二维各向异性谐振子的第三个独立守恒量及其对称性

The third independent conserved quantity and its symmetry of the two-dimensional anisotropic harmonic oscillator

二维各向异性谐振子和两分振子的能量是守恒的,但三个守恒量中只有其中两个是独立的.当频率比ω1/ω2为有理数时,系统存在第三个独立的守恒量.本文用扩展Prelle-Singer法得到五个典型谐振子的第三个独立守恒量,并讨论了与守恒量相应的Noether对称性与Lie对称性.

The energy and the two partial energies of two-dimensional anisotropic harmonic oscillator are conserved quantities, but only two of them are independent. The system possesses the third independent conserved quantity when the ω1/ω2 is a rational number. The extended Prelle-Singer method is used to find the third independent conserved quantity for the five typical two-dimensional anisotropic harmonic oscillators. The Noether symmetry and the Lie symmetry of the third independent conserved quantities are also discussed.