© Wuhan University 2024

0 Introduction

Noether proposed the Noether symmetry method in her paper "Invariante Variationsprobleme"^[1] in 1918. The theorem for seeking conserved quantities through the Noether symmetry method is called Noether's theorem. It unveiled the correlation between the conserved quantities of mechanical systems and their inherent dynamical symmetries. Since then, the symmetry of mechanical systems has become one of the most effective ways to find the invariants of more complex mechanical systems, so the study of Noether's theory has long been a widely discussed subject^[2-5].

Birkhoff mechanics is the subsequent evolution of classical mechanics following the advent of quantum mechanics, marking the fifth phase of the development of classical mechanics^[6]. Therefore, the theory of Birkhoffian mechanics can be utilized in Hamilton mechanics, Lagrange mechanics, Newton mechanics^[7], statistical mechanics, biophysics, and other domains^[8]. However, the usual Birkhoff equation is challenging to construct. In contrast, the generalized Birkhoff's equation with only one additional term is easy to implement and has more degrees of freedom. Therefore, studying the dynamics of generalized Birkhoffian systems is vital. Mei et al generalized the Pfaff-Birkhoff principle and derived the generalized Birkhoff's equation^[9,10] from it. In 2013, they concluded a study on the dynamics of generalized Birkhoffian systems, providing a comprehensive and systematic discussion of the system^[11]. Subsequently, much progress in studying the symmetry and exact invariants of generalized Birkhoffian systems^[12-14] followed.

Time scale is a mathematical theory proposed by German scholar Hilger in 1988^[15]. It not only aids in uncovering the resemblances and disparities between continuous and discrete systems but also helps us understand the essential problems of complex dynamic systems more accurately and clearly, so it has become a popular tool in various scientific and engineering domains. In recent years, research on time-scale operation rules, variational problems, symmetries, and conserved quantities has attracted widespread attention^[16-24]. The perturbation of symmetries and their adiabatic invariants in dynamic systems under small perturbation actions are closely related to the system's symmetry and specific dynamic characteristics. Hence, its research is also essential. Research on the perturbation of symmetries and adiabatic invariants on time scales is just beginning, and the research results on Noether symmetry are even rarer^[25-28].

However, Anerot et al^[29] highlighted the derivation of the second Euler-Lagrange equation on time scales in Ref. [30] is incorrect and provided their method to obtain the Noether theorem for Lagrangian system on time scales. Therefore, the correctness of the conclusions obtained by applying the second Euler-Lagrange equation in the references still needs to be explored. This paper restudy exact invariants of generalized Birkhoffian systems on time scales using the method given in Ref. [29], and the perturbations and adiabatic invariants are discussed on this basis.

1 Preliminaries

The definitions and properties listed below will be utilized throughout the entirety of this paper. A more detailed content and proof process of time-scale calculus, please see Ref. [16-18].

Time scale is an arbitrary closed subset of a real number set that is not empty, usually denoted as $\mathbb{T}$. Real number set $\mathbb{R}$, natural number set $\mathbb{N}$, integer set $\mathbb{Z}$, and so on are all time scales.

Definition 1   On time scale $\mathbb{T}$, for $s\in \mathbb{T}$, we declare the forward jump operator $\sigma :\mathbb{T}\to \mathbb{T}$ with $\sigma \left(s\right)=\mathrm{i}\mathrm{n}\mathrm{f}\left\{r\in \mathbb{T}:r>s\right\}$ and the backward jump operator $\rho :\mathbb{T}\to \mathbb{T}$ with $\rho \left(s\right)=\mathrm{s}\mathrm{u}\mathrm{p}\left\{r\in \mathbb{T}:r<s\right\}$. In particular, we let $\mathrm{i}\mathrm{n}\mathrm{f}\mathrm{\varnothing }=\mathrm{s}\mathrm{u}\mathrm{p}\mathbb{T}$ and $\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{\varnothing }=\mathrm{i}\mathrm{n}\mathrm{f}\mathbb{T}$, with $\mathrm{\varnothing }$ representing the empty set. If $s>\mathrm{i}\mathrm{n}\mathrm{f}\mathbb{T}$ and $\rho \left(s\right)=s$, we say $s$ is left-dense. Also, if $s<\mathrm{s}\mathrm{u}\mathrm{p}\mathbb{T}$ and $\sigma \left(s\right)=s$, then we say $s$ is right-dense. Besides, when $\rho \left(s\right)=s=\sigma \left(s\right)$ holds, $s$ is called dense.

Definition 2   Define the graininess function $\mu :\mathbb{T}\to [\mathrm{0},\mathrm{\infty })$ as the difference between the jump operator $\sigma \left(s\right)$ and $s$, that is $\mu \left(s\right)=\sigma \left(s\right)-s$.

Definition 3   We define the set ${\mathbb{T}}^{\kappa }$ as

${\mathbb{T}}^{\kappa }=\{\begin{array}{lll}\mathbb{T}\backslash (\rho \left(\mathrm{s}\mathrm{u}\mathrm{p}\mathbb{T}\right),\mathrm{s}\mathrm{u}\mathrm{p}\mathbb{T}],& & \mathrm{s}\mathrm{u}\mathrm{p}\mathbb{T}<\mathbb{\infty },\\ \mathbb{T},& & \mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{w}\mathrm{i}\mathrm{s}\mathrm{e}.\end{array}$

Definition 4   We call a function $f:\mathbb{T}\to \mathbb{R}$ a rd-continuous function if $f$ is continuous at right-dense points and has a (finite) boundary at left-dense points in $\mathbb{T}$, with the set of $f:\mathbb{T}\to \mathbb{R}$ being represented as $C_{\mathrm{r}\mathrm{d}}\left(\mathbb{T},\mathbb{R}\right)$, $C_{\mathrm{r}\mathrm{d}}\left(\mathbb{T}\right)$ or $C_{\mathrm{r}\mathrm{d}}$. The set of functions $f:\mathbb{T}\to \mathbb{R}$, which are differentiable and have a derivative that is rd-continuous, can be represented as $C_{\mathrm{r}\mathrm{d}}^{\mathrm{1}}\left(\mathbb{T},\mathbb{R}\right)$, $C_{\mathrm{r}\mathrm{d}}^{\mathrm{1}}\left(\mathbb{T}\right)$, or $C_{\mathrm{r}\mathrm{d}}^{\mathrm{1}}$.

