Klassik mexanikada **Appell harakat tenglamasi** (**Gibbs-Appell harakat tenglamasi**) 1879-yilda Josia Uillard Gibbs  va 1900-yilda Pol Emil Appell  tomonidan tasvirlangan klassik mexanikaning muqobil umumiy formulasidir.

Bayonot
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Gibbs-Appell tenglamasi quyidagicha

{\displaystyle Q\_{r}={\frac {\partial S}{\partial \alpha \_{r}}},}
bu yerda {\displaystyle \alpha \_{r}={\ddot {q\_{r}}}} ixtiyoriy umumlashtirilgan tezlanish yoki umumlashtirilgan koordinatalarning ikkinchi marta hosilasidir. {\displaystyle q\_{r}}, va {\displaystyle Q\_{r}} uning tegishli umumlashgan kuchidir. Umumlashtirilgan kuch bajarilgan ishni beradi

{\displaystyle dW=\sum \_{r=1}^{D}Q\_{r}dq\_{r},}
indeks bu yerda {\displaystyle r} ustidan yuguradi {\displaystyle D} umumlashtirilgan koordinatalar {\displaystyle q\_{r}}, bu odatda tizimning erkinlik darajalariga mos keladi. Funktsiya {\displaystyle S} zarracha tezlanishlarining massaviy yigʻindisi kvadrati sifatida aniqlanadi,

{\displaystyle S={\frac {1}{2}}\sum \_{k=1}^{N}m\_{k}\mathbf {a} \_{k}^{2}\,,}
indeks bu yerda {\displaystyle k} ustidan yuguradi {\displaystyle K} zarralar va

{\displaystyle \mathbf {a} \_{k}={\ddot {\mathbf {r} }}\_{k}={\frac {d^{2}\mathbf {r} \_{k}}{dt^{2}}}}
ning tezlashishi hisoblanadi {\displaystyle k} -chi zarra, uning pozitsiya vektorining ikkinchi marta hosilasi {\displaystyle \mathbf {r} \_{k}}. Har biri {\displaystyle \mathbf {r} \_{k}} umumlashtirilgan koordinatalar bilan ifodalanadi, va {\displaystyle \mathbf {a} \_{k}} umumlashgan tezlanishlar bilan ifodalanadi.

Klassik mexanikaning boshqa formulalari bilan aloqalari
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Appellning formulasi klassik mexanikaga hech qanday yangi fizikani kiritmaydi va shuning uchun Lagrang mexanikasi va Gamilton mexanikasi kabi klassik mexanikaning boshqa reformulalariga tengdir. Barcha klassik mexanika Nyutonning harakat qonunlarida mavjud. Baʼzi hollarda Appellning harakat tenglamasi odatda ishlatiladigan Lagranj mexanikasiga qaraganda qulayroq boʻlishi mumkin, ayniqsa golonomik boʻlmagan cheklovlar ishtirok etganda. Aslida, Appell tenglamasi toʻgʻridan-toʻgʻri Lagranjning harakat tenglamalariga olib keladi.  Bundan tashqari, u murakkab kosmik kemalarning harakatini tavsiflash uchun juda mos keladigan Keyn tenglamalarini olish uchun ishlatilishi mumkin.  Appell formulasi Gaussning eng kam cheklash printsipining qoʻllanilishidir. 

### Chiqarish

*D* umumlashtirilgan koordinatalarining cheksiz kichik oʻzgarishi uchun zarrachalarning joylashuvi **r** *k* oʻzgarishi

{\displaystyle d\mathbf {r} \_{k}=\sum \_{r=1}^{D}dq\_{r}{\frac {\partial \mathbf {r} \_{k}}{\partial q\_{r}}}}
Vaqtga nisbatan ikkita hosila olish tezlashuvlar uchun ekvivalent tenglamani beradi.

