potentsial energiya, {\displaystyle V=q\varphi } .

Simplektik geometriyadan Gamilton tenglamalarigacha
---------------------------------------------------

### Gamilton sistemalarining geometriyasi

Gamiltonian bir necha ekvivalent usullar bilan silliq juft oʻlchamli *M*2*n* koʻp qirrali simplektik strukturani keltirib chiqarishi mumkin, eng yaxshi maʼlum boʻlganlari quyidagilardir:

Yopiq degenerativ simplektik 2-forma sifatida ō. Darbu teoremasiga koʻra, M ning istalgan nuqtasi atrofidagi kichik mahallada tegishli mahalliy koordinatalar mavjud. {\displaystyle p\_{1},\cdots ,p\_{n},\ q\_{1},\cdots ,q\_{n}} (*kanonik* yoki *simplektik* koordinatalar), bunda simplektik shakl quyidagicha boʻladi:



ω
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{\displaystyle \omega =\sum \_{i=1}^{n}dp\_{i}\wedge dq\_{i}\,.}

{\displaystyle \omega =\sum _{i=1}^{n}dp_{i}\wedge dq_{i}\,.}Shakl {\displaystyle \omega } kotangent fazo bilan tangens fazoning tabiiy izomorfizmini keltirib chiqaradi: {\displaystyle T\_{x}M\cong T\_{x}^{\*}M.} Bu vektorni xaritalash orqali amalga oshiriladi {\displaystyle \xi \in T\_{x}M} 1-shaklga {\displaystyle \omega \_{\xi }\in T\_{x}^{\*}M,} bu yerda {\displaystyle \omega \_{\xi }(\eta )=\omega (\eta ,\xi )} Barcha uchun {\displaystyle \eta \in T\_{x}M.} Bilinearligi va degeneratsiyasi tufayli {\displaystyle \omega ,} va bu haqiqat {\displaystyle \dim T\_{x}M=\dim T\_{x}^{\*}M,} xaritalash {\displaystyle \xi \to \omega \_{\xi }} Bu, albatta, chiziqli izomorfizmdir. Bu izomorfizm *tabiiydir*, chunki u koordinatalar oʻzgarishi bilan oʻzgarmaydi {\displaystyle M.} Hammasini takrorlash {\displaystyle x\in M,} biz izomorfizm bilan yakunlaymiz {\displaystyle J^{-1}:{\text{Vect}}(M)\to \Omega ^{1}(M)} silliq vektor maydonlarining cheksiz oʻlchovli fazosi va silliq 1-shakllar orasidagi. Har biri uchun {\displaystyle f,g\in C^{\infty }(M,\mathbb {R} )} va {\displaystyle \xi ,\eta \in {\text{Vect}}(M),}


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{\displaystyle J^{-1}(f\xi +g\eta )=fJ^{-1}(\xi )+gJ^{-1}(\eta ).}

{\displaystyle J^{-1}(f\xi +g\eta )=fJ^{-1}(\xi )+gJ^{-1}(\eta ).}
(Algebraik nuqtai nazardan, shuni aytish mumkin {\displaystyle C^{\infty }(M,\mathbb {R} )} -modullar {\displaystyle {\text{Vect}}(M)} va {\displaystyle \Omega ^{1}(M)} izomorf). Agar {\displaystyle H\in C^{\infty }(M\times \mathbb {R} \_{t},\mathbb {R} ),} keyin, har bir sobit uchun {\displaystyle t\in \mathbb {R} \_{t},}{\displaystyle dH\in \Omega ^{1}(M),} va {\displaystyle J(dH)\in {\text{Vect}}(M).}{\displaystyle J(dH)} Gamilton vektor maydoni sifatida tanilgan. {\displaystyle M}da tegishli differensial tenglama yoqilgan.