Definition 5   Assume a function $f:\mathbb{T}\to \mathbb{R}$ and $s\in {\mathbb{T}}^{\kappa }$ is given. If there exists $U=\left(s-\delta ,s+\delta \right)\bigcap \mathbb{T}$ for $\forall \epsilon >\mathrm{0}$ and some $\delta >\mathrm{0}$ such that

$\left|\left[f\left(\sigma \left(s\right)\right)-f\left(t\right)\right]-f^{\mathrm{\Delta }}\left(s\right)\left[\sigma \left(s\right)-t\right]\right|\le \epsilon \left|\sigma \left(s\right)-t\right|$

holds true for all $t\in U$, then we call $f^{\mathrm{\Delta }}\left(s\right)$ the delta derivative of $f$ at $s$. Additionally, if $f^{\mathrm{\Delta }}\left(s\right)$ is present for all $s\in {\mathbb{T}}^{\kappa }$, we assert that $f$ has differentiability on ${\mathbb{T}}^{\kappa }$.

Definition 6   A function $F:\mathbb{T}\to \mathbb{R}$ is referred to as an antiderivative of $f:\mathbb{T}\to \mathbb{R}$ when $F^{\mathrm{\Delta }}\left(s\right)=f\left(s\right)$ is satisfied for every $s\in {\mathbb{T}}^{\kappa }$. And antiderivatives are present in every rd-continuous function $f:\mathbb{T}\to \mathbb{R}$. Furthermore, if $s_{\mathrm{0}}\in \mathbb{T}$, then $F$ defined by $F\left(s\right)={\int }_{s_{\mathrm{0}}}^{s}f\left(\tau \right)\mathrm{\Delta }\tau$ for $s\in \mathbb{T}$, acts as an antiderivative of $f$.

Besides, assume $H$, $F:\mathbb{T}\to \mathbb{R}$ are differentiable at $t\in {\mathbb{T}}^{\kappa }$, the following formulas valid.

$H^{\sigma }=H+\mu H^{\mathrm{\Delta }},{\left(H+F\right)}^{\mathrm{\Delta }}=H^{\mathrm{\Delta }}+F^{\mathrm{\Delta }},{\left(HF\right)}^{\mathrm{\Delta }}=H^{\mathrm{\Delta }}{F}^{\sigma }+H{F}^{\mathrm{\Delta }}=H^{\sigma }{F}^{\mathrm{\Delta }}+H^{\mathrm{\Delta }}F,{\int }_a^{b}H\left(\alpha \left(t\right)\right){\alpha }^{\mathrm{\Delta }}\left(t\right)\mathrm{\Delta }t={\int }_{\alpha \left(a\right)}^{\alpha \left(b\right)}H\left(\overline{t}\right)\overline{\mathrm{\Delta }}\overline{t},$

where $H^{\sigma }=H\circ \sigma$, the map $\alpha :\left[a,b\right]\bigcap \mathbb{T}\to \mathbb{R}$ is an increasing $C_{\mathrm{r}\mathrm{d}}^{\mathrm{1}}$ function, and its image is a new time scale with $\overline{\mathrm{\Delta }}$ symbolizes delta operator.

Lemma 1   (Dubois-Reymond) Let $h\in C_{\mathrm{r}\mathrm{d}}$, and $h:\left[a_{\mathrm{1}},a_{\mathrm{2}}\right]\to {\mathbb{R}}^n$. Following that, ${\int }_{a_{\mathrm{1}}}^{a_{\mathrm{2}}}h\left(t\right){\zeta }^{\mathrm{\Delta }}\left(t\right)\mathrm{\Delta }t=\mathrm{0}$ for all $\zeta \in C_{\mathrm{r}\mathrm{d}}^{\mathrm{1}}$ with $\zeta \left(a_{\mathrm{1}}\right)=\zeta \left(a_{\mathrm{2}}\right)=\mathrm{0}$ holds if and only if $h\left(t\right)\equiv d$ on ${\left[a_{\mathrm{1}},a_{\mathrm{2}}\right]}^{\kappa }$ for certain $d\in {\mathbb{R}}^n$.

2 Exact Invariants

In this section, the equations and exact invariants of the generalized Birkhoffian system and the constrained Birkhoffian system on time scales will be presented.

2.1 Exact Invariant for Generalized Birkhoffian System

Generalized Pfaff-Birkhoff principle is

${\int }_{t_{\mathrm{1}}}^{t_{\mathrm{2}}}\left\{\delta \left[R_l\left(t,a_{\nu }^{\sigma }\right)a_l^{\mathrm{\Delta }}-B\left(t,a_{\nu }^{\sigma }\right)\right]+{\mathrm{\Lambda }}_l\left(t,a_{\nu }^{\sigma }\right)\delta a_l^{\sigma }\right\}\mathrm{\Delta }t=\mathrm{0},$(1)

$\delta a_l|{}_{t=t_{\mathrm{1}}}=\delta a_l|{}_{t=t_{\mathrm{2}}}=\mathrm{0},$(2)

$\delta a_l^{\mathrm{\Delta }}={\left(\delta a_l\right)}^{\mathrm{\Delta }},\delta a_l^{\sigma }={\left(\delta a_l\right)}^{\sigma },$(3)

where Eq. (2) is the boundary condition and Eq. (3) is the commutative relationship, and the Birkhoffian $B\left(t,a_{\nu }^{\sigma }\right):\mathbb{T}\times {\mathbb{R}}^{\mathrm{2}n}\to \mathbb{R}$, the Birkhoff's functions $R_l\left(t,a_{\nu }^{\sigma }\right):\mathbb{T}\times {\mathbb{R}}^{\mathrm{2}n}\to \mathbb{R}$ and the additional items ${\mathrm{\Lambda }}_l\left(t,a_{\nu }^{\sigma }\right):\mathbb{T}\times {\mathbb{R}}^{\mathrm{2}n}\to \mathbb{R}$ all belong to $C_{\mathrm{r}\mathrm{d}}^{\mathrm{1}}$, $a_{\nu }^{\sigma }\left(t\right)=\left(a_{\nu }\circ \sigma \right)\left(t\right)$, $a_l^{\mathrm{\Delta }}\left(t\right)=\mathrm{\Delta }{a}_l\left(t\right)/\mathrm{\Delta }t$, $t\in {\left[t_{\mathrm{1}},t_{\mathrm{2}}\right]}^{\kappa }$, $l,\nu =$$\mathrm{1,2},\cdots ,\mathrm{2}n$.

