Conserved Quantity for Fractional Constrained Hamiltonian System

Chuanjing SONG; Jiahang WANG

doi:10.1051/wujns/2022273201

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Volume 27 / No 3 (June 2022)

Wuhan Univ. J. Nat. Sci., 27 3 (2022) 201-210

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


Page(s)		201 - 210
DOI		https://doi.org/10.1051/wujns/2022273201
Published online		24 August 2022

Wuhan University Journal of Natural Sciences, 2022, Vol.27 No.3, 201-210

Mathematics

CLC number: O 316

Conserved Quantity for Fractional Constrained Hamiltonian System

Chuanjing SONG¹^† and Jiahang WANG²

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 215009, Jiangsu, China

^† To whom correspondence should be addressed. E-mail: songchuanjingsun@126.com

Received: 8 February 2022

Abstract

Singular system has great relationship with gauge field theory, condensed matter theory and some other research areas. Based on the mixed integer and Riemann-Liouville fractional derivatives, the fractional singular system is studied. Firstly, the fractional constrained Hamilton equation and the fractional inherent constraint are presented. Secondly, Lie symmetry and conserved quantity are analyzed, including determined equation, limited equation, additional limited equation and structural equation. And finally, an example is given to illustrate the methods and results.

Key words: fractional constrained Hamilton equation / inherent constraint / Lie symmetry / conserved quantity

Biography: SONG Chuanjing, female, Ph. D., Associate professor, research direction: mathematical methods in classical mechanics. E-mail: songchuanjing sun@126.com

Foundation item: Supported by the National Natural Science Foundation of China (12172241, 11802193, 11972241), the Natural Science Foundation of Jiangsu Province (BK20191454) and the "Qinglan Project" of Jiangsu Province

© 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

As we all know, there are two ways to describe a mechanical system, the Lagrangian Method and the Hamiltonian Method. The two methods are transformed through Legendre transformation. However, if the system studied is a singular one, there will exist inherent constraints when it is transformed from the Lagrangian to the Hamiltonian, and the system is called a constrained Hamiltonian system (also called constrained canonical system). The constrained Hamiltonian system has great relationship with gauge field theory, which dominates particle physics. Besides, the constrained Hamiltonian systems can also be applied to the theory of condensed matter and some other research areas. Therefore, the study of the constrained Hamiltonian system is a hot topic, and much progress has been made, especially on the aspect of symmetry. For example, the standard constrained canonical systems and their quantization, standard canonical symmetries, symmetries of the path integral, canonical symmetries for higher-order singular systems, etc. Readers who are interested, please see Refs. [1-3] for details.

Recently, fractional calculus has gained much attention and has been applied to many fields ^[4-10]. Because fractional derivatives are elegant tools to deal with the dissipative forces for the nonconservative systems ^[11,12], fractional calculus of variations was studied by many scholars, such as Klimek^[13], Agrawal^[14,15], Muslih et al ^[16], Rabei et al ^[17], Malinowska and Torres^[18], and so on^[19-21]. Especially, Frederico and Torres ^[22] studied the fractional variational problems within mixed integer and Riemann-Liouville fractional derivatives and established the following fractional Euler-Lagrange equation

$\frac{\partial L}{\partial q_{i}} - \frac{d}{d t} \frac{\partial L}{\partial {\dot{q}}_{i}} + {}_{t}D D_{t_{2}}^{α} \frac{\partial L}{\partial {}_{t_{1}}D D_{t}^{α} q_{i}} = 0$ (1)

where $L = L (t, q_{j}, {\dot{q}}_{j}, {}_{t_{1}}D D_{t}^{α} q_{j})$ , $0 < α < 1$ , $q_{j}$ is the generalized coordinate, ${\dot{q}}_{j} = d q_{j} / d t$ is the generalized velocity, ${}_{t_{1}} D_{t}^{α} q_{j}$ means the Riemann-Liouville fractional derivative of $q_{j}$ , $q_{j} \in C^{2} ([t_{1}, t_{2}]; R)$ , $L (\cdot, \cdot, \cdot, \cdot) \in C^{2} ([t_{1}, t_{2}] \times R^{n} \times R^{n} \times R^{n}; R)$ , $i, j = 1,2, \dots, n$ .

After the differential equations of motion are established, one needs to find the solutions to them. Symmetry methods play the key role in solving differential equations ^[23-52]. Particularly, Lie symmetry, which means the invariance of the differential equations of motion, is one of the most important methods to help reduce the freedoms of differential equations (see Refs.[33-39] for a review). Therefore, in this paper, we intend to establish the Lagrange and Hamilton equations for the fractional singular Lagrangian $L = L (t, q_{j}, {\dot{q}}_{j}, {}_{t_{1}}D D_{t}^{α} q_{j})$ , and then study Lie symmetry and conserved quantity for the corresponding dynamic system.

It is noted that Eq. (1) in Ref. [22] is not correct for the Lagrangian $L = L (t, q_{j}, {\dot{q}}_{j}, {}_{t_{1}}D D_{t}^{α} q_{j})$ . Therefore, we begin with the fractional calculus of variations within mixed integer and Riemann-Liouville fractional derivatives. In this paper, we set $0 < α < 1$ .

1 Preliminaries

Several definitions and properties of fractional calculus are listed ^{[10, 14, 21]}.

Let $f (t)$ be a function, $t \in (t_{1}, t_{2})$ , then the left and the right Riemann-Liouville fractional integrals are defined as

${}_{t_{1}} D_{t}^{- α} f (t) = \frac{1}{Γ (α)} \int_{t_{1}}^{t} {(t - τ)}^{α - 1} f (τ) d τ, α > 0$ (2)

${}_{t} D_{t_{2}}^{- α} f (t) = \frac{1}{Γ (α)} \int_{t}^{t_{2}} {(τ - t)}^{α - 1} f (τ) d τ, α > 0$ (3)

the left and the right Riemann-Liouville fractional derivatives are defined as

${}_{t_{1}} D_{t}^{α} f (t) = \frac{1}{Γ (n - α)} {(\frac{d}{d t})}^{n} \int_{t_{1}}^{t} {(t - τ)}^{n - α - 1} f (τ) d τ, n - 1 \leq α < n$ (4)

