Appell harakat tenglamasi

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
Gibbs-Appell tenglamasi quyidagicha

: Q_{r} = \frac{\partial S}{\partial \alpha_{r}}, 

bu yerda \alpha_r=\ddot{q_r} ixtiyoriy umumlashtirilgan tezlanish yoki umumlashtirilgan koordinatalarning ikkinchi marta hosilasidir. q_r , va Q_r  uning tegishli umumlashgan kuchidir . Umumlashtirilgan kuch bajarilgan ishni beradi

: dW = \sum_{r=1}^{D} Q_{r} dq_{r}, 

indeks bu yerda r  ustidan yuguradi D  umumlashtirilgan koordinatalar q_r , bu odatda tizimning erkinlik darajalariga mos keladi. Funktsiya S  zarracha tezlanishlarining massaviy yigʻindisi kvadrati sifatida aniqlanadi,

: S = \frac{1}{2} \sum_{k=1}^{N} m_{k} \mathbf{a}_{k}^{2}\,,

indeks bu yerda k  ustidan yuguradi K  zarralar va

: \mathbf{a}_k = \ddot{\mathbf{r}}_k = \frac{d^2 \mathbf{r}_k}{dt^2} 

ning tezlashishi hisoblanadi 
k
 -chi zarra, uning pozitsiya vektorining ikkinchi marta hosilasi \mathbf{r}_k  . Har biri \mathbf{r}_k  umumlashtirilgan koordinatalar bilan ifodalanadi, va \mathbf{a}_k  umumlashgan tezlanishlar bilan ifodalanadi.

## Klassik mexanikaning boshqa formulalari bilan aloqalari
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

: 
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.

: 
\frac{\partial \mathbf{a}_{k}}{\partial \alpha_{r}} = \frac{\partial \mathbf{r}_{k}}{\partial q_{r}}


Umumlashtirilgan koordinatalarda cheksiz kichik oʻzgarish *dq <sub>r</sub>* tomonidan bajarilgan ish

: 
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

: \mathbf{F}_k = m_k\mathbf{a}_k

ishlatilgan. Formulani *d* **r** *k* oʻrniga qoʻyish va ikkita yigʻindining tartibini almashtirsak, formulalar hosil boʻladi.

: 
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

: 
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

: 
\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

: 
\frac{\partial S}{\partial \alpha_{r}} = Q_{r}.


## Misollar

### 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 \boldsymbol\omega, va mos keladigan burchak tezlanish vektori

: 
\boldsymbol\alpha = \frac{d\boldsymbol\omega}{dt}


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

: 
\mathbf{v}_{k} = \boldsymbol\omega \times \mathbf{r}_{k}


bu yerda 
\mathbf{r}_{k}
 zarrachaning Dekart koordinatalaridagi holati; uning mos keladigan tezlanishi

: 
\mathbf{a}_{k} = \frac{d\mathbf{v}_{k}}{dt} = 
\boldsymbol\alpha \times \mathbf{r}_{k} + \boldsymbol\omega \times \mathbf{v}_{k}


Shuning uchun, funktsiya 
S
 sifatida yozilishi mumkin

: 
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 \boldsymbol\alpha momentga teng Eyler tenglamalarini beradi

: 
I_{xx} \alpha_{x} - \left( I_{yy} - I_{zz} \right)\omega_{y} \omega_{z} = N_{x}


: 
I_{yy} \alpha_{y} - \left( I_{zz} - I_{xx} \right)\omega_{z} \omega_{x} = N_{y}


: 
I_{zz} \alpha_{z} - \left( I_{xx} - I_{yy} \right)\omega_{x} \omega_{y} = N_{z}


## Manbalar
* 
* 
* 
*  Connection of Appell’s formulation with the principle of least action.

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