English Dictionary

Liouville-Arnold theorem

固有名詞

In dynamical systems theory, a theorem stating that if, in a Hamiltonian dynamical system with n degrees of freedom, there are also known n first integrals of motion that are independent and in involution, then there exists a canonical transformation to action-angle coordinates in which the transformed Hamiltonian is dependent only upon the action coordinates and the angle coordinates evolve linearly in time. Thus the equations of motion for the system can be solved in quadratures if the canonical transform is explicitly known.

日本語の意味

リウヴィル・アーノルドの定理とは、力学系理論において、n自由度のハミルトン系で、独立かつ可換なn個の第一積分が存在する場合に、正準変換を用いて作用角座標系に変換できることを示す定理です。変換後のハミルトニアンが作用座標のみに依存し、角座標が時間に対して線形に進むため、運動方程式が積分可能になるという主張がなされています。

The Liouville-Arnold theorem ensures that an integrable Hamiltonian system with n degrees of freedom and n independent, involutive first integrals admits action-angle coordinates, reducing the equations of motion to linear flow on invariant tori.

リウヴィル＝アーノルドの定理は、自由度 n のハミルトン系が n 個の独立かつ互いに可換な第一積分を持つ場合に作用角座標が存在することを保証し、運動方程式を不変トーラス上での線形流に還元する。

意味（1）

In dynamical systems theory, a theorem stating that if, in a Hamiltonian dynamical system with n degrees of freedom, there are also known n first integrals of motion that are independent and in involution, then there exists a canonical transformation to action-angle coordinates in which the transformed Hamiltonian is dependent only upon the action coordinates and the angle coordinates evolve linearly in time. Thus the equations of motion for the system can be solved in quadratures if the canonical transform is explicitly known.

the Liouville-Arnold theorem （ canonical ）