From Eqs. (1)-(3), we have

$\begin{array}{l}\mathrm{0}={\int }_{t_{\mathrm{1}}}^{t_{\mathrm{2}}}\left[\delta R_l\left(t,a_{\nu }^{\sigma }\right)\cdot a_l^{\mathrm{\Delta }}+R_l\left(t,a_{\nu }^{\sigma }\right)\delta a_l^{\mathrm{\Delta }}-\delta B\left(t,a_{\nu }^{\sigma }\right)+{\mathrm{\Lambda }}_l\left(t,a_{\nu }^{\sigma }\right)\delta a_l^{\sigma }\right]\mathrm{\Delta }t={\int }_{t_{\mathrm{1}}}^{t_{\mathrm{2}}}\left[\frac{\partial R_l}{\partial a_{\nu }^{\sigma }}\cdot \delta a_{\nu }^{\sigma }\cdot a_l^{\mathrm{\Delta }}+R_l\delta a_l^{\mathrm{\Delta }}-\frac{\partial B}{\partial a_{\nu }^{\sigma }}\cdot \delta a_{\nu }^{\sigma }+{\mathrm{\Lambda }}_l\delta a_l^{\sigma }\right]\mathrm{\Delta }t\\ ={\int }_{t_{\mathrm{1}}}^{t_{\mathrm{2}}}\{{\left[{\int }_{t_{\mathrm{1}}}^t\left(\frac{\partial R_l}{\partial a_{\nu }^{\sigma }}\cdot a_l^{\mathrm{\Delta }}\right)\mathrm{\Delta }\tau \cdot \delta a_{\nu }\right]}^{\mathrm{\Delta }}+R_{\nu }\cdot {\left(\delta a_{\nu }\right)}^{\mathrm{\Delta }}-{\int }_{t_{\mathrm{1}}}^t\left(\frac{\partial R_l}{\partial a_{\nu }^{\sigma }}\cdot a_l^{\mathrm{\Delta }}\right)\mathrm{\Delta }\tau \cdot {\left(\delta a_{\nu }\right)}^{\mathrm{\Delta }}-{\left({\int }_{t_{\mathrm{1}}}^t\frac{\partial B}{\partial a_{\nu }^{\sigma }}\mathrm{\Delta }\tau \cdot \delta a_{\nu }\right)}^{\mathrm{\Delta }}\\ +{\int }_{t_{\mathrm{1}}}^t\frac{\partial B}{\partial a_{\nu }^{\sigma }}\mathrm{\Delta }\tau \cdot {\left(\delta a_{\nu }\right)}^{\mathrm{\Delta }}+{\left({\int }_{t_{\mathrm{1}}}^t{\mathrm{\Lambda }}_{\nu }\mathrm{\Delta }\tau \cdot \delta a_{\nu }\right)}^{\mathrm{\Delta }}-{\int }_{t_{\mathrm{1}}}^t{\mathrm{\Lambda }}_{\nu }\mathrm{\Delta }\tau \cdot {\left(\delta a_{\nu }\right)}^{\mathrm{\Delta }}\}\mathrm{\Delta }t\\ ={\int }_{t_{\mathrm{1}}}^{t_{\mathrm{2}}}\left\{\left[-{\int }_{t_{\mathrm{1}}}^t\left(\frac{\partial R_l}{\partial a_{\nu }^{\sigma }}\cdot a_l^{\mathrm{\Delta }}\right)\mathrm{\Delta }\tau +R_{\nu }+{\int }_{t_{\mathrm{1}}}^t\frac{\partial B}{\partial a_{\nu }^{\sigma }}\mathrm{\Delta }\tau -{\int }_{t_{\mathrm{1}}}^t{\mathrm{\Lambda }}_{\nu }\mathrm{\Delta }\tau \right]{\left(\delta a_{\nu }\right)}^{\mathrm{\Delta }}\right\}\mathrm{\Delta }t\end{array}$

Then from Lemma 1, we get

$-{\int }_{t_{\mathrm{1}}}^t\left(\frac{\partial R_l}{\partial a_{\nu }^{\sigma }}\cdot a_l^{\mathrm{\Delta }}\right)\mathrm{\Delta }\tau +R_{\nu }+{\int }_{t_{\mathrm{1}}}^t\frac{\partial B}{\partial a_{\nu }^{\sigma }}\mathrm{\Delta }\tau -{\int }_{t_{\mathrm{1}}}^t{\mathrm{\Lambda }}_{\nu }\mathrm{\Delta }\tau =\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}$(4)

and then taking the derivative of Eq. (4), the equations of the generalized Birkhoffian system on time scales can be obtained as^[24]

$\frac{\partial R_{\nu }\left(t,a_{\varpi }^{\sigma }\right)}{\partial a_l^{\sigma }}\cdot a_{\nu }^{\mathrm{\Delta }}-\frac{\partial B\left(t,a_{\varpi }^{\sigma }\right)}{\partial a_l^{\sigma }}-R_l^{\mathrm{\Delta }}\left(t,a_{\varpi }^{\sigma }\right)+{\mathrm{\Lambda }}_l\left(t,a_{\varpi }^{\sigma }\right)=\mathrm{0},$(5)

where $t\in {\left[t_{\mathrm{1}},t_{\mathrm{2}}\right]}^{\kappa }$, $\nu ,l,\varpi =\mathrm{1,2},\cdots ,\mathrm{2}n$.

We introduce the following infinitesimal transformations regarding $t$ and $a_{\nu }$

$P_B\left(t,a_{\varpi },\upsilon \right)=\overline{t}=t+\upsilon {\xi }_{B\mathrm{0}}^{\mathrm{0}}\left(t,a_{\varpi }\right)+o\left(\upsilon \right),$(6)

$Q_{B\nu }\left(t,a_{\varpi },\upsilon \right)={\overline{a}}_{\nu }\left(\overline{t}\right)=a_{\nu }\left(t\right)+\upsilon {\xi }_{B\nu }^{\mathrm{0}}\left(t,a_{\varpi }\right)+o\left(\upsilon \right),$(7)

where $\upsilon$ is an infinitesimal parameter, ${\xi }_{B\mathrm{0}}^{\mathrm{0}}$ and ${\xi }_{B\nu }^{\mathrm{0}}$ are the generators of infinitesimal transformations.

