Noether Theorem for Fractional Singular Systems

Chuanjing SONG; Xianghua ZHAI

doi:10.1051/wujns/2023283207

Open Access

Issue		Wuhan Univ. J. Nat. Sci. Volume 28, Number 3, June 2023


Page(s)		207 - 216
DOI		https://doi.org/10.1051/wujns/2023283207
Published online		13 July 2023

Wuhan University Journal of Natural Sciences, 2023, Vol.28 No.3, 207-216

Mathematics

CLC number: O316

Noether Theorem for Fractional Singular Systems

Chuanjing SONG¹ and Xianghua ZHAI²

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

Received: 11 June 2022

Abstract

Noether theorems for two fractional singular systems are discussed. One system involves mixed integer and Caputo fractional derivatives, and the other involves only Caputo fractional derivatives. Firstly, the fractional primary constraints and the fractional constrained Hamilton equations are given. Then, the fractional Noether theorems of the two fractional singular systems are established, including the fractional Noether identities, the fractional Noether quasi-identities and the fractional conserved quantities. Finally, the results obtained are illustrated by two examples.

Key words: singular system / fractional primary constraint / fractional constrained Hamilton equation / Noether theorem / conserved quantity

Biography: SONG Chuanjing, female, Ph. D., Associate professor, research direction: mathematical methods of classical mechanics. E-mail: This email address is being protected from spambots. You need JavaScript enabled to view it.

Fundation item: Supported by the National Natural Science Foundation of China (12172241, 12002228, 12272248, 11972241) and Qing Lan Project of Colleges and Universities in Jiangsu Province

© Wuhan University 2023

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.

DOI https://doi.org/10.1051/wujns/2023283207

0 Introduction

Given a Lagrangian $L = L (t, q, \dot{q})$ Mathematical equation , $q = (q_{1}, q_{2}, \dots, q_{n})$ , $\dot{q} = ({\dot{q}}_{1}, {\dot{q}}_{2}, \dots, {\dot{q}}_{n})$ , we define

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

where the matrix $[H_{i j}]$ Mathematical equation is called a Hessian matrix. If $d e t [H_{i j}] \neq 0$ , then the Lagrangian is called regular. If $d e t [H_{i j}] = 0$ , then the Lagrangian is called singular. For example, the Parra's Lagrangian^[1]

$L = \frac{1}{2} m ({\dot{q}}_{1}^{2} + {\dot{q}}_{2}^{2} + l^{2} {\dot{q}}_{3}^{2} + 2 l {\dot{q}}_{1} {\dot{q}}_{3} c o s q_{3} + 2 l {\dot{q}}_{2} {\dot{q}}_{3} s i n q_{3}) + V (q_{1}, q_{2}, q_{3})$ Mathematical equation (2)

the Deriglazov's Lagrangian^[2]

$L = q_{2}^{2} {\dot{q}}_{1}^{2} + q_{1}^{2} {\dot{q}}_{2}^{2} + 2 q_{1} q_{2} {\dot{q}}_{1} {\dot{q}}_{2} + V (q_{1}, q_{2}) (V = q_{1}^{2} + q_{2}^{2})$ Mathematical equation (3)

the Cawley's Lagrangian^[3]

$L = {\dot{q}}_{1} {\dot{q}}_{2} + V (q_{1}, q_{2}, q_{3}) (V = \frac{1}{2} q_{2} q_{3}^{2})$ Mathematical equation (4)

and the Mittelstaedt's Lagrangian^[4]

$L = \frac{1}{2 m} {({\dot{q}}_{1} + {\dot{q}}_{2})}^{2} + \frac{1}{2 μ} {\dot{q}}_{3}^{2} + V (q_{1}, q_{2}, q_{3})$ Mathematical equation (5)

where $l$ Mathematical equation , $m$ and $μ$ are constants and $V$ represents potential energy, they are all singular. In fact, singular systems have a close relationship with the condensed matter theories, the gauge field theories, the quantum field theories of anyons, the particle physics and so forth^[5-7]. Regarding the singular systems, Dirac^[8] was the first one to study their canonical equation. Then, singular systems were also investigated in several classical mechanics textbooks^[5-7], including both the canonical equation and the Noether theorem.

The Noether theorem was introduced by the German female mathematician Noether^[9]. The Noether symmetry method is one of the methods used to solve the differential equations of motion. There are many results on the Noether theorem^[10-12]. Fractional calculus has been popular recently. The fractional derivatives that are used most often are the Riemann-Liouville, Caputo and Riesz fractional derivatives. In 2007, Frederico and Torres^[13] initiated the study of the fractional Noether theorem. Based on a bilinear operator $D$ Mathematical equation ( $D (I) = 0$ ), they defined the fractional conserved quantity. Using this definition, fractional Noether theorems and their applications were discussed for several mechanics systems, such as the Lagrangian system^[14-17], the Birkhoffian system^[18,19], the Hamiltonian system^[20], and the multidimensional Lagrangian system^[21]. Two years later, with the idea of the definition of the classical conserved quantity, Atanacković et al^[22] introduced another definition of the fractional conserved quantity ( $d I / d t = 0$ Mathematical equation ). They held the point that this definition is more reasonable than the former definition. Then, fractional Noether theorems of the different mechanics systems, such as the Birkhoffian system^[23-25], the Hamiltonian system^[26,27] and the nonconservative system^[28,29] were obtained.

At present, two fractional singular systems, one concerning the mixed integer and Caputo fractional derivatives and the other concerning the Caputo fractional derivatives, have been established, including the fractional primary constraints and the fractional constrained Hamilton equations^[30]. The next task is to find the solutions to them. Therefore, in this paper, we intend to make use of the Noether symmetry method to complete this study.

