Perturbation to Symmetry and Adiabatic Invariants of General Discrete Holonomic Dynamical Systems

Symmetries play important roles in mathematics, physics and mechanics. Since Noether unveiled the profound relations between symmetries and conservation laws, many researches on them were done [1–15]. Recently, symmetry theories have been extended to discrete mechanics and equations [16–25]. As we know, even tiny changes in symmetry, named as perturbation to symmetry, are of great importance for physical systems. Based on the definition of adiabatic invariants, the relationship of perturbation to symmetry with adiabatic invariants are constructed. It offers an opportunity for the quasi-integrability in dynamical systems [26, 27]. Therefore, perturbation to symmetry and adiabatic invariants became a popular subject recently. The notion of approximate conservation laws was introduced with regards to approximate Noether symmetry by Baikov et al. [28]; Kara et al. [29, 30] extended Baikov’s ideas. Fu and Chen et al. [31, 32] studied the perturbation to the Lie symmetry and adiabatic invariants. Zhang et al. [33] deduced a new type of adiabatic invariants from perturbation to the Lie symmetry in 2006. Luo [34] gave another new type of adiabatic invariants called the Lutzky adiabatic invariants lately. These studies further inspire interests in research about adiabatic invariants [35–37]. However, researches about perturbation to symmetry and adiabatic invariants are all considered in continuous