It is noteworthy that the generating function is usually a function of time and coordinates, and the transformation of coordinates and time that keep the motion equation unchanged forms Lie group. Sarlet and Cantrijn^[31] discussed in detail the issue of function dependency in generating functions. Due to our research on the invariance of dynamic systems on time scales, we consider allowing for velocity-dependent transformations to expand the dimension of the generators.

Suppose that the map $t\in \left[t_{\mathrm{1}},t_{\mathrm{2}}\right]\to \alpha \left(t\right):=P_B(t,$$a_{\varpi },\upsilon )\in \mathbb{R}$ is an increasing $C_{\mathrm{r}\mathrm{d}}^{\mathrm{1}}$ function, and its image is a new time scale with $\overline{\mathrm{\Delta }}$ and $\overline{\sigma }$ representing the delta operator and forward jump operator, respectively. And $\overline{\sigma }\circ \alpha =\alpha \circ \sigma$ holds. Eqs. (6) and (7) are said to be Noether symmetric transformations for the generalized Birkhoffian system if and only if

${\int }_{t_{\mathrm{1}}}^{t_{\mathrm{2}}}\left[R_l\left(t,a_{\nu }^{\sigma }\right)a_l^{\mathrm{\Delta }}-B\left(t,a_{\nu }^{\sigma }\right)\right]\mathrm{\Delta }t-{\int }_{t_{\mathrm{1}}}^{t_{\mathrm{2}}}{\mathrm{\Lambda }}_l\left(t,a_{\nu }^{\sigma }\right)\delta a_l^{\sigma }\mathrm{\Delta }t={\int }_{{\overline{t}}_{\mathrm{1}}}^{{\overline{t}}_{\mathrm{2}}}\left[R_l\left(\overline{t},{\overline{a}}_{\nu }^{\overline{\sigma }}\left(\overline{t}\right)\right){\overline{a}}_l^{\overline{\mathrm{\Delta }}}\left(\overline{t}\right)-B\left(\overline{t},{\overline{a}}_{\nu }^{\overline{\sigma }}\left(\overline{t}\right)\right)\right]\overline{\mathrm{\Delta }}\overline{t}.$(8)

holds. From Eq. (8), there is

$\begin{array}{l}{\int }_{{\overline{t}}_{\mathrm{1}}}^{{\overline{t}}_{\mathrm{2}}}\left[R_l\left(\overline{t},{\overline{a}}_{\nu }^{\overline{\sigma }}\left(\overline{t}\right)\right){\overline{a}}_l^{\overline{\mathrm{\Delta }}}\left(\overline{t}\right)-B\left(\overline{t},{\overline{a}}_{\nu }^{\overline{\sigma }}\left(\overline{t}\right)\right)\right]\overline{\mathrm{\Delta }}\overline{t}={\int }_{t_{\mathrm{1}}}^{t_{\mathrm{2}}}\left[R_l\left(\alpha \left(t\right),\left({\overline{a}}_{\nu }\circ \overline{\sigma }\circ \alpha \right)\left(t\right)\right){\overline{a}}_l^{\overline{\mathrm{\Delta }}}\left(\alpha \left(t\right)\right)-B\left(\alpha \left(t\right),\left({\overline{a}}_{\nu }\circ \overline{\sigma }\circ \alpha \right)\left(t\right)\right)\right]{\alpha }^{\mathrm{\Delta }}\left(t\right)\mathrm{\Delta }t\\ ={\int }_{t_{\mathrm{1}}}^{t_{\mathrm{2}}}\left[R_l\left(\alpha \left(t\right),\left({\overline{a}}_{\nu }\circ \alpha \circ \sigma \right)\left(t\right)\right)\frac{{\left({\overline{a}}_l\circ \alpha \right)}^{\mathrm{\Delta }}\left(t\right)}{{\alpha }^{\mathrm{\Delta }}\left(t\right)}-B\left(\alpha \left(t\right),\left({\overline{a}}_{\nu }\circ \alpha \circ \sigma \right)\left(t\right)\right)\right]{\alpha }^{\mathrm{\Delta }}\left(t\right)\mathrm{\Delta }t\\ ={\int }_{t_{\mathrm{1}}}^{t_{\mathrm{2}}}\left[R_l\left(P_B,Q_{B\nu }^{\sigma }\right)\frac{Q_{Bl}^{\mathrm{\Delta }}}{P_B^{\mathrm{\Delta }}}-B\left(P_B,Q_{B\nu }^{\sigma }\right)\right]P_B^{\mathrm{\Delta }}\mathrm{\Delta }t\end{array}$

which is to say,

$R_l\left(t,a_{\nu }^{\sigma }\right)a_l^{\mathrm{\Delta }}-B\left(t,a_{\nu }^{\sigma }\right)-{\mathrm{\Lambda }}_l\left(t,a_{\nu }^{\sigma }\right)\delta a_l^{\sigma }=R_l\left(P_B,Q_{B\nu }^{\sigma }\right)Q_{Bl}^{\mathrm{\Delta }}-B\left(P_B,Q_{B\nu }^{\sigma }\right)P_B^{\mathrm{\Delta }}$(9)

By differentiating both sides of Eq. (9) with respect to the infinitesimal parameter $\upsilon$, let $\upsilon =\mathrm{0}$, and applying $\delta a_l=\mathrm{\Delta }{a}_l-a_l^{\mathrm{\Delta }}\mathrm{\Delta }t$^[20], we obtain the Noether identity

$\left(\frac{\partial R_{\nu }}{\partial t}\cdot a_{\nu }^{\mathrm{\Delta }}-\frac{\partial B}{\partial t}\right){\xi }_{B\mathrm{0}}^{\mathrm{0}}+\left(\frac{\partial R_{\nu }}{\partial a_l^{\sigma }}\cdot a_{\nu }^{\mathrm{\Delta }}-\frac{\partial B}{\partial a_l^{\sigma }}\right){\xi }_{Bl}^{\mathrm{0}\sigma }+R_{\nu }{\xi }_{B\nu }^{\mathrm{0}\mathrm{\Delta }}-B{\xi }_{B\mathrm{0}}^{\mathrm{0}\mathrm{\Delta }}+{\mathrm{\Lambda }}_l\left({\xi }_{Bl}^{\mathrm{0}\sigma }-a_l^{\mathrm{\Delta }\sigma }{\xi }_{B\mathrm{0}}^{\mathrm{0}\sigma }\right)=\mathrm{0}$(10)