${}_{t} D_{t_{2}}^{α} f (t) = \frac{1}{Γ (n - α)} {(- \frac{d}{d t})}^{n} \int_{t}^{t_{2}} {(τ - t)}^{n - α - 1} f (τ) d τ, n - 1 \leq α < n$ (5)

the left and the right Caputo fractional derivatives are defined as

${}_{t_{1}}^{C} D_{t}^{α} f (t) = \frac{1}{Γ (n - α)} \int_{t_{1}}^{t} {(t - τ)}^{n - α - 1} {(\frac{d}{d τ})}^{n} f (τ) d τ, n - 1 \leq α < n$ (6)

${}_{t}^{C} D_{t_{2}}^{α} f (t) = \frac{1}{Γ (n - α)} \int_{t}^{t_{2}} {(τ - t)}^{n - α - 1} {(- \frac{d}{d τ})}^{n} f (τ) d τ, n - 1 \leq α < n$ (7)

The left and the right Riemann-Liouville fractional derivatives satisfy the following formulae for integration by parts

$\int_{t_{1}}^{t_{2}} g (t) {}_{t_{1}}D D_{t}^{α} f (t) d t = \int_{t_{1}}^{t_{2}} f (t) {}_{t}^{C}D D_{t_{2}}^{α} g (t) d t + {\sum_{j = 0}^{n - 1} {}_{t_{1}} D_{t}^{j + α - n} f (t) {(- D)}^{n - 1 - j} g (t) |}_{t_{1}}^{t_{2}}$ (8)

$\int_{t_{1}}^{t_{2}} g (t) {}_{t}D D_{t_{2}}^{α} f (t) d t = \int_{t_{1}}^{t_{2}} f (t) {}_{t_{1}}^{C}D D_{t}^{α} g (t) d t - {\sum_{j = 0}^{n - 1} {}_{t} D_{t_{2}}^{j + α - n} f (t) {(D)}^{n - 1 - j} g (t) |}_{t_{1}}^{t_{2}}$ (9)

where $D = d / d t$ .

Specially, when $α \to 1$ , we have

${}_{t_{1}} D_{t}^{1} f (t) = {}_{t_{1}}^{C}D D_{t}^{1} f (t) = d f (t) / d t = \dot{f} (t), {}_{t}D D_{t_{2}}^{1} f (t) = {}_{t}^{C}D D_{t_{2}}^{1} f (t) = - d f (t) / d t = - \dot{f} (t)$ (10)

2 Fractional Constrained Hamiltonian System

2.1 Fractional Euler-Lagrange Equation

Hamilton action within mixed integer and Riemann-Liouville fractional derivatives is

$I [q_{j} (\cdot)] = \int_{t_{1}}^{t_{2}} L (t, q_{j}, {\dot{q}}_{j}, {}_{t_{1}}D D_{t}^{α} q_{j}) d t, j = 1,2, \dots, n$ (11)

The isochronous variation

$δ I = 0$ (12)

with $q_{j} (t_{1}) = q_{j 1}$ , $q_{j} (t_{2}) = q_{j 2}$ and $δ {}_{t_{1}}D D_{t}^{α} q_{j} = {}_{t_{1}}D D_{t}^{α} δ q_{j}$ is called Hamilton principle within mixed integer and Riemann-Liouville fractional derivatives, where $δ$ means the isochronous variation.

From Eq. (12), using the integration by parts formula (Eq. (8)), we have

$δ I = \int_{t_{1}}^{t_{2}} (\frac{\partial L}{\partial q_{i}} \cdot δ q_{i} + \frac{\partial L}{\partial {\dot{q}}_{i}} \cdot δ {\dot{q}}_{i} + \frac{\partial L}{\partial {}_{t_{1}}D D_{t}^{α} q_{i}} \cdot δ {}_{t_{1}}D D_{t}^{α} q_{i}) d t = 0$ (13)

where

$\int_{t_{1}}^{t_{2}} (\frac{\partial L}{\partial {\dot{q}}_{i}} \cdot δ {\dot{q}}_{i}) d t = \int_{t_{1}}^{t_{2}} (\frac{\partial L}{\partial {\dot{q}}_{i}} \cdot \frac{d}{d t} δ q_{i}) d t = \int_{t_{1}}^{t_{2}} [\frac{d}{d t} (\frac{\partial L}{\partial {\dot{q}}_{i}} \cdot δ q_{i}) - (\frac{d}{d t} \frac{\partial L}{\partial {\dot{q}}_{i}}) \cdot δ q_{i}] d t$

$= - \int_{t_{1}}^{t_{2}} [(\frac{d}{d t} \frac{\partial L}{\partial {\dot{q}}_{i}}) \cdot δ q_{i}] d t + {(\frac{\partial L}{\partial {\dot{q}}_{i}} \cdot δ q_{i}) |}_{t_{1}}^{t_{2}} = - \int_{t_{1}}^{t_{2}} [(\frac{d}{d t} \frac{\partial L}{\partial {\dot{q}}_{i}}) \cdot δ q_{i}] d t$ (14)

$\int_{t_{1}}^{t_{2}} (\frac{\partial L}{\partial {}_{t_{1}}D D_{t}^{α} q_{i}} \cdot δ {}_{t_{1}}D D_{t}^{α} q_{i}) d t = \int_{t_{1}}^{t_{2}} (\frac{\partial L}{\partial {}_{t_{1}}D D_{t}^{α} q_{i}} \cdot {}_{t_{1}}D D_{t}^{α} δ q_{i}) d t$

$= \int_{t_{1}}^{t_{2}} (δ q_{i} \cdot {}_{t}^{C}D D_{t_{2}}^{α} \frac{\partial L}{\partial {}_{t_{1}}D D_{t}^{α} q_{i}}) d t + {({}_{t_{1}} D_{t}^{α - 1} δ q_{i} \cdot \frac{\partial L}{\partial {}_{t_{1}}D D_{t}^{α} q_{i}}) |}_{t_{1}}^{t_{2}}$

