Lie Symmetry Theorem for Nonshifted Birkhoffian Systems on Time Scales

Jinyue CHEN; Yi ZHANG

doi:10.1051/wujns/2022273211

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Volume 27 / No 3 (June 2022)

Wuhan Univ. J. Nat. Sci., 27 3 (2022) 211-217

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Issue		Wuhan Univ. J. Nat. Sci. Volume 27, Number 3, June 2022


Page(s)		211 - 217
DOI		https://doi.org/10.1051/wujns/2022273211
Published online		24 August 2022

Wuhan University Journal of Natural Sciences, 2022, Vol.27 No.3, 211-217

Mathematics

CLC number: O 316

Lie Symmetry Theorem for Nonshifted Birkhoffian Systems on Time Scales

Jinyue CHEN¹ and Yi ZHANG²^†

¹ School of Mathematical Sciences, Suzhou University of Science and Technology, Suzhou 215009, Jiangsu, China
² College of Civil Engineering, Suzhou University of Science and Technology, Suzhou 215011, Jiangsu, China

^† To whom correspondence should be addressed. E-mail: zhy@mail.usts.edu.cn

Received: 6 January 2022

Abstract

The Lie theorem for Birkhoffian systems with time-scale nonshifted variational problems are studied, including free Birkhoffian system, generalized Birkhoffian system and constrained Birkhoffian system. First, the time-scale nonshifted generalized Pfaff-Birkhoff principle is established, and the dynamical equations for three Birkhoffian systems under nonshifted variational problems are deduced. Afterwards, in the time-scale nonshifted variational problems, by introducing the infinitesimal transformations, Lie symmetry for free Birkhoffian system, generalized Birkhoffian system and constrained Birkhoffian system are defined respectively. Then Lie symmetry theorems for three kinds of Birkhoffian systems are deduced and proved. In the end, three examples are given to explain the applications for the results.

Key words: time scales / Lie symmetry / conserved quantity / nonshifted Birkhoffian system

Biography: CHEN Jinyue, female, Master candidate, research direction: analytical mechanics. E-mail: 1291385301@qq.com

Foundation item: Supported by the National Natural Science Foundation of China (11972241, 11572212), the Natural Science Foundation of Jiangsu Province of China (BK20191454) and the Innovation Program for Postgraduate in Higher Education Institutions of Jiangsu Province of China (KYCX20_2744)

© Wuhan University 2022

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

0 Introduction

Time scale provides effect mathematical tools for the research of complex dynamical systems. German scholar Hilger^[1] first unified and extended the theory of continuity and discreteness ― an analytical theory on time scales. Since then, many scholars^[2-10]have conducted detailed studies on time scales. Time scale is also widely used in practical problems. For example, in a simple series circuit composed of resistance, capacitance and self-inductive coil, when the capacitor is periodically closed at a fixed frequency, the current change rate in the circuit can be precisely described by the derivative on time scales^[11], nonlinear partially defined systems on an arbitrary unbounded time scale are studied^[12,13].

The research for symmetry and conservation law is an important direction in the development of modern analysis mechanics. Noether^[14]studied the invariance of action integrals in finite continuous groups and revealed the internal relationship between symmetries and conserved quantities in dynamic systems. Lutzky^[15] introduced the Lie symmetry method of mechanical system, that is, the equation remains unchanged after the introduction of infinitesimal group transformation. Mei^[16,17] studied the dynamics equations for holonomic systems, nonholonomic systems and Birkhoffian systems in detail, as well as the corresponding symmetries and conserved quantities for these systems. Afterwards, some progress has been made in the study of symmetry and conserved quantities for different mechanical systems^[18-22].

The symmetries and conserved quantities for different dynamical systems under time-scale shifted variational problems are firstly studied. Take the Lagrange system as an example. In classical dynamical system, assuming that the configurations for the dynamics systems were determined by $n$ generalized coordinates $q_{s} (s = 1,2, \dots, n)$ , then Lagrangian functions were written as $L = L (t, q_{s} (t), {\dot{q}}_{s} (t))$ . On time scales, the Lagrangian functions were written as $L = L (t, q_{s}^{σ} (t), q_{s}^{Δ} (t))$ . On the shifted variational problem, Bartosiewicz and Torres^[23] derived Noether's theorem under the $Δ$ -derivative by using the time reparameter method, and proved the variational problem of mechanical system. Bartosiewicz and his co-workers^[24]studied the time-scale second Euler-Lagrange equation, and derived Noether's conserved quantities on time scales by using this equation. Subsequently, many scholars^[25-34]began to carry out a series of studies on time-scale the variational principle for constrained mechanical systems and Noether's theorem of different dynamic systems. However, on a time scale, there are few studies on Lie symmetry in mechanical systems. Cai et al ^[35]preliminarily studied Lie symmetry and time-scale conserved quantity for Lagrangian systems. Zhang and his team^[36-39] also discussed the time-scale for different dynamic systems and its Lie theorem.

Recently, Bourdin^[40] studied the nonshifted problem on time scales. In the nonshifted variational principle, the classical Lagrangian function was written as $L = L (t, q_{s} (t), q_{s}^{Δ} (t))$ on time scales. They also found the nonshifted variational problem in discrete conditions related to the structure-preserving algorithm. Anerot et al ^[41] rederived time-scale Noether's theorem with shifted and nonshifted variational problems, and the correctness of the results was illustrated by numerical simulation. The calculation of the nonshifted variational principle is better than that of the shifted variational principle in the dynamic system. Song and Cheng^[42]studied Noether's symmetry for free Birkhoffian systems and Hamiltonian systems about time-scale with nonshifted variational problems. Song^[43] also studied the adiabatic invariants for three kinds of dynamical systems under the nonshifted variational problem. Then Chen and Zhang^[44] researched Noether's quasi-symmetry theorem of three kinds of Birkhoffian systems about time-scale with nonshifted variational problems. Therefore, this paper will further research Lie symmetry theorem of three Birkhoffian systems about time-scale with nonshifted variational problems, including free Birkhoffian system, generalized Birkhoffian system and constrained Birkhoffian system.