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{\displaystyle {\dot {x}}=J(dH)(x)}

{\displaystyle {\dot {x}}=J(dH)(x)}
*Hamilton tenglamasi* deyiladi. Bu yerda {\displaystyle x=x(t)} va {\displaystyle J(dH)(x)\in T\_{x}M} vektor maydonining (vaqtga bogʻliq) qiymati {\displaystyle J(dH)} da {\displaystyle x\in M.}

Gamilton tizimini R vaqt davomida tolalar toʻplami E deb tushunish mumkin, Et tolasi *t* ∈ *R* vaqtidagi joylashuv maydonidir. Shunday qilib, Lagrangian jet toʻplamining J ustida E ustidagi funksiyasi; Lagranjning tolali Legendre konvertatsiyasini olib, vaqt oʻtishi bilan dual toʻplamda funksiya hosil qiladi, uning tolasi t dagi kotangent fazosi *T*∗*Et* boʻlib, u tabiiy simplektik shakl bilan jihozlangan va bu oxirgi funksiya Gamiltoniandir. Lagranj va Gamilton mexanikasi oʻrtasidagi yozishmalarga tavtologik bir shakl bilan erishiladi.

Simplektik manifolddagi har qanday silliq real qiymatli H funksiyasidan Gamilton tizimini aniqlash uchun foydalanish mumkin. H funksiyasi „Gamiltonian" yoki „energiya funksiyasi" deb nomlanadi. Simplektik manifold keyin fazo fazosi deb ataladi. Gamiltonian simplektik manifoldda Gamilton vektor maydoni deb nomlanuvchi maxsus vektor maydonini induksiya qiladi.

Gamilton vektor maydoni manifoldda Gamilton oqimini keltirib chiqaradi. Bu kollektor oʻzgarishlarining bir parametrli oilasi (egri chiziqlar parametri odatda „vaqt" deb ataladi); boshqacha qilib aytganda, simplektomorfizmlarning izotopi, identifikatsiyadan boshlab. Liuvil teoremasiga koʻra, har bir simplektomorfizm faza fazosida hajm shaklini saqlaydi. Gamilton oqimi tomonidan qoʻzgʻatilgan simplektomorfizmlar toʻplami odatda Gamilton tizimining „Gamilton mexanikasi" deb ataladi.

Simplektik tuzilma Puasson qavsni keltirib chiqaradi. Puasson qavs Lie algebrasining strukturasi manifoldidagi funksiyalar maydonini beradi.

Agar F va G M da silliq funksiyalar boʻlsa, silliq funksiya ω2(*IdG*, *IdF*) toʻgʻri aniqlanadi; u F va G funksiyalarning *Puasson qavsi* deyiladi va {*F*, *G*} bilan belgilanadi. Poisson qavs quyidagi xususiyatlarga ega:

1. ikkilik
2. antisimmetriya
3. Leybnits qoidasi : {\displaystyle \{F\_{1}\cdot F\_{2},G\}=F\_{1}\{F\_{2},G\}+F\_{2}\{F\_{1},G\}}
4. Yakobi identifikatori : {\displaystyle \{\{H,F\},G\}+\{\{F,G\},H\}+\{\{G,H\},F\}\equiv 0}
5. degeneratsiya emas: agar M dagi x nuqta F uchun muhim boʻlmasa, u holda silliq G funktsiyasi mavjud boʻladi {\displaystyle \{F,G\}(x)\neq 0} .

f funksiya berilgan



d

d

t

f
=

∂

∂
t

f
+

{

f
,

H

}

,

{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}f={\frac {\partial }{\partial t}}f+\left\{f,{\mathcal {H}}\right\},}

{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}f={\frac {\partial }{\partial t}}f+\left\{f,{\mathcal {H}}\right\},}

  

agar ehtimollik taqsimoti mavjud boʻlsa ρ, u holda (fazo fazo tezligidan beri {\displaystyle ({\dot {p}}\_{i},{\dot {q}}\_{i})} nol divergensiyaga ega va ehtimollik saqlanib qolgan) uning konvektiv hosilasi nolga teng ekanligini koʻrsatish mumkin va shuning uchun



∂

∂
t

ρ
=
−

{

ρ
,

H

}

{\displaystyle {\frac {\partial }{\partial t}}\rho =-\left\{\rho ,{\mathcal {H}}\right\}}

{\displaystyle {\frac {\partial }{\partial t}}\rho =-\left\{\rho ,{\mathcal {H}}\right\}}
Bu Liuvil teoremasi deb ataladi. Simplektik manifold ustidagi har bir silliq funksiya G simplektomorfizmlarning bir parametrli oilasini hosil qiladi va agar {*G*, *H*} = 0 boʻlsa, u holda G saqlanib qoladi va simplektomorfizmlar simmetriya oʻzgarishlari hisoblanadi.