1 Preliminaries

We give the definitions of the Riemann-Liouville and the Caputo fractional derivatives as follows. Given a function $f (t)$ Mathematical equation and two constants $α$ and $β$ that satisfy $n - 1 \leq α, β < n$ , where $n$ is an integer, the Riemann-Liouville fractional derivative and the Caputo fractional derivative have the forms^[31]

${}_{t_{1}}^{R L} D_{t}^{α} f (t) = \frac{1}{Γ (n - α)} {(\frac{d}{d t})}^{n} \int_{t_{1}}^{t} {(t - ξ)}^{n - α - 1} f (ξ) d ξ$ Mathematical equation (6)

${}_{t}^{R L} D_{t_{2}}^{β} f (t) = \frac{1}{Γ (n - β)} {(- \frac{d}{d t})}^{n} \int_{t}^{t_{2}} {(ξ - t)}^{n - β - 1} f (ξ) d ξ$ Mathematical equation (7)

${}_{t_{1}}^{C} D_{t}^{α} f (t) = \frac{1}{Γ (n - α)} \int_{t_{1}}^{t} {(t - ξ)}^{n - α - 1} {(\frac{d}{d ξ})}^{n} f (ξ) d ξ$ Mathematical equation (8)

${}_{t}^{C} D_{t_{2}}^{β} f (t) = \frac{1}{Γ (n - β)} \int_{t}^{t_{2}} {(ξ - t)}^{n - β - 1} {(- \frac{d}{d ξ})}^{n} f (ξ) d ξ$ Mathematical equation (9)

here, $α$ Mathematical equation and $β$ represent the orders of the fractional derivatives. When $α, β \to 1$ , the fractional derivative operators reduce to the classical integer derivative operators, namely,

${}_{t_{1}}^{R L} D_{t}^{1} = {}_{t_{1}}^{C}D D_{t}^{1} = \frac{d}{d t}, {}_{t}^{R L}D D_{t_{2}}^{1} = {}_{t}^{C}D D_{t_{2}}^{1} = - \frac{d}{d t}$ Mathematical equation (10)

Throughout this paper, we assume that $0 < α, β < 1$ Mathematical equation .

For the Lagrangian $L_{M} = L_{M} (t, q_{M}, {\dot{q}}_{M}, {}_{t_{1}}^{C}D D_{t}^{α} q_{M})$ Mathematical equation , where $q_{M} = (q_{M 1}, q_{M 2}, \dots, q_{M n})$ , ${\dot{q}}_{M} = ({\dot{q}}_{M 1}, {\dot{q}}_{M 2}, \dots, {\dot{q}}_{M n})$ , ${}_{t_{1}}^{C} D_{t}^{α} q_{M} = ({}_{t_{1}}^{C} D_{t}^{α} q_{M 1}, {}_{t_{1}}^{C}D D_{t}^{α} q_{M 2}, \dots, {}_{t_{1}}^{C}D D_{t}^{α} q_{M n})$ Mathematical equation , $q_{M j}$ are the generalized coordinates, ${\dot{q}}_{M j} = d q_{M j} / d t$ are the generalized velocities, ${}_{t_{1}}^{C} D_{t}^{α} q_{M j}$ are the Caputo derivatives of $q_{M j}$ , $q_{M j} (\cdot) \in C^{2} ([t_{1}, t_{2}]; R)$ , $j = 1,2, \dots, n$ , $L_{M} (\cdot, \cdot, \cdot, \cdot) \in C^{2} ([t_{1}, t_{2}] \times R^{n} \times R^{n} \times R^{n}; R)$ Mathematical equation , and $0 < α < 1$ , we define the corresponding generalized momenta and the Hamiltonian as

$\begin{array}{l} p_{M i} = \frac{\partial L_{M} (t, q_{M}, {\dot{q}}_{M}, {}_{t_{1}}^{C}D D_{t}^{α} q_{M})}{\partial {\dot{q}}_{M i}}, p_{M i}^{α} = \frac{\partial L_{M} (t, q_{M}, {\dot{q}}_{M}, {}_{t_{1}}^{C}D D_{t}^{α} q_{M})}{\partial {}_{t_{1}}^{C}D D_{t}^{α} q_{M i}}, \\ H_{M} = p_{M i} {\dot{q}}_{M i} + p_{M i}^{α} \cdot {}_{t_{1}}^{C}D D_{t}^{α} q_{M i} - L_{M} (t, q_{M}, {\dot{q}}_{M}, {}_{t_{1}}^{C}D D_{t}^{α} q_{M}), i = 1,2, \dots, n \end{array}$ Mathematical equation (11)

Specifically, we assume that ${}_{t_{1}}^{C} D_{t}^{α} q_{M i}$ Mathematical equation can always be described by a function that depends on the elements of $t$ , $q_{M j}$ , ${\dot{q}}_{M j}$ and $p_{M j}^{α}$ , namely, ${}_{t_{1}}^{C} D_{t}^{α} q_{M i} = h_{M i} (t, q_{M j}, {\dot{q}}_{M j}, p_{M j}^{α})$ , $i, j = 1,2, \dots, n$ .

In this case, we define the elements of the Hessian matrix as

$H_{M i j} = \frac{\partial^{2} L_{M} (t, q_{M}, {\dot{q}}_{M}, {}_{t_{1}}^{C}D D_{t}^{α} q_{M})}{\partial {\dot{q}}_{M i} \partial {\dot{q}}_{M j}}, i, j = 1,2, \dots, n$ Mathematical equation (12)

if $r a n k [H_{M i j}] = R$ Mathematical equation , $0 \leq R < n$ , then the fractional primary constraints with the mixed derivatives have the forms^[30]

$ϕ_{M a} (t, q_{M j}, p_{M j}, p_{M j}^{α}) = 0$ Mathematical equation (13)

where $a = 1,2, \dots, n - R$ Mathematical equation , $0 \leq R < n$ , $j = 1,2, \dots, n$ . The fractional constrained Hamilton equations with the mixed derivatives have the forms ^[30]