1 The Dynamical Equations

About time-scale calculus and its properties, please refer to Ref. [2].

1.1 Free Birkhoffian System

The time-scale nonshifted Pfaff-Birkhoff action is written as

$A = \int_{t_{1}}^{t_{2}} [R_{υ} (t, a_{ρ}) a_{υ}^{Δ} - B (t, a_{ρ})] Δ t$ (1)

where $a_{υ}^{Δ}$ is the delta derivative of $a_{υ}$ with respect to $t$ . The Birkhoffian $B : T \times R^{2 n} \to R$ , Birkhoff's functions $R_{υ} : T \times R^{2 n} \to R$ . Let all functions are differential and their derivatives are rd-continuous^[2], $ρ, υ = 1,2, \dots, 2 n$ .

The isochronous variation principle

$δ A = 0$ (2)

and exchange relationship

$δ a_{υ}^{Δ} = {(δ a_{υ})}^{Δ}$ (3)

with the endpoint conditions

$δ a_{υ} |_{t = t_{1}} = δ a_{υ} |_{t = t_{2}} = 0$ (4)

are called the nonshifted Pfaff-Birkhoff principle.

We can obtain the time-scale nonshifted Birkhoff's equations ^[42]:

$\begin{array}{l} R_{μ}^{\nabla} = σ^{\nabla} (t) [\frac{\partial R_{υ} (t, a_{ρ})}{\partial a_{μ}} a_{υ}^{Δ} - \frac{\partial B (t, a_{ρ})}{\partial a_{μ}}], \\ μ, υ, ρ = 1,2, \dots, 2 n \end{array}$ (5)

1.2 Generalized Birkhoffian System

We extend the principle (2)-(4) to the time-scale nonshifted generalized Pfaff-Birkhoff principle

$\int_{t_{1}}^{t_{2}} {δ (R_{υ} a_{υ}^{Δ} - B) + Λ_{υ} (t, a_{ρ}) δ a_{υ}} Δ t = 0$ (6)

We have the time-scale nonshifted generalized Birkhoff's equations ^[44]

$σ^{\nabla} (t) [\frac{\partial R_{υ}}{\partial a_{μ}} a_{υ}^{Δ} - \frac{\partial B}{\partial a_{μ}} + Λ_{μ}] = R_{μ}^{\nabla}, μ, υ = 1,2, \dots, 2 n$ (7)

1.3 Constrained Birkhoffian System

If the constraint equations are shown as

$f_{β} (t, a_{ρ}) = 0, β = 1,2, \dots, 2 n$ (8)

To calculate the variation in equation (8)

$\frac{\partial f_{β}}{\partial a_{μ}} δ a_{μ} = 0$ (9)

Form Eq. (9), we have

$\int_{t_{1}}^{t_{2}} λ_{β} \frac{\partial f_{β}}{\partial a_{μ}} δ a_{μ} Δ t = 0$ (10)

Integration by parts with equation (10), we get

$\int_{t_{1}}^{t_{2}} (\int_{t_{1}}^{σ (t)} λ_{β} \frac{\partial f_{β}}{\partial a_{μ}} \nabla τ) {(δ a_{μ})}^{Δ} Δ t = 0$ (11)

By the principle and Eq. (11), we have the time-scale nonshifted constrained Birkhoff's equations ^[44]

$\begin{array}{l} σ^{\nabla} (t) [\frac{\partial R_{υ}}{\partial a_{μ}} a_{υ}^{Δ} - \frac{\partial B}{\partial a_{μ}} - λ_{β} \frac{\partial f_{β}}{\partial a_{μ}}] - R_{μ}^{\nabla} = 0, \\ μ, υ, β = 1,2, \dots, 2 n \end{array}$ (12)

If the systems are nonsingular, by using Eq. (8) and (12), we can solve $λ_{β} = λ_{β} (t, a_{ρ})$ , the equation (12) can be written

$σ^{\nabla} (t) [\frac{\partial R_{υ}}{\partial a_{μ}} a_{υ}^{Δ} - \frac{\partial B}{\partial a_{μ}} - P_{μ}] = R_{μ}^{\nabla}$ (13)

where

$P_{μ} = λ_{β} \frac{\partial f_{β}}{\partial a_{μ}}$ (14)

Equation (13) should be called as the corresponding nonshifted free Birkhoffian system (5).

2 Structural Equations and Conserved Quantities

The infinitesimal transformations

$t^{*} = t + ε ξ_{0} (t, a_{ρ}), a_{μ}^{*} = a_{μ} + ε ξ_{μ} (t, a_{ρ})$ (15)

where $ε$ is an infinitesimal parameter, $ξ_{0}$ and $ξ_{μ}$ are the infinitesimal generators.

Introduce the infinitesimal generated vectors ^[17]

$X^{(0)} = ξ_{0} \frac{\partial}{\partial t} + ξ_{μ} \frac{\partial}{\partial a_{μ}}$ (16)

And an extension of infinitesimal generator

$X^{(1)} = X^{(0)} + (ξ_{μ}^{Δ} - a_{μ}^{Δ} ξ_{0}^{Δ}) \frac{\partial}{\partial a_{μ}^{Δ}}$ (17)

2.1 Free Birkhoffian System

Definition 1 If the infinitesimal transformations (15) satisfy deterministic equation

$σ^{\nabla} X^{(1)} [(\frac{\partial R_{υ}}{\partial a_{μ}}) a_{υ}^{Δ} - \frac{\partial B}{\partial a_{μ}}] + σ^{\nabla} (ξ_{υ}^{Δ} - a_{υ}^{Δ}) \frac{\partial R_{υ}}{\partial a_{μ}} - X^{(1)} (R_{μ}^{\nabla}) = 0$ (18)

and corresponding symmetry is said the Lie symmetry for nonshifted free Birkhoffian systems (5).