$= \int_{t_{1}}^{t_{2}} (δ q_{i} \cdot {}_{t}^{C}D D_{t_{2}}^{α} \frac{\partial L}{\partial {}_{t_{1}}D D_{t}^{α} q_{i}}) d t + \frac{\partial L (t_{2})}{\partial {}_{t_{1}}D D_{t}^{α} q_{i}} \cdot \int_{t_{1}}^{t_{2}} [\frac{{(t_{2} - t)}^{- α}}{Γ (1 - α)} \cdot δ q_{i} (t)] d t$ (15)

Substituting Eqs. (14) and (15) into Eq. (13), considering the independence of $δ q_{i}$ , $i = 1,2, \dots, n$ and the arbitrariness of the interval $[t_{1}, t_{2}]$ , we get

$\frac{\partial L}{\partial q_{i}} - \frac{d}{d t} \frac{\partial L}{\partial {\dot{q}}_{i}} + {}_{t}^{C}D D_{t_{2}}^{α} \frac{\partial L}{\partial {}_{t_{1}}D D_{t}^{α} q_{i}} + \frac{\partial L (t_{2})}{\partial {}_{t_{1}}D D_{t}^{α} q_{i}} \cdot \frac{{(t_{2} - t)}^{- α}}{Γ (1 - α)} = 0, i = 1,2, \dots, n$ (16)

where $\frac{\partial L (t_{2})}{\partial {}_{t_{1}}D D_{t}^{α} q_{i}} = \frac{\partial L}{\partial {}_{t_{1}}D D_{t}^{α} q_{i}} (t_{2}, q_{j} (t_{2}), {\dot{q}}_{j} (t_{2}), {}_{t_{1}}D D_{t}^{α} q_{j} (t_{2}))$ . Eq. (16) is called the fractional Euler-Lagrange equation within mixed integer and Riemann-Liouville fractional derivatives.

Remark 1 Equation (16) is different from Eq. (1), which discards the second terms of Eq. (15).

For the fractional Lagrangian system (Eq. (16)), define the element $H_{i j}$ of the Hessian matrix $[H_{i j}]$ as

$H_{i j} = \frac{\partial^{2} L}{\partial {\dot{q}}_{i} \partial {\dot{q}}_{j}}, i, j = 1,2, \dots, n$ (17)

If $d e t [H_{i j}] = 0$ , the Hessian matrix is said to be degenerated and the corresponding Lagrangian is called singular. Furthermore, if the fractional Lagrangian system (Eq. (16)) is a singular one, then there exist inherent constraints when it is transformed into a fractional constrained Hamiltonian system.

2.2 Fractional Inherent Constraint

Define the generalized momenta and the Hamiltonian as

$p_{i} = \frac{\partial L (t, q_{j}, {\dot{q}}_{j}, {}_{t_{1}}D D_{t}^{α} q_{j})}{\partial {\dot{q}}_{i}}, p_{i}^{α} = \frac{\partial L (t, q_{j}, {\dot{q}}_{j}, {}_{t_{1}}D D_{t}^{α} q_{j})}{\partial {}_{t_{1}}D D_{t}^{α} q_{i}}$ (18)

$H_{C} = p_{i} \cdot {\dot{q}}_{i} + p_{i}^{α} \cdot ({}_{t_{1}} D_{t}^{α} q_{i}) - L (t, q_{j}, {\dot{q}}_{j}, {}_{t_{1}}D D_{t}^{α} q_{j}), i, j = 1,2, \dots, n$ (19)

In this paper, we assume that $_{t_{1}} D_{t}^{α} q_{i} = x_{i} (t, q_{j}, {\dot{q}}_{j}, p_{j}^{α})$ , $i, j = 1,2, \dots, n$ and $r a n k [H_{i j}] = R$ , $0 \leq R < n$ .

When $1 \leq R < n$ , we have

${\dot{q}}_{A} = f^{A} (t, q, p^{α}, p_{A}, {\dot{q}}_{B})$ (20)

where ${\dot{q}}_{A} = ({\dot{q}}_{1}, {\dot{q}}_{2}, \dots, {\dot{q}}_{R})$ , $f^{A} = (f^{1}, f^{2}, \dots, f^{R})$ , $q = (q_{1}, q_{2}, \dots, q_{n})$ , $p_{A} = (p_{1}, p_{2}, \dots, p_{R})$ , ${\dot{q}}_{B} = ({\dot{q}}_{R + 1}, {\dot{q}}_{R + 2}, \dots, {\dot{q}}_{n})$ and $p^{α} = (p_{1}^{α}, p_{2}^{α}, \dots, p_{n}^{α})$ . From Eq. (18), we get

$p_{i} = \frac{\partial L (t, q_{j}, {\dot{q}}_{j}, {}_{t_{1}}D D_{t}^{α} q_{j})}{\partial {\dot{q}}_{i}} = g_{i} (t, q, {\dot{q}}_{A}, {\dot{q}}_{B}, p^{α}) = g_{i} (t, q, p_{A}, {\dot{q}}_{B}, p^{α}), i = 1,2, \dots, n$ (21)

If $i = 1,2, \dots, R$ , then Eq. (21) is reasonable. When $i = R + 1, R + 2, \dots, n$ , $g_{i}$ will not depend on ${\dot{q}}_{B}$ , i.e.,

$p_{B} = g_{B} (t, q, p_{A}, p^{α})$ (22)

where $p_{B} = (p_{R + 1}, p_{R + 2}, \dots, p_{n})$ and $g_{B} = (g_{R + 1}, g_{R + 2}, \dots, g_{n})$ . We write Eq. (22) as

$ϕ_{B} (t, q, p, p^{α}) = p_{B} - g_{B} (t, q, p_{A}, p^{α}) = 0$ (23)

where $ϕ_{B} = (ϕ_{R + 1}, ϕ_{R + 2}, \dots, ϕ_{n})$ , $p = (p_{1}, p_{2}, \dots, p_{n})$ . Renumber Eq. (23), we can obtain

$ϕ_{a} (t, q, p, p^{α}) = 0, a = 1,2, \dots, n - R, 1 \leq R < n$ (24)

Equation (24) is the fractional inherent constraint, and we call it the fractional primary constraint. It is noted that the fractional primary constraint comes from the definition of the generalized momenta (Eq. (18)) rather than the fractional Euler-Lagrange equation (Eq. (16)).