${\dot{q}}_{M i} = \frac{\partial H_{M}}{\partial p_{M i}} + λ_{M a} \frac{\partial ϕ_{M a}}{\partial p_{M i}}, {\dot{p}}_{M i} = - \frac{\partial H_{M}}{\partial q_{M i}} + {}_{t}^{R L}D D_{t_{2}}^{α} p_{M i}^{α} - λ_{M a} \frac{\partial ϕ_{M a}}{\partial q_{M i}}, {}_{t_{1}}^{C}D D_{t}^{α} q_{M i} = \frac{\partial H_{M}}{\partial p_{M i}^{α}} + λ_{M a} \frac{\partial ϕ_{M a}}{\partial p_{M i}^{α}}$ Mathematical equation (14)

where $H_{M} = H_{M} (t, q_{M}, p_{M}, p_{M}^{α})$ Mathematical equation , $q_{M} = (q_{M 1}, q_{M 2}, \dots, q_{M n})$ , $p_{M} = (p_{M 1}, p_{M 2}, \dots, p_{M n})$ , $p_{M}^{α} = (p_{M 1}^{α}, p_{M 2}^{α}, \dots, p_{M n}^{α})$ , $λ_{M a} (t)$ are the Lagrange multipliers, $a = 1,2, \dots, n - R$ , $0 \leq R < n$ , and $i = 1,2, \dots, n$ Mathematical equation .

However, when the Lagrange multipliers cannot be solved, Eq. (14) is invalid, and the fractional constrained Hamilton equations with the mixed derivatives have another forms, which have been investigated in Ref. [30]. In this paper, we discuss only the case in which the Lagrange multipliers can be solved.

For the Lagrangian $L_{C} = L_{C} (t, q_{C}, {}_{t_{1}}^{C}D D_{t}^{α} q_{C})$ Mathematical equation , where $q_{C} = (q_{C 1}, q_{C 2}, \dots, q_{C n})$ , ${}_{t_{1}}^{C} D_{t}^{α} q_{C} = ({}_{t_{1}}^{C} D_{t}^{α} q_{C 1}, {}_{t_{1}}^{C}D D_{t}^{α} q_{C 2}, \dots, {}_{t_{1}}^{C}D D_{t}^{α} q_{C n})$ , $q_{C j}$ are the generalized coordinates, ${}_{t_{1}}^{C} D_{t}^{α} q_{C j}$ Mathematical equation are the Caputo derivatives of $q_{C j}$ , $q_{C j} (\cdot) \in C^{2} ([t_{1}, t_{2}]; R)$ , $j = 1,2, \dots, n$ , $L_{C} (\cdot, \cdot, \cdot) \in C^{2} ([t_{1}, t_{2}] \times R^{n} \times R^{n}; R)$ , and $0 < α < 1$ , we define the corresponding generalized momenta and the Hamiltonian as

$p_{C i} = \frac{\partial L_{C} (t, q_{C}, {}_{t_{1}}^{C}D D_{t}^{α} q_{C})}{\partial {}_{t_{1}}^{C}D D_{t}^{α} q_{C i}}, H_{C} = p_{C i} \cdot {}_{t_{1}}^{C}D D_{t}^{α} q_{C i} - L_{C} (t, q_{C}, {}_{t_{1}}^{C}D D_{t}^{α} q_{C}), i = 1,2, \dots, n$ Mathematical equation (15)

We assume that the Lagrangian $L_{C} (t, q_{C}, {}_{t_{1}}^{C}D D_{t}^{α} q_{C})$ Mathematical equation is singular, i.e., only $R$ elements of ${}_{t_{1}}^{C} D_{t}^{α} q_{C i}$ , $i = 1,2, \dots, n$ , can be solved, where $0 \leq R < n$ .

In this case, the fractional primary constraints with the Caputo fractional derivatives have the forms^[30]

$ϕ_{C a} (t, q_{C j}, p_{C j}) = 0, a = 1,2, \dots, n - R, 0 \leq R < n, j = 1,2, \dots, n$ Mathematical equation (16)

The fractional constrained Hamilton equations with the Caputo fractional derivatives have the forms^[30]

${}_{t_{1}}^{C} D_{t}^{α} q_{C i} = \frac{\partial H_{C}}{\partial p_{C i}} + λ_{C a} \frac{\partial ϕ_{C a}}{\partial p_{C i}}, {}_{t}^{R L}D D_{t_{2}}^{α} p_{C i} = \frac{\partial H_{C}}{\partial q_{C i}} + λ_{C a} \frac{\partial ϕ_{C a}}{\partial q_{C i}}$ Mathematical equation (17)

where $H_{C} = H_{C} (t, q_{C}, p_{C})$ Mathematical equation , $q_{C} = (q_{C 1}, q_{C 2}, \dots, q_{C n})$ , $p_{C} = (p_{C 1}, p_{C 2}, \dots, p_{C n})$ , $λ_{C a}$ are the Lagrange multipliers, $a = 1,2, \dots, n -$ $R$ , $0 \leq R < n$ , and $i = 1,2, \dots, n$ .

Similarly, when the Lagrange multipliers cannot be solved, Eq. (17) is invalid, and the fractional constrained Hamilton equations with the Caputo fractional derivatives have another forms, which have been investigated in Ref. [30]. In this paper, we discuss only the case in which the Lagrange multipliers can be solved.

2 Fractional Noether Theorem with Mixed Derivatives

Noether symmetry with the mixed derivatives is determined by the Noether symmetric transformations, under which the fractional Hamilton action with the mixed derivatives

$I_{M} = \int_{t_{1}}^{t_{2}} L_{M} (t, q_{M}, {\dot{q}}_{M}, {}_{t_{1}}^{C}D D_{t}^{α} q_{M}) d t = \int_{t_{1}}^{t_{2}} (p_{M i} {\dot{q}}_{M i} + p_{M i}^{α} \cdot {}_{t_{1}}^{C}D D_{t}^{α} q_{M i} - H_{M} (t, q_{M}, p_{M}, p_{M}^{α})) d t$ Mathematical equation (18)

remains invariant. Therefore, if we want to study the Noether theorem, we first need to give the infinitesimal transformations with the mixed derivatives. Then, we discuss the change of the fractional Hamilton action (Eq. (18)) under the given infinitesimal transformations. Finally, the condition which is called the fractional Noether identity with the mixed derivatives is obtained.