Gamiltonian bir nechta saqlangan miqdorlarga ega boʻlishi mumkin Gi. Agar simplektik manifold 2*n* oʻlchamga ega boʻlsa va involyutsiyada boʻlgan n funksional mustaqil saqlangan Gi kattaliklari mavjud boʻlsa (yaʼni, {*G**i*, *G**j*} = 0), u holda Gamiltonian Liuvil integrallanishi mumkin. Liuvil-Arnold teoremasi shuni koʻrsatadiki, lokal ravishda har qanday Liuvil integrallanadigan Gamiltonian simplektomorfizm orqali yangi Gamiltonianga koordinatalar sifatida Gi saqlanib qolgan kattaliklarga aylantirilishi mumkin; yangi koordinatalar *harakat burchagi koordinatalari* deb ataladi. Oʻzgartirilgan Gamiltonian faqat Gi ga bogʻliq va shuning uchun harakat tenglamalari oddiy shaklga ega.




G
˙

i

=
0

,


φ
˙

i

=

F

i

(
G
)

{\displaystyle {\dot {G}}\_{i}=0\quad ,\quad {\dot {\varphi }}\_{i}=F\_{i}(G)}

{\displaystyle {\dot {G}}_{i}=0\quad ,\quad {\dot {\varphi }}_{i}=F_{i}(G)}baʼzi funksiyalar uchun F. KAM(Kolmogorov-Arnold-Moser) teoremasi tomonidan boshqariladigan integral tizimlardan kichik ogʻishlarga qaratilgan butun bir maydon mavjud.

Gamilton vektor maydonlarining integralligi ochiq savol. Umuman olganda, Gamilton tizimlari xaotikdir; oʻlchov, toʻliqlik, yaxlitlik va barqarorlik tushunchalari kam taʼriflangan.

### Riman manifoldlari

Muhim maxsus holat kvadrat shakllar boʻlgan Gamiltonianlardan iborat, yaʼni shunday yozilishi mumkin boʻlgan Gamiltonianlar.



H

(
q
,
p
)
=

1
2

⟨
p
,
p

⟩

q

{\displaystyle {\mathcal {H}}(q,p)={\tfrac {1}{2}}\langle p,p\rangle \_{q}}

{\displaystyle {\mathcal {H}}(q,p)={\tfrac {1}{2}}\langle p,p\rangle _{q}}
Bu yerda ⟨ , ⟩*q* — *T**Q* tolalaridagi silliq oʻzgaruvchan ichki mahsulot, konfiguratsiya maydonidagi q nuqtasiga kotangent boʻshliq, baʼzan komerik deb ataladi. Bu Gamiltonian butunlay kinetik atamadan iborat.

Agar Rieman manifoldu yoki psevdo-Riman manifoldini hisobga oladigan boʻlsak, Rieman metrikasi tangens va kotangens toʻplamlari oʻrtasida chiziqli izomorfizmni keltirib chiqaradi. (Qarang: Musiqiy izomorfizm). Ushbu izomorfizmdan foydalanib, komerikni aniqlash mumkin. (Koordinatalarda belgilovchi matritsa metrikani belgilovchi matritsaga teskarisidir.) Bu Gamiltonian uchun Gamilton-Jakobi tenglamalarining yechimlari manifolddagi geodeziya bilan bir xil boʻladi. Xususan, bu holda Gamilton oqimi geodezik oqim bilan bir xil narsadir. Bunday yechimlarning mavjudligi va yechimlar toʻplamining toʻliqligi geodeziya maqolasida batafsil muhokama qilinadi. Hamilton oqimlari sifatida Geodeziyaga qarang.