Similarly, when $R = 0$ , we can get

$ϕ_{a} (t, q, p, p^{α}) = 0, a = 1,2, \dots, n$ (25)

Therefore, from Eqs. (24) and (25), we have

$ϕ_{a} (t, q, p, p^{α}) = 0, a = 1,2, \dots, n - R, 0 \leq R < n$ (26)

Equation (26) is the inherent constraint and is called fractional primary constraint.

2.3 Fractional Constrained Hamilton Equation

From Eqs. (18) and (19), we obtain

$δ H_{C} = {\dot{q}}_{i} \cdot δ p_{i} + δ p_{i}^{α} \cdot {}_{t_{1}}D D_{t}^{α} q_{i} - \frac{\partial L}{\partial q_{i}} δ q_{i}$ (27)

where

$\frac{\partial L}{\partial q_{i}} = \frac{d}{d t} \frac{\partial L}{\partial {\dot{q}}_{i}} - {}_{t}^{C}D D_{t_{2}}^{α} \frac{\partial L}{\partial {}_{t_{1}}D D_{t}^{α} q_{i}} - \frac{\partial L (t_{2})}{\partial {}_{t_{1}}D D_{t}^{α} q_{i}} \cdot \frac{{(t_{2} - t)}^{- α}}{Γ (1 - α)} = {\dot{p}}_{i} - {}_{t}^{C}D D_{t_{2}}^{α} p_{i}^{α} - p_{i}^{α} (t_{2}) \cdot \frac{{(t_{2} - t)}^{- α}}{Γ (1 - α)}$ (28)

Because $H_{C} = H_{C} (t, q, p, p^{α})$ , we have

$δ H_{C} = \frac{\partial H_{C}}{\partial p_{i}} δ p_{i} + \frac{\partial H_{C}}{\partial q_{i}} δ q_{i} + \frac{\partial H_{C}}{\partial p_{i}^{α}} δ p_{i}^{α}$ (29)

From Eq. (26), we have

$δ ϕ_{a} (t, q, p, p^{α}) = \frac{\partial ϕ_{a}}{\partial q_{i}} δ q_{i} + \frac{\partial ϕ_{a}}{\partial p_{i}} δ p_{i} + \frac{\partial ϕ_{a}}{\partial p_{i}^{α}} δ p_{i}^{α} = 0$ (30)

Making use of the Lagrange multipliers $λ_{a} (t)$ , $a = 1,2, \dots, n - R$ , $0 \leq R < n$ , from Eqs. (27) - (30), we get

${\dot{q}}_{i} = \frac{\partial H_{C}}{\partial p_{i}} + λ_{a} \frac{\partial ϕ_{a}}{\partial p_{i}}, {}_{t_{1}}D D_{t}^{α} q_{i} = \frac{\partial H_{C}}{\partial p_{i}^{α}} + λ_{a} \frac{\partial ϕ_{a}}{\partial p_{i}^{α}}, {\dot{p}}_{i} = {}_{t}^{C}D D_{t_{2}}^{α} p_{i}^{α} + p_{i}^{α} (t_{2}) \cdot \frac{{(t_{2} - t)}^{- α}}{Γ (1 - α)} - \frac{\partial H_{C}}{\partial q_{i}} - λ_{a} \frac{\partial ϕ_{a}}{\partial q_{i}}$ (31)

Equation (31) is called the fractional constrained Hamilton equation (also called fractional constrained canonical equation) within mixed integer and Riemann-Liouville fractional derivatives.

Remark 2 When the element ${}_{t_{1}} D_{t}^{α} q_{j}$ in $L (t, q_{j}, {\dot{q}}_{j}, {}_{t_{1}}D D_{t}^{α} q_{j})$ is omitted, the fractional constrained Hamilton equation (Eq. (31)) will reduce to the classical constrained Hamilton equation in Ref. [2].

2.4 Consistency Condition

In this section, the main purpose is to solve the Lagrange multipliers $λ_{a} (t)$ , $a = 1,2, \dots, n - R$ , $0 \leq R < n$ .

Let $F = F (t, q, p, p^{α})$ , $G = G (t, q, p, p^{α})$ , then Poisson bracket is defined as

${F, G} = \frac{\partial F}{\partial q_{i}} \frac{\partial G}{\partial p_{i}} - \frac{\partial F}{\partial p_{i}} \frac{\partial G}{\partial q_{i}}$ (32)

Let $H_{T} = H_{C} + λ_{a} ϕ_{a}$ , $a = 1,2, \dots, n - R$ , $0 \leq R < n$ , then we have

${ϕ_{a}, H_{T}} = \frac{\partial ϕ_{a}}{\partial q_{i}} {\dot{q}}_{i} - \frac{\partial ϕ_{a}}{\partial p_{i}} ({}_{t}^{C} D_{t_{2}}^{α} p_{i}^{α} + p_{i}^{α} (t_{2}) \cdot \frac{{(t_{2} - t)}^{- α}}{Γ (1 - α)} - {\dot{p}}_{i})$ (33)

It follows from ${\dot{ϕ}}_{a} = 0$ , $a = 1,2, \dots, n - R$ , $0 \leq R < n$ , that

${ϕ_{a}, H_{C}} + λ_{b} {ϕ_{a}, ϕ_{b}} + \frac{\partial ϕ_{a}}{\partial t} + \frac{\partial ϕ_{a}}{\partial p_{i}^{α}} \cdot {\dot{p}}_{i}^{α} + \frac{\partial ϕ_{a}}{\partial p_{i}} \cdot ({}_{t}^{C} D_{t_{2}}^{α} p_{i}^{α} + p_{i}^{α} (t_{2}) \cdot \frac{{(t_{2} - t)}^{- α}}{Γ (1 - α)}) = 0$ (34)

where $a, b = 1,2, \dots, n - R$ , $0 \leq R < n$ , $i = 1,2, \dots, n$ . Eq. (34) is called the consistency condition of the fractional primary constraint.