Theorem 1 If $ξ_{0}$ and $ξ_{μ}$ satisfy the deterministic equation (18), and there exists $G (t, a_{ρ})$ satisfying structural equation

$(R_{υ} a_{υ}^{Δ} - B) ξ_{0}^{Δ} + X^{(1)} (R_{υ} a_{υ}^{Δ} - B) + G^{Δ} = 0$ (19)

This system (5) has the conserved quantity, that is

$I = R_{υ} ξ_{υ}^{σ} + \int_{t_{1}}^{t} [(\frac{\partial R_{υ}}{\partial τ} a_{υ}^{Δ} - \frac{\partial B}{\partial τ}) ξ_{0} - B ξ_{0}^{Δ}] σ^{\nabla} \nabla τ + G^{σ} = c o n s t .$ (20)

Proof Take the $\nabla$ - derivative for equation (20)

$\frac{\nabla}{\nabla t} I = {(R_{υ} ξ_{υ}^{σ})}^{\nabla} - {(B ξ_{0}^{σ})}^{\nabla} + σ^{\nabla} ξ_{0} \frac{\partial R_{υ}}{\partial t} a_{υ}^{Δ} - σ^{\nabla} ξ_{0} \frac{\partial B}{\partial t} + G^{Δ} σ^{\nabla}$

$= σ^{\nabla} ξ_{0} (\frac{\partial R_{υ}}{\partial t} a_{υ}^{Δ} - \frac{\partial B}{\partial t}) + R_{υ}^{\nabla} ξ_{υ} + σ^{\nabla} R_{υ} ξ_{υ}^{Δ} - σ^{\nabla} B ξ_{0}^{Δ} + G^{Δ} σ^{\nabla}$ (21)

According to

$\begin{array}{l} X^{(1)} (R_{υ} a_{υ}^{Δ} - B) = ξ_{0} (\frac{\partial R_{υ}}{\partial t} a_{υ}^{Δ} - \frac{\partial B}{\partial t}) + ξ_{μ} (\frac{\partial R_{υ}}{\partial a_{μ}} a_{υ}^{Δ} - \frac{\partial B}{\partial a_{μ}}) \\ + (ξ_{μ}^{Δ} - a_{μ}^{Δ} ξ_{0}^{Δ}) R_{μ} \end{array}$ (22)

Substituting equation (22) into equation (21), we can get

$\begin{array}{l} \frac{\nabla}{\nabla t} I = σ^{\nabla} ξ_{0} (\frac{\partial R_{υ}}{\partial t} a_{υ}^{Δ} - \frac{\partial B}{\partial t}) + R_{μ}^{\nabla} ξ_{μ} + σ^{\nabla} R_{υ} ξ_{υ}^{Δ} \\ - σ^{\nabla} B ξ_{0}^{Δ} - σ^{\nabla} (R_{υ} a_{υ}^{Δ} - B) ξ_{0}^{Δ} - σ^{\nabla} ξ_{0} (\frac{\partial R_{υ}}{\partial t} a_{υ}^{Δ} - \frac{\partial B}{\partial t}) \\ - σ^{\nabla} ξ_{μ} (\frac{\partial R_{υ}}{\partial a_{μ}} a_{υ}^{Δ} - \frac{\partial B}{\partial a_{μ}}) - σ (ξ_{μ}^{Δ} - a_{μ}^{Δ} ξ_{0}^{Δ}) R_{μ} \end{array}$

$= [R_{μ}^{\nabla} - σ^{\nabla} (\frac{\partial R_{υ}}{\partial a_{μ}} a_{υ}^{Δ} - \frac{\partial B}{\partial a_{μ}})] ξ_{μ} = 0$ (23)

Therefore, the proof is completed.

2.2 Generalized Birkhoffian System

Definition 2 If $ξ_{0}$ and $ξ_{μ}$ of the infinitesimal transformations (15) satisfy deterministic equation

$\begin{array}{l} σ^{\nabla} X^{(1)} [(\frac{\partial R_{υ}}{\partial a_{μ}}) a_{υ}^{Δ} - \frac{\partial B}{\partial a_{μ}}] + σ^{\nabla} (ξ_{υ}^{Δ} - a_{υ}^{Δ}) \frac{\partial R_{υ}}{\partial a_{μ}} - X^{(1)} (R_{μ}^{\nabla}) \\ + σ^{\nabla} X^{(1)} (Λ_{μ}) = 0 \end{array}$ (24)

then corresponding symmetry is said the Lie symmetry for the nonshifted generalized Birkhoffian system (7).

Theorem 2 If $ξ_{0}$ and $ξ_{μ}$ satisfy the deterministic equation (24), and there exists $G (t, a_{ρ})$ satisfying the structural equation

$(R_{υ} a_{υ}^{Δ} - B) ξ_{0}^{Δ} + X^{(1)} (R_{υ} a_{υ}^{Δ} - B) + G^{Δ} + (ξ_{υ} - a_{υ}^{Δ} ξ_{0}) Λ_{υ} = 0$ (25)

This system (7) has the conserved quantity, that is

$\begin{array}{l} I = R_{υ} ξ_{υ}^{σ} + \int_{t_{1}}^{t} [(\frac{\partial R_{υ}}{\partial τ} a_{υ}^{Δ} - \frac{\partial B}{\partial τ} - Λ_{υ} a_{υ}^{Δ}) ξ_{0} - B ξ_{0}^{Δ}] σ^{\nabla} \nabla τ \\ + G^{σ} = c o n s t \end{array}$ (26)

Proof Take the $\nabla$ - derivative for equation (26)

