Euler's equation of motion

Japanese: オイラーの運動方程式 - オイラーのうんどうほうていしき(英語表記)Euler's equation of motion
Euler's equation of motion
(1) Equation of motion for a perfect fluid. If the fluid velocity vector is u , pressure is p , density is ρ, and external force vector acting per unit mass of the fluid is K , the following equation of motion holds for a perfect fluid.
This is called Euler's equation of motion. For a perfect fluid, this equation can be solved simultaneously with the continuity equation and the fluid state equation ρ = f ( p ) under given initial and boundary conditions to obtain the unknown functions u , p , and ρ, which determine the fluid's motion.
(2) Equations of motion that determine the rotational motion of a rigid body around a fixed point or center of gravity. The equations of rotational motion of a rigid body are very complicated, but are relatively simple when expressed in a coordinate system that is fixed to the rigid body and rotates with the rigid body. In particular, the simplest equations expressed in a rotating coordinate system with three principal axes of inertia relative to the center of rotation are the Euler equations of motion. If the principal moments of inertia are I 1 , I 2 , and I 3 , the components of the rotational angular velocity are ω 1 , ω 2 , and ω 3 , and the components of the moment of external forces are N 1 , N 2 , and N 3 , then the Euler equations of motion are given by the following three symmetrical equations:

I 1 ( d ω 1 / dt )-( I 2 - I 32 ω 3 = N 1



I 2 ( d ω 2 / dt )-( I 3 - I 13 ω 1 = N 2



I 3 ( d ω 3 / dt )-( I 1 - I 21 ω 2 = N 3

Source: Encyclopaedia Britannica Concise Encyclopedia About Encyclopaedia Britannica Concise Encyclopedia Information

Japanese:
(1) 完全流体に対する運動方程式。流体の速度ベクトルを u ,圧力を p ,密度を ρ ,流体の単位質量あたりに働く外力ベクトルを K とすれば,完全流体に対して次の運動方程式が成り立つ。
これをオイラーの運動方程式という。完全流体においては,この方程式と,連続の方程式および流体の状態方程式 ρ=f(p) とを連立させて,与えられた初期条件および境界条件のもとで解けば,未知関数 up ,ρ が求められ,それにより流体の運動が決定される。
(2) 固定点または重心のまわりの,剛体の回転運動を決める運動方程式。剛体の回転運動の方程式は非常に複雑であるが,剛体に固定されて剛体とともに回転する座標系で表わすと比較的簡単になる。特に回転の中心に関する3つの慣性主軸を座標軸とする回転座標系で表わした最も単純な方程式がオイラーの運動方程式である。主慣性モーメントを I1I2I3 ,回転角速度の成分を ω1 ,ω2 ,ω3 ,外力のモーメントの成分を N1N2N3 とすれば,オイラーの運動方程式は対称的な次の3式で与えられる。

I1(dω1/dt)-(I2I32ω3N1



I2(dω2/dt)-(I3I13ω1N2



I3(dω3/dt)-(I1I21ω2N3

出典 ブリタニカ国際大百科事典 小項目事典ブリタニカ国際大百科事典 小項目事典について 情報

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