3 Lie Symmetry and Conserved Quantity

3.1 Lie Symmetry

Lie symmetry means the invariance of the differential equations of motion (Eq. (31)) under the infinitesimal transformations

$\bar{t} = t + Δ t, {\bar{q}}_{i} (\bar{t}) = q_{i} (t) + Δ q_{i}, {\bar{p}}_{i} (\bar{t}) = p_{i} (t) + Δ p_{i}, {\bar{p}}_{i}^{α} (\bar{t}) = p_{i}^{α} (t) + Δ p_{i}^{α}$ (35)

whose expanding expressions are

$\bar{t} = t + θ ξ_{0} (t, q, p, p^{α}) + ο (θ), {\bar{q}}_{i} (\bar{t}) = q_{i} (t) + θ ξ_{i} (t, q, p, p^{α}) + ο (θ)$

${\bar{p}}_{i} (\bar{t}) = p_{i} (t) + θ η_{i} (t, q, p, p^{α}) + ο (θ), {\bar{p}}_{i}^{α} (\bar{t}) = p_{i}^{α} (t) + θ η_{i}^{α} (t, q, p, p^{α}) + ο (θ)$ (36)

where $θ$ is a small parameter, $ξ_{0}$ , $ξ_{i}$ , $η_{i}$ and $η_{i}^{α}$ are called infinitesimal generators of the infinitesimal transformations. In this paper, we only consider the linear part of $θ$ and neglect its higher order.

Expanding Eq. (31) and denoting

${\dot{q}}_{i} = s_{i} (t, q, p, p^{α})$ (37)

${}_{t_{1}} D_{t}^{α} q_{i} = h_{i} (t, q, p, p^{α})$ (38)

${\dot{p}}_{i} = {}_{t}^{C}D D_{t_{2}}^{α} p_{i}^{α} + f_{i} (t, q, p, p^{α})$ (39)

For Eq. (37), we have

${\dot{\bar{q}}}_{i} - s_{i} (\bar{t}, \bar{q}, \bar{p}, {\bar{p}}^{α}) = \frac{d {\bar{q}}_{i}}{d \bar{t}} - s_{i} (t + Δ t, q + Δ q, p + Δ p, p^{α} + Δ p^{α})$

$= {\dot{q}}_{i} - s_{i} (t, q, p, p^{α}) + θ [{\dot{ξ}}_{i} - {\dot{q}}_{i} {\dot{ξ}}_{0} - X^{(0)} (s_{i})]$ (40)

where $X^{(0)} = ξ_{0} \frac{\partial}{\partial t} + ξ_{k} \frac{\partial}{\partial q_{k}} + η_{k} \frac{\partial}{\partial p_{k}} + η_{k}^{α} \frac{\partial}{\partial p_{k}^{α}}$ , $k = 1,2, \dots, n$ .

For Eq. (38), we have

${}_{{\bar{t}}_{1}} D_{\bar{t}}^{α} {\bar{q}}_{i} - h_{i} (\bar{t}, \bar{q}, \bar{p}, {\bar{p}}^{α}) = {}_{t_{1}}D D_{t}^{α} q_{i} - h_{i} (t, q, p, p^{α})$

$+ θ [{}_{t_{1}} D_{t}^{α} (ξ_{i} - {\dot{q}}_{i} ξ_{0}) - X^{(0)} (h_{i}) + ξ_{0} \frac{d}{d t} {}_{t_{1}}D D_{t}^{α} q_{i} - \frac{1}{Γ (1 - α)} \frac{d}{d t} ({(t - t_{1})}^{- α} q_{i} (t_{1}) ξ_{0} (t_{1}))]$ (41)

where $ξ_{0} (t_{1}) = ξ_{0} (t_{1}, q (t_{1}), p (t_{1}), p^{α} (t_{1}))$ .

For Eq. (39), we have

${\dot{\bar{p}}}_{i} - {}_{\bar{t}}^{C}D D_{{\bar{t}}_{2}}^{α} {\bar{p}}_{i}^{α} - f_{i} (\bar{t}, \bar{q}, \bar{p}, {\bar{p}}^{α}) = {\dot{p}}_{i} - {}_{t}^{C}D D_{t_{2}}^{α} p_{i}^{α} - f_{i} (t, q, p, p^{α}) + θ [{\dot{η}}_{i} - ξ_{0} \frac{d}{d t} {}_{t}^{C}D D_{t_{2}}^{α} p_{i}^{α}$

$- {\dot{p}}_{i} {\dot{ξ}}_{0} - {}_{t}^{C}D D_{t_{2}}^{α} (η_{i}^{α} - {\dot{p}}_{i}^{α} ξ_{0}) + \frac{1}{Γ (1 - α)} {(t_{2} - t)}^{- α} {\dot{p}}_{i}^{α} (t_{2}) ξ_{0} (t_{2}) - X^{(0)} (f_{i})]$ (42)

where $ξ_{0} (t_{2}) = ξ_{0} (t_{2}, q (t_{2}), p (t_{2}), p^{α} (t_{2}))$ .

For the fractional primary constraint Eq. (26), we have

$ϕ_{a} (\bar{t}, \bar{q}, \bar{p}, {\bar{p}}^{α}) = ϕ_{a} (t, q, p, p^{α}) + θ X^{(0)} (ϕ_{a})$ (43)

Lie symmetry requires that the coefficients of $θ$ are zero in Eqs. (40) - (43), then we obtain

${\dot{ξ}}_{i} - {\dot{q}}_{i} {\dot{ξ}}_{0} = X^{(0)} (s_{i}), {}_{t_{1}}D D_{t}^{α} (ξ_{i} - {\dot{q}}_{i} ξ_{0}) + ξ_{0} \frac{d}{d t} {}_{t_{1}}D D_{t}^{α} q_{i} - \frac{q_{i} (t_{1}) ξ_{0} (t_{1})}{Γ (1 - α)} \frac{d}{d t} {(t - t_{1})}^{- α} = X^{(0)} (h_{i}),$

${\dot{η}}_{i} - {\dot{p}}_{i} {\dot{ξ}}_{0} - ξ_{0} \frac{d}{d t} {}_{t}^{C}D D_{t_{2}}^{α} p_{i}^{α} - {}_{t}^{C}D D_{t_{2}}^{α} (η_{i}^{α} - {\dot{p}}_{i}^{α} ξ_{0}) + \frac{1}{Γ (1 - α)} {(t_{2} - t)}^{- α} {\dot{p}}_{i}^{α} (t_{2}) ξ_{0} (t_{2}) = X^{(0)} (f_{i})$ (44)

and

$X^{(0)} (ϕ_{a}) = 0$ (45)

Equation (44) is called determined equation and Eq. (45) is called limited equation.