$\begin{array}{l} \frac{\nabla}{\nabla t} I = {(R_{υ} ξ_{υ}^{σ})}^{\nabla} - {(B ξ_{0}^{σ})}^{\nabla} + σ^{\nabla} ξ_{0} \frac{\partial R_{υ}}{\partial t} a_{υ}^{Δ} - σ^{\nabla} ξ_{0} \frac{\partial B}{\partial t} \\ - σ^{\nabla} ξ_{0} Λ_{υ} a_{υ}^{Δ} + G^{Δ} σ^{\nabla} \end{array}$

$\begin{array}{l} = σ^{\nabla} ξ_{0} (\frac{\partial R_{υ}}{\partial t} a_{υ}^{Δ} - \frac{\partial B}{\partial t}) + R_{υ}^{\nabla} ξ_{υ} + σ^{\nabla} R_{υ} ξ_{υ}^{Δ} \\ - σ^{\nabla} B ξ_{0}^{Δ} - σ^{\nabla} ξ_{0} Λ_{υ} a_{υ}^{Δ} + G^{Δ} σ^{\nabla} \end{array}$ (27)

And because

$\begin{array}{l} X^{(1)} (R_{υ} a_{υ}^{Δ} - B) = ξ_{0} (\frac{\partial R_{υ}}{\partial t} a_{υ}^{Δ} - \frac{\partial B}{\partial t}) + ξ_{μ} (\frac{\partial R_{υ}}{\partial a_{μ}} a_{υ}^{Δ} - \frac{\partial B}{\partial a_{μ}}) \\ + (ξ_{μ}^{Δ} - a_{μ}^{Δ} ξ_{0}^{Δ}) R_{μ} \end{array}$ (28)

Substituting Eqs. (7) and (28) into equation (27), we can get

$\begin{array}{l} \frac{\nabla}{\nabla t} I = σ^{\nabla} ξ_{0} (\frac{\partial R_{υ}}{\partial t} a_{υ}^{Δ} - \frac{\partial B}{\partial t}) + R_{μ}^{\nabla} ξ_{μ} + σ^{\nabla} R_{υ} ξ_{υ}^{Δ} - σ^{\nabla} B ξ_{0}^{Δ} \\ - σ^{\nabla} ξ_{0} Λ_{υ} a_{υ}^{Δ} - σ^{\nabla} (R_{υ} a_{υ}^{Δ} - B) ξ_{0}^{Δ} \\ - σ^{\nabla} ξ_{0} (\frac{\partial R_{υ}}{\partial t} a_{υ}^{Δ} - \frac{\partial B}{\partial t}) - σ^{\nabla} ξ_{μ} (\frac{\partial R_{υ}}{\partial a_{μ}} a_{υ}^{Δ} - \frac{\partial B}{\partial a_{μ}}) \\ - σ^{\nabla} (ξ_{μ}^{Δ} - a_{μ}^{Δ} ξ_{0}^{Δ}) R_{μ} - σ^{\nabla} Λ_{υ} (ξ_{υ} - a_{υ}^{Δ} ξ_{0}) \end{array}$

$= [R_{μ}^{\nabla} - σ^{\nabla} (\frac{\partial R_{υ}}{\partial a_{μ}} a_{υ}^{Δ} - \frac{\partial B}{\partial a_{μ}} + Λ_{μ})] ξ_{μ} = 0$ (29)

Therefore, the proof is completed.

2.3 Constrained Birkhoffian System

Definition 3 If $ξ_{0}$ and $ξ_{μ}$ of the infinitesimal transformations (15) satisfy the Lie symmetry deterministic equation

$\begin{array}{l} σ^{\nabla} X^{(1)} [(\frac{\partial R_{υ}}{\partial a_{μ}}) a_{υ}^{Δ} - \frac{\partial B}{\partial a_{μ}}] + σ^{\nabla} (ξ_{υ}^{Δ} - a_{υ}^{Δ}) \frac{\partial R_{υ}}{\partial a_{μ}} - X^{(1)} (R_{μ}^{\nabla}) \\ - σ^{\nabla} X^{(1)} (P_{μ}) = 0 \end{array}$ (30)

then corresponding symmetry is said the Lie symmetry for nonshifted constrained Birkhoffian system (13).

Theorem 3 If $ξ_{0}$ and $ξ_{μ}$ satisfy the deterministic equation (30), and exist $G (t, a_{ρ})$ satisfying the structural equation

$(R_{υ} a_{υ}^{Δ} - B) ξ_{0}^{Δ} + X^{(1)} (R_{υ} a_{υ}^{Δ} - B) + G^{Δ} - (ξ_{υ} - a_{υ}^{Δ} ξ_{0}) P_{υ} = 0$ (31)

then system (13) has the conserved quantity, that is

$\begin{array}{l} I = R_{υ} ξ_{υ}^{σ} + \int_{t_{1}}^{t} [(\frac{\partial R_{υ}}{\partial τ} a_{υ}^{Δ} - \frac{\partial B}{\partial τ} + P_{υ} a_{υ}^{Δ}) ξ_{0} - B ξ_{0}^{Δ}] σ^{\nabla} \nabla τ + G^{σ} \\ = c o n s t \end{array}$ (32)

3 Example

Example 1 On time scales, the Hojman-Urrutia problem can be written

${\begin{array}{l} B = \frac{1}{2} {{(a_{3})}^{2} + 2 a_{2} a_{3} - {(a_{4})}^{2}} \\ R_{1} = a_{2} + a_{3}, R_{2} = 0, R_{3} = a_{4}, R_{4} = 0 \end{array}$ (33)

According to the study of Santilli^[45], the Hojman-Urrutia problem admits a Birkhoffian representation. But since the equation itself is not self-adjoint, it has no Hamiltonian representation.