Here, the infinitesimal transformations have the forms

$\bar{t} = t + Δ t, {\bar{q}}_{M i} (\bar{t}) = q_{M i} (t) + Δ q_{M i}, {\bar{p}}_{M i} (\bar{t}) = p_{M i} (t) + Δ p_{M i}, {\bar{p}}_{M i}^{α} (\bar{t}) = p_{M i}^{α} (t) + Δ p_{M i}^{α}$ Mathematical equation (19)

whose expansions are

$\begin{array}{l} \bar{t} = t + θ_{M} ξ_{M 0} (t, q_{M}, p_{M}, p_{M}^{α}) + ο (θ_{M}), {\bar{q}}_{M i} (\bar{t}) = q_{M i} (t) + θ_{M} ξ_{M i} (t, q_{M}, p_{M}, p_{M}^{α}) + ο (θ_{M}), \\ {\bar{p}}_{M i} (\bar{t}) = p_{M i} (t) + θ_{M} η_{M i} (t, q_{M}, p_{M}, p_{M}^{α}) + ο (θ_{M}), {\bar{p}}_{M i}^{α} (\bar{t}) = p_{M i}^{α} (t) + θ_{M} η_{M i}^{α} (t, q_{M}, p_{M}, p_{M}^{α}) + ο (θ_{M}) \end{array}$ Mathematical equation (20)

where $ξ_{M 0}$ Mathematical equation , $ξ_{M i}$ , $η_{M i}$ and $η_{M i}^{α}$ are called the infinitesimal generators with the mixed derivatives, $θ_{M}$ is a small parameter, and $i = 1,2, \dots, n$ .

We denote the change of the fractional Hamilton action as $Δ I_{M}$ Mathematical equation , namely, $Δ I_{M} = {\bar{I}}_{M} - I_{M}$ . If we consider only the linear part of $θ_{M}$ , then we have

$\begin{array}{l} Δ I_{M} = \int_{{\bar{t}}_{1}}^{{\bar{t}}_{2}} ({\bar{p}}_{M i} {\dot{\bar{q}}}_{M i} + {\bar{p}}_{M i}^{α} {}_{{\bar{t}}_{1}}^{C}D D_{\bar{t}}^{α} {\bar{q}}_{M i} - H_{M} (t, {\bar{q}}_{M}, {\bar{p}}_{M}, {\bar{p}}_{M}^{α})) d \bar{t} - \int_{t_{1}}^{t_{2}} (p_{M i} {\dot{q}}_{M i} + p_{M i}^{α} {}_{t_{1}}^{C}D D_{t}^{α} q_{M i} - H_{M} (t, q_{M}, p_{M}, p_{M}^{α})) d t \\ = \int_{t_{1}}^{t_{2}} [(p_{M i} + Δ p_{M i}) ({\dot{q}}_{M i} + Δ {\dot{q}}_{M i}) + (p_{M i}^{α} + Δ p_{M i}^{α}) \cdot ({}_{t_{1}}^{C} D_{t}^{α} q_{M i} + {}_{t_{1}}^{C}D D_{t}^{α} δ q_{M i} + Δ t \frac{d}{d t} {}_{t_{1}}^{C}D D_{t}^{α} q_{M i} - \frac{1}{Γ (1 - α)} ({(t - t_{1})}^{- α} {\dot{q}}_{M i} (t_{1}) Δ t_{1})) \\ - H_{M} (t + Δ t, q_{M j} + Δ q_{M j}, p_{M j} + Δ p_{M j}, p_{M j}^{α} + Δ p_{M j}^{α})] \cdot (1 + \frac{d}{d t} Δ t) d t - \int_{t_{1}}^{t_{2}} (p_{M i} {\dot{q}}_{M i} + p_{M i}^{α} {}_{t_{1}}^{C}D D_{t}^{α} q_{M i} - H_{M} (t, q_{M}, p_{M}, p_{M}^{α})) d t \\ = \int_{t_{1}}^{t_{2}} [p_{M i} {\dot{q}}_{M i} + p_{M i} Δ {\dot{q}}_{M i} + Δ p_{M i} {\dot{q}}_{M i} + p_{M i}^{α} {}_{t_{1}}^{C}D D_{t}^{α} q_{M i} + p_{M i}^{α} {}_{t_{1}}^{C}D D_{t}^{α} δ q_{M i} + p_{M i}^{α} Δ t \frac{d}{d t} {}_{t_{1}}^{C}D D_{t}^{α} q_{M i} + Δ p_{M i}^{α} {}_{t_{1}}^{C}D D_{t}^{α} q_{M i} - \frac{p_{M i}^{α}}{Γ (1 - α)} {(t - t_{1})}^{- α} {\dot{q}}_{M i} (t_{1}) Δ t_{1} \\ - H_{M} (t, q_{M}, p_{M}, p_{M}^{α}) - \frac{\partial H_{M}}{\partial t} Δ t - \frac{\partial H_{M}}{\partial q_{M i}} Δ q_{M i} - \frac{\partial H_{M}}{\partial p_{M i}} Δ p_{M i} - \frac{\partial H_{M}}{\partial p_{M i}^{α}} Δ p_{M i}^{α} + (p_{M i} {\dot{q}}_{M i} + p_{M i}^{α} {}_{t_{1}}^{C}D D_{t}^{α} q_{M i} - H_{M}) \frac{d}{d t} Δ t] d t \\ - \int_{t_{1}}^{t_{2}} (p_{M i} {\dot{q}}_{M i} + p_{M i}^{α} {}_{t_{1}}^{C}D D_{t}^{α} q_{M i} - H_{M} (t, q_{M}, p_{M}, p_{M}^{α})) d t \\ = θ_{M} \int_{t_{1}}^{t_{2}} [p_{M i} {\dot{ξ}}_{M i} + p_{M i}^{α} {}_{t_{1}}^{C}D D_{t}^{α} (ξ_{M i} - {\dot{q}}_{M i} ξ_{M 0}) + (p_{M i}^{α} \frac{d}{d t} {}_{t_{1}}^{C}D D_{t}^{α} q_{M i} - \frac{\partial H_{M}}{\partial t}) ξ_{M 0} + λ_{M a} \frac{\partial ϕ_{M a}}{\partial p_{M i}^{α}} η_{M i}^{α} \\ - \frac{p_{M i}^{α}}{Γ (1 - α)} {(t - t_{1})}^{- α} {\dot{q}}_{M i} (t_{1}) ξ_{M 0} (t_{1}) + (p_{M i}^{α} {}_{t_{1}}^{C}D D_{t}^{α} q_{M i} - H_{M}) {\dot{ξ}}_{M 0} - \frac{\partial H_{M}}{\partial q_{M i}} ξ_{M i} + λ_{M a} \frac{\partial ϕ_{M a}}{\partial p_{M i}} η_{M i}] d t \end{array}$ Mathematical equation (21)

