The time-scale version of Noether symmetry and conservation laws for three Birkhoffian mechanics, namely, nonshifted Birkhoffian systems, nonshifted generalized Birkhoffian systems, and nonshitfed constrained Birkhoffian systems, are studied. Firstly, on the basis of the nonshifted Pfaff-Birkhoff principle on time scales, Birkhoff’s equations for nonshifted variables are deduced; then, Noether’s quasi-symmetry for the nonshifted Birkhoffian system is proved and time-scale conserved quantity is presented. Secondly, the nonshifted generalized Pfaff-Birkhoff principle on time scales is proposed, the generalized Birkhoff’s equations for nonshifted variables are derived, and Noether’s symmetry for the nonshifted generalized Birkhoffian system is established. Finally, for the nonshifted constrained Birkhoffian system, Noether’s symmetry and time-scale conserved quantity are proposed and proved. The validity of the result is proved by examples.

中文翻译：

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广义伯克霍夫力学的时标形式，对称性和Noether型守恒量

研究了三种Birkhoffian力学的Noether对称和守恒律的时标版本，即非平移Birkhoffian系统，非平移广义Birkhoffian系统和非固定约束Birkhoffian系统。首先，基于时间尺度上的非移位Pfaff-Birkhoff原理，推导了非移位变量的Birkhoff方程。然后证明了非移位Birkhoffian系统的Noether准对称性，并给出了时标守恒量。其次，提出了时标上的非移位广义Pfaff-Birkhoff原理，推导了非移位变量的广义Birkhoff方程，建立了非移位广义Birkhoffian系统的Noether对称性。最后，对于非平移约束的伯克霍夫系统，提出并证明了Noether的对称性和时标守恒量。实例证明了该结果的有效性。