By the nonshifted Birkhoff's equation (5), we have

${\begin{array}{l} {(a_{2} + a_{3})}^{\nabla} = 0 \\ σ^{\nabla} (a_{1}^{Δ} - a_{3}) = 0 \\ σ^{\nabla} (a_{1}^{Δ} - a_{3} - a_{2}) = a_{4}^{\nabla} \\ σ^{\nabla} (a_{3}^{Δ} + a_{4}) = 0 \end{array}$ (34)

We take

$ξ_{0} = 0, ξ_{1} = t, ξ_{2} = 0, ξ_{3} = 1, ξ_{4} = 0, X^{(1)} = t \frac{\partial}{\partial a_{1}} + \frac{\partial}{\partial a_{3}} + \frac{\partial}{\partial a_{1}^{Δ}}$ (35)

The generator (35) satisfies determination equation (18), so the generator (35) corresponds to Lie symmetry in the system (5).

Form the structural equation (19), we can get

$\begin{array}{l} {(a_{2} + a_{3}) a_{1}^{Δ} + a_{4} a_{3}^{Δ} - \frac{1}{2} [{(a_{3})}^{2} + 2 a_{2} a_{3} - {(a_{4})}^{2}]} ξ_{0}^{Δ} \\ + ξ_{2} a_{1}^{Δ} - ξ_{2} a_{3} + ξ_{3} (a_{1}^{Δ} - a_{2} - a_{3}) + ξ_{4} (a_{3}^{Δ} + a_{4}) \\ + (ξ_{1}^{Δ} - a_{1}^{Δ} ξ_{0}^{Δ}) (a_{2} + a_{3}) + (ξ_{3}^{Δ} - a_{3}^{Δ} ξ_{0}^{Δ}) a_{4} + G^{Δ} = 0 \end{array}$ (36)

Substituting (35) into (36), we have $G = - a_{1}$ .

By Theorem 1, we can get conserved quantity

$I = σ (t) (a_{2} + a_{3}) + a_{4} - a_{1}^{σ} = c o n s t$ (37)

Let $T = {2^{n} : n \in N ⋃ {0}}$ , we have $σ (t) = 2 t, μ (t) = σ (t) - t = t$ , then the conserved quantity

$I = 2 t (a_{2} + a_{3}) + a_{4} - a_{1} - t a_{1}^{Δ} = c o n s t$ (38)

If the initial condition are $a_{1} (1) = 1, a_{2} (1) = 1, a_{2} (2) = 2, a_{3} (1) = - 2, a_{4} (1) = 1$ , let $T = {2^{n} : n \in N ⋃ {0}}$ the trajectory of motion $a_{1}, a_{2}, a_{3}, a_{4}$ and the conserved quantity $I$ are calculated, the results are shown in Fig. 1.

Fig. 1

Simulations of a₁, a₂, a₃, a₄, I on time scale 𝕋 = {2ⁿ: n ∈ ℕ ∪ {0}}

Let $T = R$ , we have $σ (t) = t, μ (t) = σ (t) - t = 0$ , then the conserved quantity

$I = t (a_{2} + a_{3}) + a_{4} - a_{1} = c o n s t$ (39)

Example 2 The nonshifted generalized Birkhoffian and Birkhoff's functions on time scales are

${\begin{array}{l} B = \frac{1}{2} {(a_{3})}^{2} + \frac{1}{2} {(a_{4})}^{2} \\ R_{1} = 0, R_{2} = 0, R_{3} = a_{1}, R_{4} = a_{2} \\ Λ_{1} = - t, Λ_{2} = - t, Λ_{3} = a_{3}, Λ_{4} = a_{4} \end{array}$ (40)

By the nonshifted generalized Birkhoff's equation (7), we have

$a_{3}^{Δ} = t, a_{4}^{Δ} = t, a_{1}^{\nabla} = a_{2}^{\nabla} = 0$ (41)

We take

$ξ_{0} = 0, ξ_{1} = 0, ξ_{2} = 0, ξ_{3} = t, ξ_{4} = t, X^{(1)} = t \frac{\partial}{\partial a_{3}} + t \frac{\partial}{\partial a_{4}}$ (42)

The generator (42) satisfies determination equation (24), so the generator (42) corresponds to Lie symmetry in this system (7).

According to the structural equation (25), we can get

$\begin{array}{l} (a_{1} a_{3}^{Δ} + a_{2} a_{4}^{Δ} - B) ξ_{0}^{Δ} + (t a_{1}^{Δ} + t a_{2}^{Δ} - a_{3} a_{3}^{Δ} - a_{4} a_{4}^{Δ}) ξ_{0} \\ + (a_{3}^{Δ} - t) ξ_{1} + (a_{4}^{Δ} - t) ξ_{2} + G^{Δ} = 0 \end{array}$ (43)

Substituting (42) into (43), we have $G = 0$ .

By Theorem 2, we can get conserved quantity

$I = σ (t) (a_{1} + a_{2}) = c o n s t$ (44)

Let $T = {3^{n} : n \in N ⋃ {0}}$ , we get $σ (t) = 3 t, μ (t) = σ (t) - t = 2 t$ , then the conserved quantity

$I = 3 t (a_{1} + a_{2}) = c o n s t$ (45)

Example 3 On time scales, let $T = {2^{n} : n \in N ⋃ {0}}$ , the nonshifted constrained Birkhoffian and Birkhoff's functions are

$B = \frac{1}{2} [{(a_{1})}^{2} + {(a_{3})}^{2} + {(a_{4})}^{2}], R_{1} = a_{3}, R_{2} = a_{4}, R_{3} = R_{4} = 0$ (46)

The constraints equations are

$f_{1} = a_{1} a_{3} - {(c_{1})}^{2} = 0, f_{2} = a_{1} + a_{4} - c_{2} = 0$ (47)