### Sub-Riman manifoldlari

Kometrika degenerativ boʻlsa, u teskari emas. Bunday holda, metrikaga ega boʻlmaganidek, Rieman manifoldiga ega emas. Biroq, Gamiltonian hali ham mavjud. Kometrik konfiguratsiya manifoldu Q ning har q nuqtasida degeneratsiyaga uchragan holatda, komerikning darajasi manifold Q oʻlchamidan kichik boʻlishi uchun, birida Riemanning pastki manifoldiga ega boʻladi.

Bu holatda Gamiltonian **sub-rimanlik Gamiltonian** sifatida tanilgan. Har bir bunday Gamiltonian komerikni oʻziga xos tarzda aniqlaydi va aksincha. Bu shuni anglatadiki, har bir sub-rimanlik manifold oʻzining sub-rimanlik Gamiltonian tomonidan noyob tarzda aniqlanadi va buning aksi toʻgʻri: har bir sub-rimanlik manifold noyob sub-rimanlik Gamiltonianga ega. Sub-Riman geodeziyasining mavjudligi Chou-Rashevskiy teoremasi bilan berilgan.

Uzluksiz, haqiqiy qiymatli Heisenberg guruhi sub-Riemann manifoldining oddiy misolini beradi. Heisenberg guruhi uchun Gamiltonian tomonidan berilgan

  



H

(

x
,
y
,
z
,

p

x

,

p

y

,

p

z

)

=

1
2

(

p

x

2

+

p

y

2

)

.

{\displaystyle {\mathcal {H}}\left(x,y,z,p\_{x},p\_{y},p\_{z}\right)={\tfrac {1}{2}}\left(p\_{x}^{2}+p\_{y}^{2}\right).}

{\displaystyle {\mathcal {H}}\left(x,y,z,p_{x},p_{y},p_{z}\right)={\tfrac {1}{2}}\left(p_{x}^{2}+p_{y}^{2}\right).}
pz Hamiltonianni ichiga kirmaydi.

### Puasson algebralari

Gamilton tizimlarini turli yoʻllar bilan umumlashtirish mumkin. Simplektik manifold ustidagi silliq funksiyalar algebrasini oddiygina koʻrib chiqish oʻrniga, Gamilton mexanikasi umumiy kommutativ birlik real Puasson algebralarida shakllantirilishi mumkin. Holat Puasson algebrasidagi uzluksiz chiziqli funksionaldir (baʼzi mos topologiyalar bilan jihozlangan), algebraning istalgan A elementi uchun *A*2 manfiy boʻlmagan haqiqiy songa mos keladi.

Yana bir umumlashtirish Nambu dinamikasi tomonidan berilgan.

### Puasson qavs orqali kvant mexanikasiga umumlashtirish

Yuqoridagi Gamilton tenglamalari klassik mexanika uchun yaxshi ishlaydi, lekin kvant mexanikasi uchun emas, chunki muhokama qilingan differensial tenglamalar bir vaqtning oʻzida istalgan vaqtda zarrachaning aniq oʻrni va momentumini belgilash mumkinligini taxmin qiladi. Shu bilan birga, tenglamalarni keyinchalik umumlashtirish mumkin, keyin esa kvant mexanikasiga, shuningdek klassik mexanikaga, Puasson algebrasini p va q nisbatan Moyal qavslar algebrasiga deformatsiya qilish orqali kengaytirish mumkin.