where

$\begin{array}{l} {}_{{\bar{t}}_{1}}^{C}D D_{\bar{t}}^{α} {\bar{q}}_{M i} = {}_{t_{1}}^{C}D D_{t}^{α} q_{M i} + {}_{t_{1}}^{C}D D_{t}^{α} δ q_{M i} + Δ t \frac{d}{d t} {}_{t_{1}}^{C}D D_{t}^{α} q_{M i} - \frac{1}{Γ (1 - α)} {(t - t_{1})}^{- α} {\dot{q}}_{M i} (t_{1}) Δ t_{1} \\ ξ_{M 0} (t_{1}) = ξ_{M 0} (t_{1}, q_{M} (t_{1}), p_{M} (t_{1}), p_{M}^{α} (t_{1})), Δ {\dot{q}}_{M i} = θ_{M} ({\dot{ξ}}_{M i} - {\dot{q}}_{M i} {\dot{ξ}}_{M 0}), \end{array}$ Mathematical equation

That fractional Hamilton action remains invariant implies $Δ I_{M} = 0$ Mathematical equation , therefore, from Eq. (21), we have

$\begin{array}{l} p_{M i} {\dot{ξ}}_{M i} + p_{M i}^{α} {}_{t_{1}}^{C}D D_{t}^{α} (ξ_{M i} - {\dot{q}}_{M i} ξ_{M 0}) + (p_{M i}^{α} \frac{d}{d t} {}_{t_{1}}^{C}D D_{t}^{α} q_{M i} - \frac{\partial H_{M}}{\partial t}) ξ_{M 0} - \frac{\partial H_{M}}{\partial q_{M i}} ξ_{M i} + λ_{M a} \frac{\partial ϕ_{M a}}{\partial p_{M i}^{α}} η_{M i}^{α} \\ + λ_{M a} \frac{\partial ϕ_{M a}}{\partial p_{M i}} η_{M i} - \frac{p_{M i}^{α}}{Γ (1 - α)} {(t - t_{1})}^{- α} {\dot{q}}_{M i} (t_{1}) ξ_{M 0} (t_{1}) + (p_{M i}^{α} {}_{t_{1}}^{C}D D_{t}^{α} q_{M i} - H_{M}) {\dot{ξ}}_{M 0} = 0, i = 1,2, \dots, n \end{array}$ Mathematical equation (22)

Equation (22) is called the fractional Noether identity with the mixed derivatives for the fractional constrained Hamiltonian system (Eq. (14)). The infinitesimal transformations in this case are called the Noether symmetric transformations, which determine the Noether symmetry.

In this paper, we adopt Atanacković's definition of the fractional conserved quantity. We review it first.

Definition 1^[22] A quantity $C$ Mathematical equation is called a fractional conserved quantity if and only if $d C / d t = 0$ holds.

Theorem 1 If the infinitesimal generators $ξ_{M 0}$ Mathematical equation , $ξ_{M i}$ , $η_{M i}$ and $η_{M i}^{α}$ satisfy the fractional Noether identity (Eq. (22)), then a fractional conserved quantity with the mixed derivatives exists for the fractional constrained Hamiltonian system (Eq. (14)), as follows:

$\begin{array}{l} C_{M} = (p_{M i}^{α} {}_{t_{1}}^{C}D D_{t}^{α} q_{M i} - H_{M}) ξ_{M 0} + \int_{t_{1}}^{t} [p_{M i}^{α} {}_{t_{1}}^{C}D D_{τ}^{α} (ξ_{M i} - {\dot{q}}_{M i} ξ_{M 0}) - (ξ_{M i} - {\dot{q}}_{M i} ξ_{M 0}) {}_{τ}^{R L}D D_{t_{2}}^{α} p_{M i}^{α}] d τ \\ + p_{M i} ξ_{M i} - \int_{t_{1}}^{t} \frac{p_{M i}^{α}}{Γ (1 - α)} {(τ - t_{1})}^{- α} {\dot{q}}_{M i} (t_{1}) ξ_{M 0} (t_{1}) d τ = c o n s t \end{array}$ Mathematical equation (23)