By Eq. (13), we can get

${\begin{array}{l} a_{3}^{\nabla} = 2 (- λ_{1} a_{3} - λ_{2} - a_{1}) \\ a_{4}^{\nabla} = 0 \\ a_{1}^{Δ} - a_{3} = λ_{1} a_{1} \\ a_{2}^{Δ} - a_{4} = λ_{2} \end{array}$ (48)

From Eqs. (47) and (48), we have

$λ_{1} = - \frac{a_{3}}{a_{1}}, λ_{2} = - a_{1} + \frac{{(a_{3})}^{2}}{a_{1}}$ (49)

Hence, we have

$P_{1} = - a_{1}, P_{2} = 0, P_{3} = - a_{3}, P_{4} = - a_{1} + \frac{{(a_{3})}^{2}}{a_{1}}$ (50)

Let

$ξ_{0} = 0, ξ_{1} = 1, ξ_{2} = 0, ξ_{3} = 0, X^{(1)} = \frac{\partial}{\partial a_{1}}$ (51)

The generator (51) satisfies determination equation (30), so the generator (51) corresponds to Lie symmetry in this system (13).

According to the structural equation (31), we can get

$\begin{array}{l} {a_{3} a_{1}^{Δ} + a_{4} a_{2}^{Δ} - \frac{1}{2} [{(a_{1})}^{2} + {(a_{2})}^{2} + {(a_{3})}^{2} + {(a_{4})}^{2}]} ξ_{0}^{Δ} \\ - ξ_{1} a_{1} - ξ_{2} a_{2} + ξ_{3} (a_{1}^{Δ} - a_{3}) \end{array}$

$\begin{array}{l} + ξ_{4} (a_{2}^{Δ} - a_{4}) + a_{1} (ξ_{1} - a_{1}^{Δ} ξ_{0}) + a_{3} (ξ_{3} - a_{3}^{Δ} ξ_{0}) \\ + (a_{1} - \frac{{(a_{3})}^{2}}{a_{1}}) (ξ_{4} - a_{4}^{Δ} ξ_{0}) + G^{Δ} = 0 \end{array}$ (52)

Substituting (51) into (52), we have $G = 0$ .

By Theorem 3, we can get conserved quantity

$I = a_{3} = c o n s t$ (53)

4 Conclusion

Time-scale theory has been generally used in many fields. When $T = R$ , the dynamic equation on time scale can be degraded into a continuous differential equation. When $T = Z$ , the dynamic equation on time scale can be degraded into discrete difference equation. This paper mainly studies the time-scale with nonshifted variational problem which Lie theorem for three kinds of Birkhoffian systems. In this paper, nonshifted generalized Birkhoff's equations and nonshifted constrained Birkhoff's equations are derived by using principle (6). By introducing infinitesimal transformation, the structural equation of Lie symmetry is derived, and then the Noether conserved quantities caused by Lie symmetries is deduced. Since the study about nonshifted variational problem on a time scale is just beginning, the research of paper has great theoretical and practical significance.

References

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Hilger S. Differential and difference calculus-unified [J]. Nonlinear Analysis: Theory, Methods and Applications, 1997, 30(5): 2683-2694. [CrossRef] [MathSciNet] [Google Scholar]
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Song C J, Zhang Y. Noether theorem for Birkhoffian systems on time scales [J]. Journal of Mathematical Physics, 2015, 56(10): 102701. [CrossRef] [MathSciNet] [Google Scholar]
Zhang Y. Noether symmetries and conserved quantities for constrained Birkhoffian systems on time scales [J]. Journal of Dynamics and Control, 2019, 17(5): 482-486. [Google Scholar]
Song C J, Zhang Y. Noether theory for Birkhoffian systems with nabla derivatives [J]. Journal of Nonlinear Sciences and Applications, 2017, 10(4): 2268-2282. [CrossRef] [Google Scholar]
Zhai X H, Zhang Y. Noether theorem for non-conservative systems with time delay on time scales [J]. Communications in Nonlinear Science and Numerical Simulation, 2017, 52: 32-43. [NASA ADS] [CrossRef] [MathSciNet] [Google Scholar]
Jin S X, Zhang Y. Methods of reduction for Lagrange systems on time scales with nabla derivatives [J]. Chinese Physics B, 2017, 26(1): 243-249. [Google Scholar]
Song J, Zhang Y. Routh method of reduction for dynamical systems with nonstandard Lagrangians on time scales [J]. Indian Journal of Physics, 2020, 94(4): 501-506. [NASA ADS] [CrossRef] [Google Scholar]
Shi Y F, Zhang Y. Noether theorem on time scales for Lagrangian systems in event space [J]. Wuhan University Journal of Natural Sciences, 2019, 24(4): 295-304. [CrossRef] [Google Scholar]
Cai P P, Fu J L, Guo Y X. Lie symmetries and conserved quantities of the constraint mechanical systems on time scales [J]. Reports on Mathematical Physics, 2017, 79(3): 279-298. [NASA ADS] [CrossRef] [MathSciNet] [Google Scholar]
Zhang Y. Adiabatic invariants and Lie symmetries on time scales for nonholonomic systems of non-Chetaev type [J]. Acta Mechanica, 2020, 231(1): 293-303. [CrossRef] [MathSciNet] [Google Scholar]
Zhai X H, Zhang Y. Lie symmetry analysis on time scales and its application on mechanical systems [J]. Journal of Vibration and Control, 2019, 25(3): 581-592. [CrossRef] [MathSciNet] [Google Scholar]
Zhang Y. Lie symmetry and invariants for a generalized Birkhoffian system on time scales [J]. Chaos, Solitons and Fractals, 2019, 128: 306-312. [NASA ADS] [CrossRef] [MathSciNet] [Google Scholar]
Zhang Y, Zhai X H. Perturbation to Lie symmetry and adiabatic invariants for Birkhoffian systems on time scales [J]. Communications in Nonlinear Science and Numerical Simulation, 2019, 75: 251-261. [NASA ADS] [CrossRef] [MathSciNet] [Google Scholar]
Bourdin L. Nonshifted calculus of variations on time scales with -differentiable [J]. Journal of Mathematical Analysis and Applications, 2014, 411(2): 543-554. [CrossRef] [MathSciNet] [Google Scholar]
Anerot B, Cresson J, Belgacem K H, et al. Noether's-type theorems on time scales [J]. Journal of Mathematical Physics, 2020, 61(11): 113502. [NASA ADS] [CrossRef] [MathSciNet] [Google Scholar]
Song C J, Cheng Y. Noether's theorems for nonshifted dynamic systems on time scales [J]. Applied Mathematics and Computation, 2020, 374: 125086. [CrossRef] [MathSciNet] [Google Scholar]
Song C J. Nonshifted adiabatic invariant on time scale [J]. Wuhan University Journal of Natural Sciences, 2020, 25(4): 301-306. [Google Scholar]
Chen J Y, Zhang Y. Time-scale version of generalized Birkhoffian mechanics and its symmetries and conserved quantities of Noether type [J]. Advances in Mathematical Physics, 2021, 2021: 9982975. [Google Scholar]
Santilli R M. Foundations of Theoretical Mechanics Ⅱ [M]. New York: Springer-Verlag, 1983. [CrossRef] [Google Scholar]