Definition 1 For the fractional constrained Hamiltonian system within mixed integer and Riemann-Liouville fractional derivatives (31), if the infinitesimal generators $ξ_{0}$ , $ξ_{i}$ , $η_{i}$ and $η_{i}^{α}$ satisfy the determined equation (44), then the corresponding symmetry is called Lie symmetry.

Definition 2 For the fractional constrained Hamiltonian system within mixed integer and Riemann-Liouville fractional derivatives (31), if the infinitesimal generators $ξ_{0}$ , $ξ_{i}$ , $η_{i}$ and $η_{i}^{α}$ satisfy both the determined equation (44) and the limited equation (45), then the corresponding symmetry is called weak Lie symmetry.

However, if we consider the deduction process of the fractional constrained Hamilton equation (31), then another condition will be exposed on the infinitesimal generators $ξ_{0}$ , $ξ_{i}$ , $η_{i}$ and $η_{i}^{α}$ , and a new Lie symmetry will be defined.

Substituting Eq. (36) into Eq. (30), we get

$\frac{\partial ϕ_{a}}{\partial q_{i}} (ξ_{i} - {\dot{q}}_{i} ξ_{0}) + \frac{\partial ϕ_{a}}{\partial p_{i}} (η_{i} - {\dot{p}}_{i} ξ_{0}) + \frac{\partial ϕ_{a}}{\partial p_{i}^{α}} (η_{i}^{α} - {\dot{p}}_{i}^{α} ξ_{0}) = 0$ (46)

Equation (46) is called additional limited equation.

Definition 3 For the fractional constrained Hamiltonian system within mixed integer and Riemann-Liouville fractional derivatives (31), if the infinitesimal generators $ξ_{0}$ , $ξ_{i}$ , $η_{i}$ and $η_{i}^{α}$ satisfy the determined equation (44), the limited equation (45) and the additional limited equation (46), then the corresponding symmetry is called strong Lie symmetry.

3.2 Conserved Quantity

Lie symmetry may not lead to a conserved quantity. However, under some conditions, a conserved quantity can be deduced from Lie symmetry. Of course, different conditions lead to different results.

Definition 4 A quantity $I$ is called a conserved quantity if and only if $d I / d t = 0$ .

Theorem 1 For the infinitesimal generators $ξ_{0}$ , $ξ_{i}$ , $η_{i}$ and $η_{i}^{α}$ , which meet the determined equation (44), if there exists a gauge function $G (t, q, p, p^{α})$ satisfying the following structural equation

$p_{i} {\dot{ξ}}_{i} + p_{i}^{α} {}_{t_{1}}D D_{t}^{α} (ξ_{i} - {\dot{q}}_{i} ξ_{0}) + (p_{i}^{α} \frac{d}{d t} {}_{t_{1}}D D_{t}^{α} q_{i} - \frac{\partial H_{C}}{\partial t}) ξ_{0} - \frac{\partial H_{C}}{\partial q_{i}} ξ_{i} + λ_{a} \frac{\partial ϕ_{a}}{\partial p_{i}^{α}} η_{i}^{α}$

$+ λ_{a} \frac{\partial ϕ_{a}}{\partial p_{i}} η_{i} - \frac{q_{i} (t_{1}) ξ_{0} (t_{1})}{Γ (1 - α)} p_{i}^{α} \frac{d}{d t} {(t - t_{1})}^{- α} + (p_{i}^{α} {}_{t_{1}}D D_{t}^{α} q_{i} - H_{C}) {\dot{ξ}}_{0} + \dot{G} = 0$ (47)

then the following Lie symmetry conserved quantity exists for the fractional constrained Hamiltonian system (31)

$I = p_{i} ξ_{i} + (p_{i}^{α} {}_{t_{1}}D D_{t}^{α} q_{i} - H_{C}) ξ_{0} + {p_{i}^{α} {}_{t_{1}}D D_{τ}^{α} (ξ_{i} - {\dot{q}}_{i} ξ_{0}) - (ξ_{i} - {\dot{q}}_{i} ξ_{0}) [{}_{τ}^{C} D_{t_{2}}^{α} p_{i}^{α} + p_{i}^{α} (t_{2}) \cdot \frac{{(t_{2} - τ)}^{- α}}{Γ (1 - α)}]} d τ$

$- \frac{q_{i} (t_{1}) ξ_{0} (t_{1})}{Γ (1 - α)} \int_{t_{1}}^{t} p_{i}^{α} \frac{d}{d τ} {(τ - t_{1})}^{- α} d τ + G = c o n s t$ (48)

Proof Using Eqs. (26), (31) and (47), we have

$\frac{d}{d t} I = {\dot{p}}_{i} ξ_{i} + p_{i} {\dot{ξ}}_{i} + ({\dot{p}}_{i}^{α} {}_{t_{1}}D D_{t}^{α} q_{i} + p_{i}^{α} \cdot \frac{d}{d t} {}_{t_{1}}D D_{t}^{α} q_{i} - \frac{\partial H_{C}}{\partial t} - \frac{\partial H_{C}}{\partial q_{i}} {\dot{q}}_{i} - \frac{\partial H_{C}}{\partial p_{i}} {\dot{p}}_{i} - \frac{\partial H_{C}}{\partial p_{i}^{α}} {\dot{p}}_{i}^{α}) ξ_{0}$

$- (ξ_{i} - {\dot{q}}_{i} ξ_{0}) [{}_{t}^{C} D_{t_{2}}^{α} p_{i}^{α} + p_{i}^{α} (t_{2}) \cdot \frac{{(t_{2} - t)}^{- α}}{Γ (1 - α)}] - \frac{q_{i} (t_{1}) ξ_{0} (t_{1})}{Γ (1 - α)} p_{i}^{α} \frac{d}{d t} {(t - t_{1})}^{- α} + \dot{G}$

$+ (p_{i}^{α} {}_{t_{1}}D D_{t}^{α} q_{i} - H_{C}) {\dot{ξ}}_{0} + p_{i}^{α} {}_{t_{1}}D D_{t}^{α} (ξ_{i} - {\dot{q}}_{i} ξ_{0})$