{\displaystyle {\frac {\partial \mathbf {a} \_{k}}{\partial \alpha \_{r}}}={\frac {\partial \mathbf {r} \_{k}}{\partial q\_{r}}}}
Umumlashtirilgan koordinatalarda cheksiz kichik oʻzgarish *dq r* tomonidan bajarilgan ish

{\displaystyle dW=\sum \_{r=1}^{D}Q\_{r}dq\_{r}=\sum \_{k=1}^{N}\mathbf {F} \_{k}\cdot d\mathbf {r} \_{k}=\sum \_{k=1}^{N}m\_{k}\mathbf {a} \_{k}\cdot d\mathbf {r} \_{k}}
bu yerda *k* -chi zarra uchun Nyutonning ikkinchi qonuni

{\displaystyle \mathbf {F} \_{k}=m\_{k}\mathbf {a} \_{k}}
ishlatilgan. Formulani *d* **r** *k* oʻrniga qoʻyish va ikkita yigʻindining tartibini almashtirsak, formulalar hosil boʻladi.

{\displaystyle dW=\sum \_{r=1}^{D}Q\_{r}dq\_{r}=\sum \_{k=1}^{N}m\_{k}\mathbf {a} \_{k}\cdot \sum \_{r=1}^{D}dq\_{r}\left({\frac {\partial \mathbf {r} \_{k}}{\partial q\_{r}}}\right)=\sum \_{r=1}^{D}dq\_{r}\sum \_{k=1}^{N}m\_{k}\mathbf {a} \_{k}\cdot \left({\frac {\partial \mathbf {r} \_{k}}{\partial q\_{r}}}\right)}
Shuning uchun umumlashgan kuchlar

{\displaystyle Q\_{r}=\sum \_{k=1}^{N}m\_{k}\mathbf {a} \_{k}\cdot \left({\frac {\partial \mathbf {r} \_{k}}{\partial q\_{r}}}\right)=\sum \_{k=1}^{N}m\_{k}\mathbf {a} \_{k}\cdot \left({\frac {\partial \mathbf {a} \_{k}}{\partial \alpha \_{r}}}\right)}
Bu umumlashtirilgan tezlanishlarga nisbatan *S* ning hosilasiga teng

{\displaystyle {\frac {\partial S}{\partial \alpha \_{r}}}={\frac {\partial }{\partial \alpha \_{r}}}{\frac {1}{2}}\sum \_{k=1}^{N}m\_{k}\left|\mathbf {a} \_{k}\right|^{2}=\sum \_{k=1}^{N}m\_{k}\mathbf {a} \_{k}\cdot \left({\frac {\partial \mathbf {a} \_{k}}{\partial \alpha \_{r}}}\right)}
Appellning harakat tenglamasi olinadi

{\displaystyle {\frac {\partial S}{\partial \alpha \_{r}}}=Q\_{r}.}

Misollar
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### Qattiq jismlar dinamikasining Eyler tenglamalari

Eyler tenglamalari Appell formulasining ajoyib tasvirini beradi.

Qattiq tayoqchalar bilan birlashtirilgan *N* zarrachaning qattiq tanasini koʻrib chiqaylik. Tananing aylanishini burchak tezligi vektori bilan tasvirlash mumkin {\displaystyle {\boldsymbol {\omega }}}, va mos keladigan burchak tezlanish vektori

{\displaystyle {\boldsymbol {\alpha }}={\frac {d{\boldsymbol {\omega }}}{dt}}}
Aylanish uchun umumlashtirilgan kuch moment hisoblanadi {\displaystyle {\textbf {N}}}, cheksiz kichik aylanish uchun bajarilgan ish beri {\displaystyle \delta {\boldsymbol {\phi }}} hisoblanadi {\displaystyle dW=\mathbf {N} \cdot \delta {\boldsymbol {\phi }}}. ning tezligi {\displaystyle k} -chi zarracha tomonidan berilgan