All Figures

	Fig. 1 Simulations of a₁, a₂, a₃, a₄, I on time scale 𝕋 = {2ⁿ: n ∈ ℕ ∪ {0}}
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[1] Hilger S. Analysis on measure chains—A unified approach to continuous and discrete calculus [J]. Results in Mathematics, 1990, 18(1-2): 18-56. [CrossRef] [Google Scholar]

[2] Bohner M, Peterson A. Dynamic Equations on Time Scale [M]. Boston: Birkhäuser, 2001. [CrossRef] [Google Scholar]

[3] Ahlbrandt C D, Bohner M, Ridenhour J. Hamiltonian systems on time scales [J]. Journal of Mathematical Analysis and Applications, 2000, 250(2): 561-578. [CrossRef] [MathSciNet] [Google Scholar]

[4] Bohner M. Calculus of variations on time scales [J]. Dynamic Systems and Applications, 2004, 13(3-4): 339-349. [MathSciNet] [Google Scholar]

[5] Hilger S. Generalized theorem of Hartman-Grobman on measure chains [J]. Journal of the Australian Mathematical Society, 1996, 60(2): 157-191. [CrossRef] [Google Scholar]

[6] Hilger S. Differential and difference calculus-unified [J]. Nonlinear Analysis: Theory, Methods and Applications, 1997, 30(5): 2683-2694. [CrossRef] [MathSciNet] [Google Scholar]

[7] Agarwal R P, Bohner M. Basic calculus on time scales and some of its applications [J]. Results in Mathematics, 1999, 35(1-2): 3-22. [CrossRef] [Google Scholar]

[8] Bohner M, Sh Guseinov G. Double integral calculus of variations on time scales [J]. Computers and Mathematics with Applications, 2007, 54(1): 45-57. [CrossRef] [MathSciNet] [Google Scholar]

[9] Hilscher R, Zeidan V. First order conditions for generalized variational problems over time scales [J]. Computers and Mathematics with Applications, 2011, 62: 3490-3503. [CrossRef] [MathSciNet] [Google Scholar]

[10] Herzallah M A E, Muslih S I, Balwanu D, et al. Hamilton-Jacobi and fractional like action with time scaling [J]. Nonlinear Dynamics, 2011, 66(4): 549-555. [CrossRef] [MathSciNet] [Google Scholar]

[11] Han Z L, Sun S R. Vibration Theory of Dynamic Equation on Time Scale [M]. Jinan: Shandong University Press, 2014 (Ch). [Google Scholar]

[12] Fausett L V, Murty K N. Controllability, observability and realizability criteria on time scale dynamical systems [J]. Nonlinear Studies, 2004, 11(4): 627-638. [MathSciNet] [Google Scholar]

[13] Bartosiewicz Z, Kotta Ü, Pawłuszewicz E. Equivalence of linear control systems on time scales [J]. Proceedings of the Estonian Academy of Sciences Physics Mathematics, 2006, 55(1): 43-52. [CrossRef] [Google Scholar]

[14] Noether A. Invariante variations probleme. Nachrichten von der Gesellschaft der Wissenschaften zuGöttingen [J]. Mathematisch-Physikalische Klasse, 1918, 2: 235-257. [Google Scholar]

[15] Lutzky M. Dynamical symmetries and conserved quantities [J]. Journal of Physics A: Mathematical General, 1979, 12(7): 973-981. [CrossRef] [MathSciNet] [Google Scholar]

[16] Mei F X. Application of Lie Group and Lie Algebra to Constrained Mechanical Systems [M]. Beijing: Science Press, 1999 (Ch) . [Google Scholar]

[17] Mei F X. Symmetry and Conserved Quantity of Constrained Mechanical System [M]. Beijing: Beijing Institute of Technology Press, 2004 (Ch) . [Google Scholar]

[18] Zhang Y, Cai J X. Noether theorem of Herglotz-type for nonconservative Hamilton systems in event space [J]. Wuhan University Journal of Natural Sciences, 2021, 26(5): 376-382. [Google Scholar]

[19] Luo S K. Mei symmetry, Noether symmetry and Lie symmetry of Hamiltonian system [J]. Acta Physica Sinica, 2003, 52(12): 2941-2944 (Ch). [CrossRef] [MathSciNet] [Google Scholar]

[20] Chen X W, Luo S K, Mei F X. A form invariance of constrained Birkhoffian system [J]. Applied Mathematics and Mechanics, 2002, 23(1): 53-57. [CrossRef] [MathSciNet] [Google Scholar]