Theorem 1   If ${\xi }_{B\mathrm{0}}^{\mathrm{0}}\left(t,a_{\varpi }\right)$ and ${\xi }_{B\nu }^{\mathrm{0}}\left(t,a_{\varpi }\right)$ satisfy Eq. (10), then the generalized Birkhoffian system (5) has the exact invariant of the form

$\begin{array}{l}{I}_{GB\mathrm{0}}\left(t,a_{\nu },a_{\nu }^{\sigma },a_{\nu }^{\mathrm{\Delta }}\right)=R_l{\xi }_{Bl}^{\mathrm{0}}-\mu \left(t\right)\left(\frac{\partial R_l}{\partial t}\cdot a_l^{\mathrm{\Delta }}-\frac{\partial B}{\partial t}\right){\xi }_{B\mathrm{0}}^{\mathrm{0}}-B{\xi }_{B\mathrm{0}}^{\mathrm{0}}\\ +{\int }_{t_{\mathrm{1}}}^t\left\{\frac{\mathrm{\Delta }}{\mathrm{\Delta }\tau }[B+\mu \left(\tau \right)\left(\frac{\partial R_l}{\partial \tau }\cdot a_l^{\mathrm{\Delta }}-\frac{\partial B}{\partial \tau }\right)]+\frac{\partial R_l}{\partial \tau }\cdot a_l^{\mathrm{\Delta }}-\frac{\partial B}{\partial \tau }-{\mathrm{\Lambda }}_l{a}_l^{\mathrm{\Delta }\sigma }\right\}{\xi }_{B\mathrm{0}}^{\mathrm{0}\sigma }\mathrm{\Delta }\tau \end{array}$(11)

Proof   Making use of Eqs. (5) and (10), we get

$\begin{array}{l}\frac{\mathrm{\Delta }}{\mathrm{\Delta }t}{I}_{GB\mathrm{0}}=R_l^{\mathrm{\Delta }}{\xi }_{Bl}^{\mathrm{0}\sigma }+R_l{\xi }_{Bl}^{\mathrm{0}\mathrm{\Delta }}-\mu \left(t\right)\left(\frac{\partial R_l}{\partial t}\cdot a_l^{\mathrm{\Delta }}-\frac{\partial B}{\partial t}\right){\xi }_{B\mathrm{0}}^{\mathrm{0}\mathrm{\Delta }}-B{\xi }_{B\mathrm{0}}^{\mathrm{0}\mathrm{\Delta }}+\frac{\partial R_l}{\partial t}\cdot a_l^{\mathrm{\Delta }}{\xi }_{B\mathrm{0}}^{\mathrm{0}\sigma }-\frac{\partial B}{\partial t}{\xi }_{B\mathrm{0}}^{\mathrm{0}\sigma }-{\mathrm{\Lambda }}_l{a}_l^{\mathrm{\Delta }\sigma }{\xi }_{B\mathrm{0}}^{\mathrm{0}\sigma }\\ =\left(\frac{\partial R_l}{\partial t}\cdot a_l^{\mathrm{\Delta }}-\frac{\partial B}{\partial t}\right){\xi }_{B\mathrm{0}}^{\mathrm{0}}+R_l^{\mathrm{\Delta }}{\xi }_{Bl}^{\mathrm{0}\sigma }+R_l{\xi }_{Bl}^{\mathrm{0}\mathrm{\Delta }}-B{\xi }_{B\mathrm{0}}^{\mathrm{0}\mathrm{\Delta }}-{\mathrm{\Lambda }}_l{a}_l^{\mathrm{\Delta }\sigma }{\xi }_{B\mathrm{0}}^{\mathrm{0}\sigma }=-\left(\frac{\partial R_{\nu }}{\partial a_l^{\sigma }}\cdot a_{\nu }^{\mathrm{\Delta }}-\frac{\partial B}{\partial a_l^{\sigma }}\right){\xi }_{Bl}^{\mathrm{0}\sigma }+R_l^{\mathrm{\Delta }}{\xi }_{Bl}^{\mathrm{0}\sigma }-{\mathrm{\Lambda }}_l{\xi }_{Bl}^{\mathrm{0}\sigma }\\ =-\left(\frac{\partial R_{\nu }}{\partial a_l^{\sigma }}\cdot a_{\nu }^{\mathrm{\Delta }}-\frac{\partial B}{\partial a_l^{\sigma }}-R_l^{\mathrm{\Delta }}+{\mathrm{\Lambda }}_l\right){\xi }_{Bl}^{\mathrm{0}\sigma }=\mathrm{0}\end{array}$

Remark 1   For generalized Birkhoffian system Eq. (5), if there is ${\mathrm{\Lambda }}_l=\mathrm{0}$, then Eq. (5) degenerates to Birkhoffian system, and the conclusion of Theorem 1 is still valid.

2.2 Exact Invariant for Constrained Birkhoffian System

If the variables $a_{\varpi }^{\sigma }$ in the Birkhoffian system have a correlation, and are constrained by the constraint equations

$f_{\beta }\left(t,a_{\varpi }^{\sigma }\right)=\mathrm{0},\beta =\mathrm{1,2},\cdots ,\mathrm{2}k,k\in \mathbb{N},$(12)

then do the isochronous variation for Eq. (12), we have

$\frac{\partial f_{\beta }}{\partial a_l^{\sigma }}\cdot \delta a_l^{\sigma }=\mathrm{0},\beta =\mathrm{1,2},\cdots ,\mathrm{2}k,k\in \mathbb{N}.$(13)

The constrained Birkhoff's equations with multiplier form on time scales can be obtained as^[22]

$\frac{\partial R_l\left(t,a_{\varpi }^{\sigma }\right)}{\partial a_{\nu }^{\sigma }}\cdot a_l^{\mathrm{\Delta }}-\frac{\partial B\left(t,a_{\varpi }^{\sigma }\right)}{\partial a_{\nu }^{\sigma }}-R_{\nu }^{\mathrm{\Delta }}\left(t,a_{\varpi }^{\sigma }\right)={\lambda }_{\beta }\frac{\partial f_{\beta }\left(t,a_{\varpi }^{\sigma }\right)}{\partial a_{\nu }^{\sigma }},$(14)

where $t\in {\left[t_{\mathrm{1}},t_{\mathrm{2}}\right]}^{\kappa }$, $\nu ,l,\varpi =\mathrm{1,2},\cdots ,\mathrm{2}n$, ${\lambda }_{\beta }\left(t,a_{\varpi }^{\sigma }\right)$ is called the constrained multiplier, $\beta =\mathrm{1,2},\cdots ,\mathrm{2}k$. Then the constrained multipliers can be calculated by Eqs. (12) and (14). Substitute the constrained multipliers that has been calculated into Eq. (14), then Eq. (14) can be written as