{\displaystyle \mathbf {v} \_{k}={\boldsymbol {\omega }}\times \mathbf {r} \_{k}}
bu yerda {\displaystyle \mathbf {r} \_{k}} zarrachaning Dekart koordinatalaridagi holati; uning mos keladigan tezlanishi

{\displaystyle \mathbf {a} \_{k}={\frac {d\mathbf {v} \_{k}}{dt}}={\boldsymbol {\alpha }}\times \mathbf {r} \_{k}+{\boldsymbol {\omega }}\times \mathbf {v} \_{k}}
Shuning uchun, funktsiya {\displaystyle S} sifatida yozilishi mumkin

{\displaystyle S={\frac {1}{2}}\sum \_{k=1}^{N}m\_{k}\left(\mathbf {a} \_{k}\cdot \mathbf {a} \_{k}\right)={\frac {1}{2}}\sum \_{k=1}^{N}m\_{k}\left\{\left({\boldsymbol {\alpha }}\times \mathbf {r} \_{k}\right)^{2}+\left({\boldsymbol {\omega }}\times \mathbf {v} \_{k}\right)^{2}+2\left({\boldsymbol {\alpha }}\times \mathbf {r} \_{k}\right)\cdot \left({\boldsymbol {\omega }}\times \mathbf {v} \_{k}\right)\right\}}
ga nisbatan *S* ning hosilasini belgilash {\displaystyle {\boldsymbol {\alpha }}} momentga teng Eyler tenglamalarini beradi

{\displaystyle I\_{xx}\alpha \_{x}-\left(I\_{yy}-I\_{zz}\right)\omega \_{y}\omega \_{z}=N\_{x}}
{\displaystyle I\_{yy}\alpha \_{y}-\left(I\_{zz}-I\_{xx}\right)\omega \_{z}\omega \_{x}=N\_{y}}
{\displaystyle I\_{zz}\alpha \_{z}-\left(I\_{xx}-I\_{yy}\right)\omega \_{x}\omega \_{y}=N\_{z}}

Manbalar
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* Pars, LA*. A Treatise on Analytical Dynamics*. Woodbridge, Connecticut: Ox Bow Press, 1965 — 197–227,631–632 bet.
* Whittaker, ET*. A Treatise on the Analytical Dynamics of Particles and Rigid Bodies, with an Introduction to the Problem of Three Bodies*, 4th, New York: Dover Publications, 1937. ISBN.
* Seeger (1930). "Appell's equations". *Journal of the Washington Academy of Science* **20**: 481–484.
* Brell, H (1913). "Nachweis der Aquivalenz des verallgemeinerten Prinzipes der kleinsten Aktion mit dem Prinzip des kleinsten Zwanges". *Wien. Sitz.* **122**: 933–944.  Connection of Appell’s formulation with the principle of least action.

Qoʻshimcha adabiyotlar
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* PDF copy of Appell’s article at Goettingen University
* PDF copy of a second article on Appell’s equations and Gauss’s principle


1. ↑  Gibbs, JW (1879). "On the Fundamental Formulae of Dynamics.". *American Journal of Mathematics* **2** (1): 49–64. doi:10.2307/2369196.
2. ↑  Appell, P (1900). "Sur une forme générale des équations de la dynamique.". *Journal für die reine und angewandte Mathematik* **121**: 310–?.
3. ↑  Deslodge, Edward A. (1988). "The Gibbs–Appell equations of motion". *American Journal of Physics* **56** (9): 841–46. doi:10.1119/1.15463. .
4. ↑  Deslodge, Edward A. (1987). "Relationship between Kane's equations and the Gibbs-Appell equations". *Journal of Guidance, Control, and Dynamics* (American Institute of Aeronautics and Astronautics) **10** (1): 120–22. doi:10.2514/3.20192.
5. ↑  Lewis, Andrew D. (August 1996). "The geometry of the Gibbs-Appell equations and Gauss' principle of least constraint". *Reports on Mathematical Physics* **38** (1): 11–28. doi:10.1016/0034-4877(96)87675-0. .