$= \frac{\partial H_{C}}{\partial q_{i}} ξ_{i} - λ_{a} \frac{\partial ϕ_{a}}{\partial p_{i}^{α}} η_{i}^{α} - λ_{a} \frac{\partial ϕ_{a}}{\partial p_{i}} η_{i} + ({\dot{p}}_{i}^{α} {}_{t_{1}}D D_{t}^{α} q_{i} - \frac{\partial H_{C}}{\partial q_{i}} {\dot{q}}_{i} - \frac{\partial H_{C}}{\partial p_{i}} {\dot{p}}_{i} - \frac{\partial H_{C}}{\partial p_{i}^{α}} {\dot{p}}_{i}^{α}) ξ_{0}$

$+ {\dot{p}}_{i} ξ_{i} - (ξ_{i} - {\dot{q}}_{i} ξ_{0}) [{}_{t}^{C} D_{t_{2}}^{α} p_{i}^{α} + p_{i}^{α} (t_{2}) \cdot \frac{{(t_{2} - t)}^{- α}}{Γ (1 - α)}]$

$= (ξ_{i} - {\dot{q}}_{i} ξ_{0}) (\frac{\partial H_{C}}{\partial q_{i}} - {}_{t}^{C}D D_{t_{2}}^{α} p_{i}^{α} - p_{i}^{α} (t_{2}) \cdot \frac{{(t_{2} - t)}^{- α}}{Γ (1 - α)} + {\dot{p}}_{i}) - λ_{a} \frac{\partial ϕ_{a}}{\partial p_{i}^{α}} δ p_{i}^{α} - λ_{a} \frac{\partial ϕ_{a}}{\partial p_{i}} δ p_{i} = 0$

This proof is completed.

Theorem 2 For the infinitesimal generators $ξ_{0}$ , $ξ_{i}$ , $η_{i}$ and $η_{i}^{α}$ , which meet the determined equation (44) and the limited equation (45), if there exists a gauge function $G (t, q, p, p^{α})$ satisfying the structural equation (47), then a weak Lie symmetry conserved quantity (Eq. (48)) exists for the fractional constrained Hamiltonian system (31).

Theorem 3 For the infinitesimal generators $ξ_{0}$ , $ξ_{i}$ , $η_{i}$ and $η_{i}^{α}$ , which meet the determined equation (44), the limited equation (45) and the additional limited equation (46), if there exists a gauge function $G (t, q, p, p^{α})$ satisfying the structural equation (47), then a strong Lie symmetry conserved quantity (Eq. (48)) exists for the fractional constrained Hamiltonian system (31).

Remark 3 When the element ${}_{t_{1}} D_{t}^{α} q_{j}$ in $L (t, q_{j}, {\dot{q}}_{j}, {}_{t_{1}}D D_{t}^{α} q_{j})$ is omitted, Theorem 1 - Theorem 3 will reduce to the integer order cases, which are consistent with the results in Ref. [39].

4 An Example

We study Lie symmetry for the following fractional singular Lagrangian system

$L = {\dot{q}}_{1} q_{2} - q_{1} {\dot{q}}_{2} + {(q_{1})}^{2} + {(q_{2})}^{2} + \frac{1}{2} [{({}_{t_{1}} D_{t}^{α} q_{1})}^{2} + {({}_{t_{1}} D_{t}^{α} q_{2})}^{2}]$ (49)

From Eq. (49) we have

$p_{1} = \frac{\partial L}{\partial {\dot{q}}_{1}} = q_{2}, p_{2} = \frac{\partial L}{\partial {\dot{q}}_{2}} = - q_{1}, p_{1}^{α} = \frac{\partial L}{\partial {}_{t_{1}}D D_{t}^{α} q_{1}} = {}_{t_{1}}D D_{t}^{α} q_{1}, p_{2}^{α} = \frac{\partial L}{\partial {}_{t_{1}}D D_{t}^{α} q_{2}} = {}_{t_{1}}D D_{t}^{α} q_{2}$ (50)

$H_{C} = p_{1} {\dot{q}}_{1} + p_{2} {\dot{q}}_{2} + p_{1}^{α} \cdot {}_{t_{1}}D D_{t}^{α} q_{1} + p_{2}^{α} \cdot {}_{t_{1}}D D_{t}^{α} q_{2} - L = \frac{1}{2} [{(p_{1}^{α})}^{2} + {(p_{2}^{α})}^{2}] - {(q_{1})}^{2} - {(q_{2})}^{2}$ (51)

Therefore, there exist two fractional primary constraints

$ϕ_{1} = p_{1} - q_{2} = 0, ϕ_{2} = p_{2} + q_{1} = 0$ (52)

From Eq. (34), we get

$λ_{1} = - q_{2} - \frac{1}{2} [{}_{t}^{C} D_{t_{2}}^{α} p_{2}^{α} + p_{2}^{α} (t_{2}) \frac{{(t_{2} - t)}^{- α}}{Γ (1 - α)}], λ_{2} = q_{1} + \frac{1}{2} [{}_{t}^{C} D_{t_{2}}^{α} p_{1}^{α} + p_{1}^{α} (t_{2}) \frac{{(t_{2} - t)}^{- α}}{Γ (1 - α)}]$ (53)

Then Eq. (31) gives the following fractional constrained Hamilton equation

${\dot{q}}_{1} = - q_{2} - \frac{1}{2} [{}_{t}^{C} D_{t_{2}}^{α} p_{2}^{α} + p_{2}^{α} (t_{2}) \frac{{(t_{2} - t)}^{- α}}{Γ (1 - α)}], {\dot{q}}_{2} = q_{1} + \frac{1}{2} [{}_{t}^{C} D_{t_{2}}^{α} p_{1}^{α} + p_{1}^{α} (t_{2}) \frac{{(t_{2} - t)}^{- α}}{Γ (1 - α)}],$

${\dot{p}}_{1} = q_{1} + \frac{1}{2} [{}_{t}^{C} D_{t_{2}}^{α} p_{1}^{α} + p_{1}^{α} (t_{2}) \frac{{(t_{2} - t)}^{- α}}{Γ (1 - α)}], {\dot{p}}_{2} = q_{2} + \frac{1}{2} [{}_{t}^{C} D_{t_{2}}^{α} p_{2}^{α} + p_{2}^{α} (t_{2}) \frac{{(t_{2} - t)}^{- α}}{Γ (1 - α)}],$