$\frac{\partial R_l\left(t,a_{\varpi }^{\sigma }\right)}{\partial a_{\nu }^{\sigma }}\cdot a_l^{\mathrm{\Delta }}-\frac{\partial B\left(t,a_{\varpi }^{\sigma }\right)}{\partial a_{\nu }^{\sigma }}-R_{\nu }^{\mathrm{\Delta }}\left(t,a_{\varpi }^{\sigma }\right)=Q_{\nu }\left(t,a_{\varpi }^{\sigma }\right)$(15)

where $Q_{\nu }={\lambda }_{\beta }\frac{\partial f_{\beta }\left(t,a_{\varpi }^{\sigma }\right)}{\partial a_{\nu }^{\sigma }}$, $t\in {\left[t_{\mathrm{1}},t_{\mathrm{2}}\right]}^{\kappa }$. Eqs. (12) and (14) together are called constrained Birkhoff equations, and Eq. (15) is called the equation of the free Birkhoffian system corresponding to the constrained Birkhoffian system.

Eqs. (6) and (7) are said to be Noether symmetric transformations for the corresponding free Birkhoffian system Eq. (15) if and only if there holds

${\int }_{t_{\mathrm{1}}}^{t_{\mathrm{2}}}\left[R_l\left(t,a_{\nu }^{\sigma }\right)a_l^{\mathrm{\Delta }}-B\left(t,a_{\nu }^{\sigma }\right)\right]\mathrm{\Delta }t+{\int }_{t_{\mathrm{1}}}^{t_{\mathrm{2}}}{Q}_l\left(t,a_{\nu }^{\sigma }\right)\delta a_l^{\sigma }\mathrm{\Delta }t={\int }_{{\overline{t}}_{\mathrm{1}}}^{{\overline{t}}_{\mathrm{2}}}\left[R_l\left(\overline{t},{\overline{a}}_{\nu }^{\overline{\sigma }}\left(\overline{t}\right)\right){\overline{a}}_l^{\overline{\mathrm{\Delta }}}\left(\overline{t}\right)-B\left(\overline{t},{\overline{a}}_{\nu }^{\overline{\sigma }}\left(\overline{t}\right)\right)\right]\overline{\mathrm{\Delta }}\overline{t}$(16)

Therefore, for $t\in {\left[t_{\mathrm{1}},t_{\mathrm{2}}\right]}^{\kappa }$, Eq. (16) gives

$\left(\frac{\partial R_{\nu }}{\partial t}\cdot a_{\nu }^{\mathrm{\Delta }}-\frac{\partial B}{\partial t}\right){\xi }_{B\mathrm{0}}^{\mathrm{0}}+\left(\frac{\partial R_{\nu }}{\partial a_l^{\sigma }}\cdot a_{\nu }^{\mathrm{\Delta }}-\frac{\partial B}{\partial a_l^{\sigma }}\right){\xi }_{Bl}^{\mathrm{0}\sigma }+R_{\nu }{\xi }_{B\nu }^{\mathrm{0}\mathrm{\Delta }}-B{\xi }_{B\mathrm{0}}^{\mathrm{0}\mathrm{\Delta }}-Q_l\left({\xi }_{Bl}^{\mathrm{0}\sigma }-a_l^{\mathrm{\Delta }\sigma }{\xi }_{B\mathrm{0}}^{\mathrm{0}\sigma }\right)=\mathrm{0}$(17)

Theorem 2   For the corresponding free Birkhoffian system (15), if ${\xi }_{B\mathrm{0}}^{\mathrm{0}}\left(t,a_{\varpi }\right)$ and ${\xi }_{B\nu }^{\mathrm{0}}\left(t,a_{\varpi }\right)$ satisfy Eq. (17), then there exists the exact invariant

$\begin{array}{l}{I}_{CB\mathrm{0}}\left(t,a_{\nu },a_{\nu }^{\sigma },a_{\nu }^{\mathrm{\Delta }}\right)=R_l{\xi }_{Bl}^{\mathrm{0}}-\mu \left(t\right)\left(\frac{\partial R_l}{\partial t}\cdot a_l^{\mathrm{\Delta }}-\frac{\partial B}{\partial t}\right){\xi }_{B\mathrm{0}}^{\mathrm{0}}-B{\xi }_{B\mathrm{0}}^{\mathrm{0}}\\ +{\int }_{t_{\mathrm{1}}}^t\left\{\frac{\mathrm{\Delta }}{\mathrm{\Delta }\tau }[B+\mu \left(\tau \right)\left(\frac{\partial R_l}{\partial \tau }\cdot a_l^{\mathrm{\Delta }}-\frac{\partial B}{\partial \tau }\right)]+\frac{\partial R_l}{\partial \tau }\cdot a_l^{\mathrm{\Delta }}-\frac{\partial B}{\partial \tau }+Q_l{a}_l^{\mathrm{\Delta }\sigma }\right\}{\xi }_{B\mathrm{0}}^{\mathrm{0}\sigma }\mathrm{\Delta }\tau \end{array}$(18)

Proof   This proof process is similar to Theorem 1.

Remark 2   For the constrained Birkhoff Eqs. (12) and (14), if ${\xi }_{B\mathrm{0}}^{\mathrm{0}}\left(t,a_{\varpi }\right)$ and ${\xi }_{B\nu }^{\mathrm{0}}\left(t,a_{\varpi }\right)$ satisfy the Noether identity (Eq. (17)) and constrained equation (Eq. (13)), then the system exists the same exact invariant (Eq. (18)).

3 Adiabatic Invariants

This section studies the perturbation to symmetries and the corresponding adiabatic invariants of the disturbed generalized and constrained Birkhoffian system. First, the notion of adiabatic invariant is introduced.

