Kirxgof tenglamalari

Suyuqlik dinamikasida, Gustav Kirchhoff nomi bilan atalgan Kirchhoff tenglamalari, ideal suyuqlikdagi qattiq jismning harakatini tasvirlaydi.

$$\begin{align} {d\over{dt}} {{\partial T}\over{\partial \vec \omega}} & = {{\partial T}\over{\partial \vec \omega}} \times \vec \omega + {{\partial T}\over{\partial \vec v}} \times \vec v + \vec Q_h + \vec Q, \\[10pt] {d\over{dt}} {{\partial T}\over{\partial \vec v}} & = {{\partial T}\over{\partial \vec v}} \times \vec \omega + \vec F_h + \vec F, \\[10pt] T & = {1 \over 2} \left( \vec \omega^T \tilde I \vec \omega + m v^2 \right) \\[10pt] \vec Q_h & = -\int p \vec x \times \hat n \, d\sigma, \\[10pt] \vec F_h & = -\int p \hat n \, d\sigma \end{align}$$

bu yerda $\vec \omega$ va $\vec v$ $\vec x$, mos ravishda; $\tilde I$ inersiya tensor momenti, $m$ tananing massasi; $\hat n$ nuqtadagi jismning sirtiga normal birlikdir $\vec x$ ; $p$ bu nuqtada bosim; $\vec Q_h$ va $\vec F_h$ mos ravishda jismga taʼsir qiluvchi gidrodinamik moment va kuch; $\vec Q$ va $\vec F$ jismga taʼsir etuvchi boshqa barcha moment va kuchlarni ham xuddi shunday bildiring. Integratsiya jismning sirtining suyuqlik bilan taʼsirlangan qismida amalga oshiriladi.

Agar tana butunlay suvga botgan boʻlsa, jism cheksiz katta hajmdagi aylanadigan, siqilmaydigan, oʻzgarmas suyuqlikda, yaʼni cheksiz dam olishda boʻlsa, u holda vektorlar. $\vec Q_h$ va $\vec F_h$ aniq integratsiya orqali topish mumkin va tananing dinamikasi Kirchhoff — Klebsh tenglamalari bilan tavsiflanadi:

$${d\over{dt}} {{\partial L}\over{\partial \vec \omega}} = {{\partial L}\over{\partial \vec \omega}} \times \vec \omega + {{\partial L}\over{\partial \vec v}} \times \vec v, \quad {d\over{dt}} {{\partial L}\over{\partial \vec v}} = {{\partial L}\over{\partial \vec v}} \times \vec \omega,$$

$$L(\vec \omega, \vec v) = {1 \over 2} (A \vec \omega,\vec \omega) + (B \vec \omega,\vec v) + {1 \over 2} (C \vec v,\vec v) + (\vec k,\vec \omega) + (\vec l,\vec v).$$

Ularning birinchi integrallari oʻqiladi

$$J_0 = \left({{\partial L}\over{\partial \vec \omega}}, \vec \omega \right) + \left({{\partial L}\over{\partial \vec v}}, \vec v \right) - L, \quad J_1 = \left({{\partial L}\over{\partial \vec \omega}},{{\partial L}\over{\partial \vec v}}\right), \quad J_2 = \left({{\partial L}\over{\partial \vec v}},{{\partial L}\over{\partial \vec v}}\right) .$$

Keyingi integratsiya pozitsiya va tezliklar uchun aniq ifodalarni ishlab chiqaradi.

Manbalar

• Kirchhoff G. R. Vorlesungen ueber Mathematische Physik, Mechanik. Lecture 19. Leipzig: Teubner. 1877.
• Lamb, H., Hydrodynamics. Sixth Edition Cambridge (UK): Cambridge University Press. 1932.