Application of quaternian algebra to classical mechanics

ABSTRACT Master of Science Thesis APPLICATION OF QUATERNION ALGEBRA TO CLASSICAL MECHANICS TÜLAY TOLAN Anadolu University Gradate School of Natural and Applied Sciences Physics Program Supervisor: Prof. Dr. Kudret ÖZDAŞ 1998, Page 44 It is known that representation of translational and rotational motions by the quaternions and dual quaternions have simpler form and clearer geometrical meaning than homogeneous matris representation which is often used by physicists. Therefore quaternion representation, gives magnitudes and direction of translation and rotation, and angular momentum, moment of inertia of a system which are an application of quaternion representation are investigated in this thesis. In addition, dual quaternion algebra is studied and representations of Serret- Frenet Formula are defined. Keywords: Quaternions, Dual QuaternionsIll içindekiler ÖZET i ABSTRACT ii İÇİNDEKİLER üi ŞEKİLLER DİZİNİ iv SİMGELER VE KISALTMALAR v 1. GİRİŞ 1 2. KUATERNİON 2 2.1. Tanım 2 2.2. Kuaternion Cebri 2 2.3. Özellikler 9 3. DÖNME VE ÖTELEMENİN KUATERNİON GÖSTERİMLERİ 11 3.1. Ötelemenin Kuaternion Gösterimi 11 3.2. Dönmenin Kuaternion Gösterimi 12 3.3. Dönme ve Ötelemenin Kuaternion Gösterimi 15 3.4. Rodrigues Formülünün İspatı 16 4. KİNEMATİK VE DİNAMİK DİFERANSİYEL DENKLEMLERİN KUATERNİON GÖSTERİMLERİ 20 4.1. Açısal Hız 20 4.2. Açısal İvme 22 4.3. Açısal Momentum. 23 5. DUAL KUATERNİON 28 5.1. Tanım 28 5.2. Dual Kuaternion Cebri 29 5.3. Özellikler 32 6. DÖNME VE ÖTELEMENİN DUAL KUATERNİON GÖSTERİMLERİ 33 7. SERRET-FRENET ÜÇYÜZLÜSÜNÜN DUAL KUATERNİON GÖSTERİMİ 38 8. TARTIŞMA VE SONUÇ 42 9. KAYNAKLAR 43