Definition 7   If $I_z\left(t,a_{\theta },a_{\theta }^{\sigma },a_{\theta }^{\mathrm{\Delta }},\epsilon \right)$ is a physical quantity on time scales of the mechanical system containing a small parameter $\epsilon$ with a maximum power of $z$, and $\mathrm{\Delta }{I}_z/\mathrm{\Delta }t$ is proportional to ${\epsilon }^{z+\mathrm{1}}$, then $I_z$ is known as the z-th-order adiabatic invariant of the mechanical system on time scales.

In particular, the adiabatic invariant degenerates to the exact invariant when $z=\mathrm{0}$.

3.1 Adiabatic Invariant for Generalized Birkhoffian System

Assuming that generalized Birkhoffian system (5) is disturbed by the minor disturbance $\epsilon W_{Bl}\left(t,a_{\varpi }^{\sigma }\right)$, thus

$\frac{\partial R_{\nu }\left(t,a_{\varpi }^{\sigma }\right)}{\partial a_l^{\sigma }}\cdot a_{\nu }^{\mathrm{\Delta }}-\frac{\partial B\left(t,a_{\varpi }^{\sigma }\right)}{\partial a_l^{\sigma }}-R_l^{\mathrm{\Delta }}+{\mathrm{\Lambda }}_l\left(t,a_{\varpi }^{\sigma }\right)=\epsilon W_{Bl}\left(t,a_{\varpi }^{\sigma }\right).$(19)

As a result of the effect of small perturbation $\epsilon W_{Bl}$, the primitive symmetry and exact invariant of the system will be altered correspondingly. Assuming that this change occurs on the basis of the system without disturbance, then

${\xi }_{B\mathrm{0}}={\xi }_{B\mathrm{0}}^{\mathrm{0}}+\epsilon {\xi }_{B\mathrm{0}}^{\mathrm{1}}+{\epsilon }^{\mathrm{2}}{\xi }_{B\mathrm{0}}^{\mathrm{2}}+\cdots ,{\xi }_{B\nu }={\xi }_{B\nu }^{\mathrm{0}}+\epsilon {\xi }_{B\nu }^{\mathrm{1}}+{\epsilon }^{\mathrm{2}}{\xi }_{B\nu }^{\mathrm{2}}+\cdots .$(20)

Therefore, we hold the theorem that follows.

Theorem 3   For disturbed generalized Birkhoffian system (19), if ${\xi }_{B\mathrm{0}}^m\left(t,a_{\varpi }\right)$ and ${\xi }_{B\nu }^m\left(t,a_{\varpi }\right)$ satisfy

$\left(\frac{\partial R_{\nu }}{\partial t}\cdot a_{\nu }^{\mathrm{\Delta }}-\frac{\partial B}{\partial t}\right){\xi }_{B\mathrm{0}}^m+\left(\frac{\partial R_{\nu }}{\partial a_l^{\sigma }}\cdot a_{\nu }^{\mathrm{\Delta }}-\frac{\partial B}{\partial a_l^{\sigma }}\right){\xi }_{Bl}^{m\sigma }-B{\xi }_{B\mathrm{0}}^{m\mathrm{\Delta }}+R_{\nu }{\xi }_{B\nu }^{m\mathrm{\Delta }}+{\mathrm{\Lambda }}_l\left({\xi }_{Bl}^{m\sigma }-a_l^{\mathrm{\Delta }\sigma }{\xi }_{B\mathrm{0}}^{m\sigma }\right)-W_{Bl}\left({\xi }_{Bl}^{\left(m-\mathrm{1}\right)\sigma }-a_l^{\mathrm{\Delta }}{\xi }_{B\mathrm{0}}^{\left(m-\mathrm{1}\right)\sigma }\right)=\mathrm{0},$(21)

then this system has a z-th-order adiabatic invariant

$\begin{array}{l}{I}_{GBz}={\displaystyle \sum _{m=\mathrm{0}}^z}{\epsilon }^m\{R_l{\xi }_{Bl}^m-\mu \left(t\right)\left(\frac{\partial R_l}{\partial t}\cdot a_l^{\mathrm{\Delta }}-\frac{\partial B}{\partial t}\right){\xi }_{B\mathrm{0}}^m-B{\xi }_{B\mathrm{0}}^m+{\int }_{t_{\mathrm{1}}}^t\{\frac{\mathrm{\Delta }}{\mathrm{\Delta }\tau }\left[B+\mu \left(\tau \right)\left(\frac{\partial R_l}{\partial \tau }\cdot a_l^{\mathrm{\Delta }}-\frac{\partial B}{\partial \tau }\right)-R_l{a}_l^{\mathrm{\Delta }}\right]\\ +R_l^{\sigma }{a}_l^{\mathrm{\Delta }\mathrm{\Delta }}+\left(\frac{\partial R_{\nu }}{\partial a_l^{\sigma }}\cdot a_{\nu }^{\mathrm{\Delta }}-\frac{\partial B}{\partial a_l^{\sigma }}+{\mathrm{\Lambda }}_l\right)a_l^{\mathrm{\Delta }}+\frac{\partial R_l}{\partial \tau }\cdot a_l^{\mathrm{\Delta }}-\frac{\partial B}{\partial \tau }-{\mathrm{\Lambda }}_l{a}_l^{\mathrm{\Delta }\sigma }\}{\xi }_{B\mathrm{0}}^{m\sigma }\mathrm{\Delta }\tau \}\end{array}$(22)

where we let ${\xi }_{Bl}^{m-\mathrm{1}}={\xi }_{B\mathrm{0}}^{m-\mathrm{1}}=\mathrm{0}$ when $m=\mathrm{0}$.