Xususan, Gamilton tenglamasining umumiy shakli oʻqiydi




d

f

d

t

=

{

f
,

H

}

+

∂
f

∂
t


{\displaystyle {\frac {\mathrm {d} f}{\mathrm {d} t}}=\left\{f,{\mathcal {H}}\right\}+{\frac {\partial f}{\partial t}}}

{\displaystyle {\frac {\mathrm {d} f}{\mathrm {d} t}}=\left\{f,{\mathcal {H}}\right\}+{\frac {\partial f}{\partial t}}}Bu yerda f p va q ning baʼzi funksiyasi, H esa Gamiltonian. Differensial tenglamalarga murojaat qilmasdan Puasson qavsni baholash qoidalarini bilish uchun Li algebrasiga qarang; Puasson qavs — bu Puasson algebrasidagi Li qavsning nomi. Keyinchalik bu Puasson qavslari Hilbrand J. Groenewold tomonidan isbotlanganidek, tengsiz Li algebrasiga mos keladigan Moyal qavslariga kengaytirilishi mumkin va shu bilan fazalar fazosida kvant mexanik diffuziyasini tasvirlaydi (Faza fazosining formulasiga va Wigner-Veyl konvertatsiyasiga qarang). Ushbu koʻproq algebraik yondashuv nafaqat fazalar boʻshligʻida ehtimollik taqsimotini Wigner kvazi-ehtimollik taqsimotiga kengaytirish imkonini beradi, balki oddiy Puasson qavs klassik sozlamalarida tizimdagi tegishli saqlangan miqdorlarni tahlil qilishda koʻproq kuch beradi.

Manbalar
--------

1. ↑  Hamilton, William Rowan, Sir*. On a general method of expressing the paths of light, & of the planets, by the coefficients of a characteristic function.*. Printed by P.D. Hardy, 1833. OCLC 68159539.
2. ↑  Landau & Lifshitz 1976, ss. 33–34 harvnb error: no target: CITEREFLandauLifshitz1976 (help)
3. ↑  This derivation is along the lines as given in Arnol'd 1989, ss. 65–66 harvnb error: no target: CITEREFArnol'd1989 (help)
4. ↑  Goldstein, Poole & Safko 2002, ss. 347–349 harvnb error: no target: CITEREFGoldsteinPooleSafko2002 (help)
5. ↑  Zinn-Justin, Jean; Guida, Riccardo (2008-12-04). "Gauge invariance" (en). *Scholarpedia* **3** (12): 8287. doi:10.4249/scholarpedia.8287. ISSN 1941-6016.
6. ↑  Arnol'd, Kozlov & Neĩshtadt 1988. sfn error: no target: CITEREFArnol'dKozlovNeĩshtadt1988 (help)
7. ↑  Arnol'd, Kozlov & Neĩshtadt 1988 harvnb error: no target: CITEREFArnol'dKozlovNeĩshtadt1988 (help)

Adabiyotlar
-----------

* Landau, Lev Davidovich; Lifshitz, Evgenii Mikhailovich (1976). Mechanics. Course of Theoretical Physics. Vol. 1. Sykes, J. B. (John Bradbury), Bell, J. S. (3rd ed.). Oxford. ISBN 0-08-021022-8. OCLC 2591126.
* Abraham, R.; Marsden, J.E. (1978). Foundations of mechanics (2d ed. rev. enl. and reset ed.). Reading, Mass.: Benjamin/Cummings Pub. Co. ISBN 0-8053-0102-X. OCLC 3516353.
* Arnol'd, V. I.; Kozlov, V. V.; Neĩshtadt, A. I. (1988). "Mathematical aspects of classical and celestial mechanics". Encyclopaedia of Mathematical Sciences, Dynamical Systems III. Vol. 3. Anosov, D. V. Berlin: Springer-Verlag. ISBN 0-387-17002-2. OCLC 16404140.
* Arnol'd, V. I. (1989). Mathematical methods of classical mechanics (2nd ed.). New York: Springer-Verlag. ISBN 0-387-96890-3. OCLC 18681352.
* Goldstein, Herbert; Poole, Charles P. Jr.; Safko, John L. (2002). Classical mechanics (3rd ed.). San Francisco: Addison Wesley. ISBN 0-201-31611-0. OCLC 47056311.
* Vinogradov, A. M.; Kupershmidt, B A (1977-08-31). "The structure of Hamiltonian mechanics". Russian Mathematical Surveys. 32 (4): 177–243.