Proof It is obtained from Eqs. (14), (23) that

$\begin{array}{l} \frac{d}{d t} C_{M} = ({\dot{p}}_{M i}^{α} {}_{t_{1}}^{C}D D_{t}^{α} q_{M i} + p_{M i}^{α} \frac{d}{d t} {}_{t_{1}}^{C}D D_{t}^{α} q_{M i} - \frac{\partial H_{M}}{\partial t} - \frac{\partial H_{M}}{\partial p_{M i}} {\dot{p}}_{M i} - \frac{\partial H_{M}}{\partial q_{M i}} {\dot{q}}_{M i} - \frac{\partial H_{M}}{\partial p_{M i}^{α}} {\dot{p}}_{M i}^{α}) ξ_{M 0} + (p_{M i}^{α} {}_{t_{1}}^{C}D D_{t}^{α} q_{M i} - H_{M}) {\dot{ξ}}_{M 0} \\ + p_{M i}^{α} {}_{t_{1}}^{C}D D_{t}^{α} (ξ_{M i} - {\dot{q}}_{M i} ξ_{M 0}) - (ξ_{M i} - {\dot{q}}_{M i} ξ_{M 0}) {}_{t}^{R L}D D_{t_{2}}^{α} p_{M i}^{α} + p_{M i} {\dot{ξ}}_{M i} - \frac{p_{M i}^{α}}{Γ (1 - α)} {(t - t_{1})}^{- α} {\dot{q}}_{M i} (t_{1}) ξ_{M 0} (t_{1}) + {\dot{p}}_{M i} ξ_{M i} \\ = \frac{\partial H_{M}}{\partial q_{M i}} ξ_{M i} + {\dot{p}}_{M i} ξ_{M i} + ({\dot{p}}_{M i}^{α} {}_{t_{1}}^{C}D D_{t}^{α} q_{M i} - \frac{\partial H_{M}}{\partial p_{M i}} {\dot{p}}_{M i} - \frac{\partial H_{M}}{\partial q_{M i}} {\dot{q}}_{M i} - \frac{\partial H_{M}}{\partial p_{M i}^{α}} {\dot{p}}_{M i}^{α}) ξ_{M 0} \\ - (ξ_{M i} - {\dot{q}}_{M i} ξ_{M 0}) {}_{t}^{R L}D D_{t_{2}}^{α} p_{M i}^{α} - λ_{M a} \frac{\partial ϕ_{M a}}{\partial p_{M i}^{α}} η_{M i}^{α} - λ_{M a} \frac{\partial ϕ_{M a}}{\partial p_{M i}} η_{M i} \\ = ({\dot{p}}_{M i} + \frac{\partial H_{M}}{\partial q_{M i}} - {}_{t}^{R L}D D_{t_{2}}^{α} p_{M i}^{α} + λ_{M a} \frac{\partial ϕ_{M a}}{\partial q_{M i}}) (ξ_{M i} - {\dot{q}}_{M i} ξ_{M 0}) - λ_{M a} \frac{\partial ϕ_{M a}}{\partial p_{M i}^{α}} η_{M i}^{α} - λ_{M a} \frac{\partial ϕ_{M a}}{\partial q_{M i}} (ξ_{M i} - {\dot{q}}_{M i} ξ_{M 0}) \\ + ({\dot{p}}_{M i}^{α} {}_{t_{1}}^{C}D D_{t}^{α} q_{M i} - {\dot{p}}_{M i}^{α} \frac{\partial H_{M}}{\partial p_{M i}^{α}} - \frac{\partial H_{M}}{\partial p_{M i}} {\dot{p}}_{M i}) ξ_{M 0} - λ_{M a} \frac{\partial ϕ_{M a}}{\partial p_{M i}} η_{M i} + {\dot{p}}_{M i} ξ_{M 0} (\frac{\partial H_{M}}{\partial p_{M i}} + λ_{M a} \frac{\partial ϕ_{M a}}{\partial p_{M i}}) \\ = - λ_{M a} \frac{\partial ϕ_{M a}}{\partial p_{M i}^{α}} η_{M i}^{α} - λ_{M a} \frac{\partial ϕ_{M a}}{\partial p_{M i}} η_{M i} + {\dot{p}}_{M i}^{α} ξ_{M 0} λ_{M a} \frac{\partial ϕ_{M a}}{\partial p_{M i}^{α}} - λ_{M a} \frac{\partial ϕ_{M a}}{\partial q_{M i}} (ξ_{M i} - {\dot{q}}_{M i} ξ_{M 0}) + {\dot{p}}_{M i} ξ_{M 0} λ_{M a} \frac{\partial ϕ_{M a}}{\partial p_{M i}} \\ = - λ_{M a} \frac{\partial ϕ_{M a}}{\partial p_{M i}^{α}} (η_{M i}^{α} - {\dot{p}}_{M i}^{α} ξ_{M 0}) - λ_{M a} \frac{\partial ϕ_{M a}}{\partial p_{M i}} (η_{M i} - {\dot{p}}_{M i} ξ_{M 0}) - λ_{M a} \frac{\partial ϕ_{M a}}{\partial q_{M i}} (ξ_{M i} - {\dot{q}}_{M i} ξ_{M 0}) = 0 \end{array}$ Mathematical equation

where

$\begin{array}{l} δ ϕ_{M a} (t, q_{M j}, p_{M j}, p_{M j}^{α}) = \frac{\partial ϕ_{M a}}{\partial q_{M i}} δ q_{M i} + \frac{\partial ϕ_{M a}}{\partial p_{M i}} δ p_{M i} + \frac{\partial ϕ_{M a}}{\partial p_{M i}^{α}} δ p_{M i}^{α} \\ = \frac{\partial ϕ_{M a}}{\partial q_{M i}} (Δ q_{M i} - {\dot{q}}_{M i} Δ t) + \frac{\partial ϕ_{M a}}{\partial p_{M i}} (Δ p_{M i} - {\dot{p}}_{M i} Δ t) + \frac{\partial ϕ_{M a}}{\partial p_{M i}^{α}} (Δ p_{M i}^{α} - {\dot{p}}_{M i}^{α} Δ t) \\ = θ_{M} [\frac{\partial ϕ_{M a}}{\partial q_{M i}} (ξ_{M i} - {\dot{q}}_{M i} ξ_{M 0}) + \frac{\partial ϕ_{M a}}{\partial p_{M i}} (η_{M i} - {\dot{p}}_{M i} ξ_{M 0}) + \frac{\partial ϕ_{M a}}{\partial p_{M i}^{α}} (η_{M i}^{α} - {\dot{p}}_{M i}^{α} ξ_{M 0})] = 0 \end{array}$ Mathematical equation

The proof is completed.