[21] Zhang H B, Chen H B. Noether's theorem of fractional Birkhoffian systems [J]. Journal of Mathematical Analysis and Applications, 2017, 456(2): 1442-1456. [CrossRef] [MathSciNet] [Google Scholar]

[22] Zhang Y. A set of conserved quantities from Lie symmetries for Birkhoffian systems [J]. Acta Physica Sinica, 2002, 51(3): 461-464 (Ch). [CrossRef] [MathSciNet] [Google Scholar]

[23] Bartosiewicz Z, Torres D F M. Noether's theorem on time scales [J]. Journal of Mathematical Analysis and Applications, 2008, 342(2): 1220-1226. [NASA ADS] [CrossRef] [MathSciNet] [Google Scholar]

[24] Bartosiewicz Z, Martins N, Torres D F M. The second Euler-Lagrange equation of variational calculus on time scales [J]. European Journal of Control, 2011, 17(1): 9-18. [CrossRef] [MathSciNet] [Google Scholar]

[25] Malinowska A B, Ammi M R S. Noether's theorem for control problems on time scales [J]. International Journal of Difference Equations, 2014, 9(1): 87-100. [Google Scholar]

[26] Cai P P, Fu J L, Guo Y X. Noether symmetries of the nonconservative and nonholonomic systems on time scales [J]. Science China Physics, Mechanics and Astronomy, 2013, 56(5): 1017-1028. [CrossRef] [Google Scholar]

[27] Peng K K, Luo Y. Dynamics symmetries of Hamiltonian system on time scales [J]. Journal of Mathematical Physics, 2014, 55(4): 2683-2694. [Google Scholar]

[28] Song C J, Zhang Y. Noether theorem for Birkhoffian systems on time scales [J]. Journal of Mathematical Physics, 2015, 56(10): 102701. [CrossRef] [MathSciNet] [Google Scholar]

[29] Zhang Y. Noether symmetries and conserved quantities for constrained Birkhoffian systems on time scales [J]. Journal of Dynamics and Control, 2019, 17(5): 482-486. [Google Scholar]

[30] Song C J, Zhang Y. Noether theory for Birkhoffian systems with nabla derivatives [J]. Journal of Nonlinear Sciences and Applications, 2017, 10(4): 2268-2282. [CrossRef] [Google Scholar]

[31] Zhai X H, Zhang Y. Noether theorem for non-conservative systems with time delay on time scales [J]. Communications in Nonlinear Science and Numerical Simulation, 2017, 52: 32-43. [NASA ADS] [CrossRef] [MathSciNet] [Google Scholar]

[32] Jin S X, Zhang Y. Methods of reduction for Lagrange systems on time scales with nabla derivatives [J]. Chinese Physics B, 2017, 26(1): 243-249. [Google Scholar]

[33] Song J, Zhang Y. Routh method of reduction for dynamical systems with nonstandard Lagrangians on time scales [J]. Indian Journal of Physics, 2020, 94(4): 501-506. [NASA ADS] [CrossRef] [Google Scholar]

[34] Shi Y F, Zhang Y. Noether theorem on time scales for Lagrangian systems in event space [J]. Wuhan University Journal of Natural Sciences, 2019, 24(4): 295-304. [CrossRef] [Google Scholar]

[35] Cai P P, Fu J L, Guo Y X. Lie symmetries and conserved quantities of the constraint mechanical systems on time scales [J]. Reports on Mathematical Physics, 2017, 79(3): 279-298. [NASA ADS] [CrossRef] [MathSciNet] [Google Scholar]

[36] Zhang Y. Adiabatic invariants and Lie symmetries on time scales for nonholonomic systems of non-Chetaev type [J]. Acta Mechanica, 2020, 231(1): 293-303. [CrossRef] [MathSciNet] [Google Scholar]

[37] Zhai X H, Zhang Y. Lie symmetry analysis on time scales and its application on mechanical systems [J]. Journal of Vibration and Control, 2019, 25(3): 581-592. [CrossRef] [MathSciNet] [Google Scholar]

[38] Zhang Y. Lie symmetry and invariants for a generalized Birkhoffian system on time scales [J]. Chaos, Solitons and Fractals, 2019, 128: 306-312. [NASA ADS] [CrossRef] [MathSciNet] [Google Scholar]

[39] Zhang Y, Zhai X H. Perturbation to Lie symmetry and adiabatic invariants for Birkhoffian systems on time scales [J]. Communications in Nonlinear Science and Numerical Simulation, 2019, 75: 251-261. [NASA ADS] [CrossRef] [MathSciNet] [Google Scholar]

[40] Bourdin L. Nonshifted calculus of variations on time scales with -differentiable [J]. Journal of Mathematical Analysis and Applications, 2014, 411(2): 543-554. [CrossRef] [MathSciNet] [Google Scholar]

[41] Anerot B, Cresson J, Belgacem K H, et al. Noether's-type theorems on time scales [J]. Journal of Mathematical Physics, 2020, 61(11): 113502. [NASA ADS] [CrossRef] [MathSciNet] [Google Scholar]

[42] Song C J, Cheng Y. Noether's theorems for nonshifted dynamic systems on time scales [J]. Applied Mathematics and Computation, 2020, 374: 125086. [CrossRef] [MathSciNet] [Google Scholar]

[43] Song C J. Nonshifted adiabatic invariant on time scale [J]. Wuhan University Journal of Natural Sciences, 2020, 25(4): 301-306. [Google Scholar]

[44] Chen J Y, Zhang Y. Time-scale version of generalized Birkhoffian mechanics and its symmetries and conserved quantities of Noether type [J]. Advances in Mathematical Physics, 2021, 2021: 9982975. [Google Scholar]

[45] Santilli R M. Foundations of Theoretical Mechanics Ⅱ [M]. New York: Springer-Verlag, 1983. [CrossRef] [Google Scholar]