Proof   Using Eqs. (19) and (21) can achieve

$\begin{array}{l}\frac{\mathrm{\Delta }}{\mathrm{\Delta }t}{I}_{GBz}={\displaystyle \sum _{m=\mathrm{0}}^z}{\epsilon }^m\{R_l{\xi }_{Bl}^{m\mathrm{\Delta }}+R_l^{\mathrm{\Delta }}{\xi }_{Bl}^{m\sigma }-\mu \left(t\right)\left(\frac{\partial R_l}{\partial t}\cdot a_l^{\mathrm{\Delta }}-\frac{\partial B}{\partial t}\right){\xi }_{B\mathrm{0}}^{m\mathrm{\Delta }}-B{\xi }_{B\mathrm{0}}^{m\mathrm{\Delta }}-R_l^{\mathrm{\Delta }}{a}_l^{\mathrm{\Delta }}{\xi }_{B\mathrm{0}}^{m\sigma }-R_l^{\sigma }{a}_l^{\mathrm{\Delta }\mathrm{\Delta }}{\xi }_{B\mathrm{0}}^{m\sigma }\\ +R_l^{\sigma }{a}_l^{\mathrm{\Delta }\mathrm{\Delta }}{\xi }_{B\mathrm{0}}^{m\sigma }+\left(\frac{\partial R_{\nu }}{\partial a_l^{\sigma }}\cdot a_{\nu }^{\mathrm{\Delta }}-\frac{\partial B}{\partial a_l^{\sigma }}+{\mathrm{\Lambda }}_l\right)a_l^{\mathrm{\Delta }}{\xi }_{B\mathrm{0}}^{m\sigma }+\frac{\partial R_l}{\partial t}\cdot a_l^{\mathrm{\Delta }}{\xi }_{B\mathrm{0}}^{m\sigma }-\frac{\partial B}{\partial t}\cdot {\xi }_{B\mathrm{0}}^{m\sigma }-{\mathrm{\Lambda }}_l{a}_l^{\mathrm{\Delta }\sigma }{\xi }_{B\mathrm{0}}^{m\sigma }\}\\ ={\displaystyle \sum _{m=\mathrm{0}}^z}{\epsilon }^m\left[R_l{\xi }_{Bl}^{m\mathrm{\Delta }}+R_l^{\mathrm{\Delta }}{\xi }_{Bl}^{m\sigma }+\left(\frac{\partial R_l}{\partial t}\cdot a_l^{\mathrm{\Delta }}-\frac{\partial B}{\partial t}\right){\xi }_{B\mathrm{0}}^m-B{\xi }_{B\mathrm{0}}^{m\mathrm{\Delta }}-R_l^{\mathrm{\Delta }}{a}_l^{\mathrm{\Delta }}{\xi }_{B\mathrm{0}}^{m\sigma }+\left(\frac{\partial R_{\nu }}{\partial a_l^{\sigma }}\cdot a_{\nu }^{\mathrm{\Delta }}-\frac{\partial B}{\partial a_l^{\sigma }}+{\mathrm{\Lambda }}_l\right)a_l^{\mathrm{\Delta }}{\xi }_{B\mathrm{0}}^{m\sigma }-{\mathrm{\Lambda }}_l{a}_l^{\mathrm{\Delta }\sigma }{\xi }_{B\mathrm{0}}^{m\sigma }\right]\\ ={\displaystyle \sum _{m=\mathrm{0}}^z}{\epsilon }^m\left[R_l^{\mathrm{\Delta }}{\xi }_{Bl}^{m\sigma }-R_l^{\mathrm{\Delta }}{a}_l^{\mathrm{\Delta }}{\xi }_{B\mathrm{0}}^{m\sigma }+\left(\frac{\partial R_{\nu }}{\partial a_l^{\sigma }}\cdot a_{\nu }^{\mathrm{\Delta }}-\frac{\partial B}{\partial a_l^{\sigma }}+{\mathrm{\Lambda }}_l\right)a_l^{\mathrm{\Delta }}{\xi }_{B\mathrm{0}}^{m\sigma }-\left(\frac{\partial R_{\nu }}{\partial a_l^{\sigma }}\cdot a_{\nu }^{\mathrm{\Delta }}-\frac{\partial B}{\partial a_l^{\sigma }}\right){\xi }_{Bl}^{m\sigma }-{\mathrm{\Lambda }}_l{\xi }_{Bl}^{m\sigma }+W_{Bl}\left({\xi }_{Bl}^{\left(m-\mathrm{1}\right)\sigma }-a_l^{\mathrm{\Delta }}{\xi }_{B\mathrm{0}}^{\left(m-\mathrm{1}\right)\sigma }\right)\right]\\ ={\displaystyle \sum _{m=\mathrm{0}}^z}{\epsilon }^m\left[R_l^{\mathrm{\Delta }}\left({\xi }_{Bl}^{m\sigma }-a_l^{\mathrm{\Delta }}{\xi }_{B\mathrm{0}}^{m\sigma }\right)-\left(\frac{\partial R_{\nu }}{\partial a_l^{\sigma }}\cdot a_{\nu }^{\mathrm{\Delta }}-\frac{\partial B}{\partial a_l^{\sigma }}+{\mathrm{\Lambda }}_l\right)\cdot \left({\xi }_{Bl}^{m\sigma }-a_l^{\mathrm{\Delta }}{\xi }_{B\mathrm{0}}^{m\sigma }\right)+W_{Bl}\left({\xi }_{Bl}^{\left(m-\mathrm{1}\right)\sigma }-a_l^{\mathrm{\Delta }}{\xi }_{B\mathrm{0}}^{\left(m-\mathrm{1}\right)\sigma }\right)\right]\\ ={\displaystyle \sum _{m=\mathrm{0}}^z}{\epsilon }^m\left[\left(R_l^{\mathrm{\Delta }}-\frac{\partial R_{\nu }}{\partial a_l^{\sigma }}\cdot a_{\nu }^{\mathrm{\Delta }}+\frac{\partial B}{\partial a_l^{\sigma }}-{\mathrm{\Lambda }}_l\right)\left({\xi }_{Bl}^{m\sigma }-a_l^{\mathrm{\Delta }}{\xi }_{B\mathrm{0}}^{m\sigma }\right)+W_{Bl}\left({\xi }_{Bl}^{\left(m-\mathrm{1}\right)\sigma }-a_l^{\mathrm{\Delta }}{\xi }_{B\mathrm{0}}^{\left(m-\mathrm{1}\right)\sigma }\right)\right]\\ ={\displaystyle \sum _{m=\mathrm{0}}^z}{\epsilon }^m\left[-\epsilon W_{Bl}\left({\xi }_{Bl}^{m\sigma }-a_l^{\mathrm{\Delta }}{\xi }_{B\mathrm{0}}^{m\sigma }\right)+W_{Bl}\left({\xi }_{Bl}^{\left(m-\mathrm{1}\right)\sigma }-a_l^{\mathrm{\Delta }}{\xi }_{B\mathrm{0}}^{\left(m-\mathrm{1}\right)\sigma }\right)\right]=-{\epsilon }^{z+\mathrm{1}}{W}_{Bl}\left({\xi }_{Bl}^{z\sigma }-a_l^{\mathrm{\Delta }}{\xi }_{B\mathrm{0}}^{z\sigma }\right)\end{array}$