Furthermore, if the fractional Hamilton action (Eq. (18)) does not remain invariant under the infinitesimal transformations with the mixed derivatives, for instance, if $Δ I_{M} = {\bar{I}}_{M} - I_{M} = - \int_{t_{1}}^{t_{2}} \frac{d}{d t} (Δ G_{M}) d t$ Mathematical equation , where $Δ G_{M} = θ_{M} G_{M}$ and $G_{M} = G_{M} (t, q_{M j}, p_{M j}^{α}, p_{M j})$ is called a gauge function with the mixed derivatives, then from Eq. (21), we obtain

$\begin{array}{l} p_{M i} {\dot{ξ}}_{M i} + p_{M i}^{α} {}_{t_{1}}^{C}D D_{t}^{α} (ξ_{M i} - {\dot{q}}_{M i} ξ_{M 0}) + (p_{M i}^{α} \frac{d}{d t} {}_{t_{1}}^{C}D D_{t}^{α} q_{M i} - \frac{\partial H_{M}}{\partial t}) ξ_{M 0} - \frac{\partial H_{M}}{\partial q_{M i}} ξ_{M i} + λ_{M a} \frac{\partial ϕ_{M a}}{\partial p_{M i}^{α}} η_{M i}^{α} \\ + λ_{M a} \frac{\partial ϕ_{M a}}{\partial p_{M i}} η_{M i} - \frac{p_{M i}^{α}}{Γ (1 - α)} {(t - t_{1})}^{- α} {\dot{q}}_{M i} (t_{1}) ξ_{M 0} (t_{1}) + (p_{M i}^{α} {}_{t_{1}}^{C}D D_{t}^{α} q_{M i} - H_{M}) {\dot{ξ}}_{M 0} + {\dot{G}}_{M} = 0 \end{array}$ Mathematical equation (24)

Equation (24) is called the fractional Noether-quasi identity with the mixed derivatives for the fractional constrained Hamiltonian system (Eq. (14)). The infinitesimal transformations in this case are called the Noether-quasi symmetric transformations with the mixed derivatives, which determine the Noether-quasi symmetry with the mixed derivatives. Then, a fractional conserved quantity can also be obtained from the Noether-quasi symmetry.

Theorem 2 If the infinitesimal generators $ξ_{M 0}$ Mathematical equation , $ξ_{M i}$ , $η_{M i}$ , $η_{M i}^{α}$ and a gauge function $G_{M}$ satisfy the fractional Noether-quasi identity (Eq. (24)), then a fractional conserved quantity with the mixed derivatives exists for the fractional constrained Hamiltonian system (Eq. (14))

$\begin{array}{l} C_{G M} = (p_{M i}^{α} {}_{t_{1}}^{C}D D_{t}^{α} q_{M i} - H_{M}) ξ_{M 0} + \int_{t_{1}}^{t} [p_{M i}^{α} {}_{t_{1}}^{C}D D_{τ}^{α} (ξ_{M i} - {\dot{q}}_{M i} ξ_{M 0}) - (ξ_{M i} - {\dot{q}}_{M i} ξ_{M 0}) {}_{τ}^{R L}D D_{t_{2}}^{α} p_{M i}^{α}] d τ \\ + p_{M i} ξ_{M i} - \int_{t_{1}}^{t} \frac{p_{M i}^{α}}{Γ (1 - α)} {(τ - t_{1})}^{- α} {\dot{q}}_{M i} (t_{1}) ξ_{M 0} (t_{1}) d τ + G_{M} = c o n s t \end{array}$ Mathematical equation (25)

Proof The intended result can be obtained from Eqs. (14), (24) and (25).

Remark 1 The Noether-quasi symmetry with the mixed derivatives is more general than the Noether symmetry with the mixed derivatives. In fact, by setting $G_{M} = 0$ Mathematical equation , Theorem 2 reduces to Theorem 1.

An example is presented to illustrate the results and methods above.

Example 1 For the Lagrangian

$L_{M} = {\dot{q}}_{M 1} q_{M 2} - q_{M 1} {\dot{q}}_{M 2} + q_{M 1}^{2} + q_{M 2}^{2} + \frac{1}{2} [{({}_{t_{1}}^{C} D_{t}^{α} q_{M 1})}^{2} + {({}_{t_{1}}^{C} D_{t}^{α} q_{M 2})}^{2}]$ Mathematical equation (26)

find its conserved quantity.

For this Lagrangian $L_{M}$ Mathematical equation , there exist two fractional primary constraints ^[30]

$ϕ_{M 1} = p_{M 1} - q_{M 2} = 0, ϕ_{M 2} = p_{M 2} + q_{M 1} = 0$ Mathematical equation (27)

In addition, all the Lagrange multipliers can be obtained ^[30]:

$λ_{M 1} = - q_{M 2} - \frac{1}{2} {}_{t}^{R L}D D_{t_{2}}^{α} p_{M 2}^{α}, λ_{M 2} = q_{M 1} + \frac{1}{2} {}_{t}^{R L}D D_{t_{2}}^{α} p_{M 1}^{α}$ Mathematical equation (28)

The fractional constrained Hamilton equations can also be established ^[30]:

$\begin{array}{l} {\dot{q}}_{M 1} = - q_{M 2} - \frac{1}{2} {}_{t}^{R L}D D_{t_{2}}^{α} p_{M 2}^{α}, {\dot{q}}_{M 2} = q_{M 1} + \frac{1}{2} {}_{t}^{R L}D D_{t_{2}}^{α} p_{M 1}^{α}, {\dot{p}}_{M 1} = q_{M 1} + \frac{1}{2} {}_{t}^{R L}D D_{t_{2}}^{α} p_{M 1}^{α}, \\ [p M 2 = q 2 + 12 R L t D α t 2 p α M 2, C t 1 D α t q M 1 = p α M 1, C t 1 D α t q M 2 = p α M 2] \end{array}$ Mathematical equation (29)