${}_{t_{1}} D_{t}^{α} q_{1} = p_{1}^{α}, {}_{t_{1}}D D_{t}^{α} q_{2} = p_{2}^{α}$ (54)

The determined equation (44) gives

${\dot{ξ}}_{1} - {\dot{q}}_{1} {\dot{ξ}}_{0} = - ξ_{0} \frac{\partial}{\partial t} [\frac{1}{2} p_{2}^{α} (t_{2}) \frac{{(t_{2} - t)}^{- α}}{Γ (1 - α)}] - ξ_{2}$

${\dot{ξ}}_{2} - {\dot{q}}_{2} {\dot{ξ}}_{0} = ξ_{0} \frac{\partial}{\partial t} [\frac{1}{2} p_{1}^{α} (t_{2}) \frac{{(t_{2} - t)}^{- α}}{Γ (1 - α)}] + ξ_{1}$

${}_{t_{1}} D_{t}^{α} (ξ_{1} - {\dot{q}}_{1} ξ_{0}) + ξ_{0} \frac{d}{d t} {}_{t_{1}}D D_{t}^{α} q_{1} - \frac{1}{Γ (1 - α)} q_{1} (t_{1}) ξ_{0} (t_{1}) \frac{d}{d t} {(t - t_{1})}^{- α} = η_{1}^{α}$

${}_{t_{1}} D_{t}^{α} (ξ_{2} - {\dot{q}}_{2} ξ_{0}) + ξ_{0} \frac{d}{d t} {}_{t_{1}}D D_{t}^{α} q_{2} - \frac{1}{Γ (1 - α)} q_{2} (t_{1}) ξ_{0} (t_{1}) \frac{d}{d t} {(t - t_{1})}^{- α} = η_{2}^{α}$

${\dot{η}}_{1} - {\dot{p}}_{1} {\dot{ξ}}_{0} - ξ_{0} \frac{d}{d t} {}_{t}^{C}D D_{t_{2}}^{α} p_{1}^{α} - {}_{t}^{C}D D_{t_{2}}^{α} (η_{1}^{α} - {\dot{p}}_{1}^{α} ξ_{0}) + \frac{1}{Γ (1 - α)} {(t_{2} - t)}^{- α} {\dot{p}}_{1}^{α} (t_{2}) ξ_{0} (t_{2})$

$= ξ_{0} \frac{\partial}{\partial t} [\frac{1}{2} p_{1}^{α} (t_{2}) \frac{{(t_{2} - t)}^{- α}}{Γ (1 - α)}] + ξ_{1}$

${\dot{η}}_{2} - {\dot{p}}_{2} {\dot{ξ}}_{0} - ξ_{0} \frac{d}{d t} {}_{t}^{C}D D_{t_{2}}^{α} p_{2}^{α} - {}_{t}^{C}D D_{t_{2}}^{α} (η_{2}^{α} - {\dot{p}}_{2}^{α} ξ_{0}) + \frac{1}{Γ (1 - α)} {(t_{2} - t)}^{- α} {\dot{p}}_{2}^{α} (t_{2}) ξ_{0} (t_{2})$

$= ξ_{0} \frac{\partial}{\partial t} [\frac{1}{2} p_{2}^{α} (t_{2}) \frac{{(t_{2} - t)}^{- α}}{Γ (1 - α)}] + ξ_{2}$ (55)

The limited equation (45) gives

$- ξ_{2} + η_{1} = 0, ξ_{1} + η_{2} = 0$ (56)

The additional limited equation (46) gives

$- ξ_{2} + η_{1} + ({\dot{q}}_{2} - {\dot{p}}_{1}) ξ_{0} = 0, ξ_{1} + η_{2} - ({\dot{p}}_{2} + {\dot{q}}_{1}) ξ_{0} = 0$ (57)

Taking calculation, we find that

$ξ_{0} = 1, ξ_{1} = ξ_{2} = 0, η_{1} = η_{2} = 0, η_{1}^{α} = η_{2}^{α} = 0, G = 0$ (58)

satisfy the structural equation (47) and they also meet the determined equation (55) under the condition $p_{1}^{α} (t_{2}) = p_{2}^{α} (t_{2}) = 0$ . Therefore, Lie symmetry conserved quantity can be obtained from Theorem 1 as

$I = \int_{t_{1}}^{t} (p_{1}^{α} \frac{d}{d τ} {}_{t_{1}}D D_{τ}^{α} q_{1} + p_{2}^{α} \frac{d}{d τ} {}_{t_{1}}D D_{τ}^{α} q_{2} - {\dot{q}}_{1} {}_{τ}^{C}D D_{t_{2}}^{α} p_{1}^{α} - {\dot{q}}_{2} {}_{τ}^{C}D D_{t_{2}}^{α} p_{2}^{α}) d τ$

$- [\frac{1}{2} {(p_{1}^{α})}^{2} + \frac{1}{2} {(p_{2}^{α})}^{2} + {(q_{1})}^{2} + {(q_{2})}^{2}]$ (59)

Because the solution (58) also meets the limited equation (56) and the additional limited equation (57), Eq. (59) is also a strong Lie symmetry conserved quantity.

Specially, if the element ${}_{t_{1}} D_{t}^{α} q_{j}$ in $L (t, q_{j}, {\dot{q}}_{j}, {}_{t_{1}}D D_{t}^{α} q_{j})$ is omitted, then this example will reduce to the integer order case, in which the strong Lie symmetry conserved quantity has the form

$I = {(q_{1})}^{2} + {(q_{2})}^{2}$ (60)

5 Conclusion

Lie symmetry and conserved quantity are studied for the fractional constrained Hamiltonian system within mixed integer and Riemann-Liouville fractional derivatives here. The study of the fractional singular system just begins, and a lot of work is deserving to be done, such as, other symmetry methods, singular systems with only fractional derivatives, singular systems on time scales, and so on.

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[1] Li Z P. Classical and Quantal Dynamics of Contrained Systems and Their Symmetrical Properties [M]. Beijing: Beijing Polytechnic University Press, 1993(Ch). [Google Scholar]

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