The fractional Noether-quasi identity (Eq. (24)) gives

$\begin{array}{l} p_{M 1} {\dot{ξ}}_{M 1} + p_{M 1}^{α} {}_{t_{1}}^{C}D D_{t}^{α} (ξ_{M 1} - {\dot{q}}_{M 1} ξ_{M 0}) + (p_{M 1}^{α} \frac{d}{d t} {}_{t_{1}}^{C}D D_{t}^{α} q_{M 1} + p_{M 2}^{α} \frac{d}{d t} {}_{t_{1}}^{C}D D_{t}^{α} q_{M 2}) ξ_{M 0} + λ_{M 1} η_{M 1} + λ_{M 2} η_{M 2} \\ + 2 q_{M 1} ξ_{M 1} + 2 q_{M 2} ξ_{M 2} - \frac{p_{M 1}^{α}}{Γ (1 - α)} {(t - t_{1})}^{- α} {\dot{q}}_{M 1} (t_{1}) ξ_{M 0} (t_{1}) + p_{M 2} {\dot{ξ}}_{M 2} - \frac{p_{M 2}^{α}}{Γ (1 - α)} {(t - t_{1})}^{- α} {\dot{q}}_{M 2} (t_{1}) ξ_{M 0} (t_{1}) \\ + p_{M 2}^{α} {}_{t_{1}}^{C}D D_{t}^{α} (ξ_{M 2} - {\dot{q}}_{M 2} ξ_{M 0}) + (p_{M 1}^{α} {}_{t_{1}}^{C}D D_{t}^{α} q_{M 1} + p_{M 2}^{α} {}_{t_{1}}^{C}D D_{t}^{α} q_{M 2} - H_{M}) {\dot{ξ}}_{M 0} + {\dot{G}}_{M} = 0 \end{array}$ Mathematical equation (30)

Through computation, we can verify that

$ξ_{M 0} = - 1, ξ_{M 1} = ξ_{M 2} = 0, η_{M 1} = η_{M 2} = 0, η_{M 1}^{α} = η_{M 2}^{α} = 0, G_{M} = 0$ Mathematical equation (31)

is a solution to Eq. (30). Finally, Theorem 2 gives the fractional conserved quantity

$C_{G M} = \int_{t_{1}}^{t} (p_{M 1}^{α} \frac{d}{d τ} {}_{t_{1}}^{C}D D_{τ}^{α} q_{M 1} + p_{M 2}^{α} \frac{d}{d τ} {}_{t_{1}}^{C}D D_{τ}^{α} q_{M 2} - {\dot{q}}_{M 1} {}_{τ}^{R L}D D_{t_{2}}^{α} p_{M 1}^{α} - {\dot{q}}_{M 2} {}_{τ}^{R L}D D_{t_{2}}^{α} p_{M 2}^{α}) d τ - [\frac{1}{2} {(p_{M 1}^{α})}^{2} + \frac{1}{2} {(p_{M 2}^{α})}^{2} + q_{M 1}^{2} + q_{M 2}^{2}] = c o n s t$ Mathematical equation (32)

3 Fractional Noether Theorem with only Caputo Fractional Derivatives

Noether symmetry with the Caputo fractional derivative is determined by the Noether symmetric transformations under which the fractional Hamilton action with the Caputo fractional derivatives

$I_{C} = \int_{t_{1}}^{t_{2}} L_{C} (t, q_{C}, {}_{t_{1}}^{C}D D_{t}^{α} q_{C}) d t = \int_{t_{1}}^{t_{2}} (p_{C i} \cdot {}_{t_{1}}^{C}D D_{t}^{α} q_{C i} - H_{C} (t, q_{C}, p_{C})) d t$ Mathematical equation (33)

remains invariant. Similarly, if we want to study the Noether theorem, we first need to give the infinitesimal transformations with the Caputo fractional derivative; then, we discuss the change of the fractional Hamilton action (Eq. (33)) under the given infinitesimal transformations. Finally, the condition called the fractional Noether identity with the Caputo fractional derivatives is obtained.

Here, the infinitesimal transformations have the forms

$\bar{t} = t + Δ t, {\bar{q}}_{C i} (\bar{t}) = q_{C i} (t) + Δ q_{C i}, {\bar{p}}_{C i} (\bar{t}) = p_{C i} (t) + Δ p_{C i}$ Mathematical equation (34)

whose expansions are

$\bar{t} = t + θ_{C} ξ_{C 0} (t, q_{C}, p_{C}) + ο (θ_{C}), {\bar{q}}_{C i} (\bar{t}) = q_{C i} (t) + θ_{C} ξ_{C i} (t, q_{C}, p_{C}) + ο (θ_{C}), {\bar{p}}_{C i} (\bar{t}) = p_{C i} (t) + θ_{C} η_{C i} (t, q_{C}, p_{C}) + ο (θ_{C})$ Mathematical equation (35)

where $ξ_{C 0}$ Mathematical equation , $ξ_{C i}$ and $η_{C i}$ are called the infinitesimal generators with the Caputo fractional derivatives, $θ_{C}$ is a small parameter, and $i = 1,2, \dots, n$ .

We denote the change of the fractional Hamilton action (Eq. (33)) as $Δ I_{C}$ Mathematical equation ; namely, $Δ I_{C} = {\bar{I}}_{C} - I_{C}$ . If we consider only the linear part of $θ_{C}$ , then we have

$\begin{array}{l} Δ I_{C} = \int_{{\bar{t}}_{1}}^{{\bar{t}}_{2}} ({\bar{p}}_{C i} {}_{{\bar{t}}_{1}}^{C}D D_{\bar{t}}^{α} {\bar{q}}_{C i} - H_{C} (t, {\bar{q}}_{C}, {\bar{p}}_{C})) d \bar{t} - \int_{t_{1}}^{t_{2}} (p_{C i} \cdot {}_{t_{1}}^{C}D D_{t}^{α} q_{C i} - H_{C} (t, q_{C}, p_{C})) d t \\ = \int_{t_{1}}^{t_{2}} [(p_{C i} + Δ p_{C i}) \cdot ({}_{t_{1}}^{C} D_{t}^{α} q_{C i} + {}_{t_{1}}^{C}D D_{t}^{α} δ q_{C i} + Δ t \frac{d}{d t} {}_{t_{1}}^{C}D D_{t}^{α} q_{C i} - \frac{1}{Γ (1 - α)} {(t - t_{1})}^{- α} {\dot{q}}_{C i} (t_{1}) Δ t_{1}) \\ - H_{C} (t + Δ t, q_{C j} + Δ q_{C j}, p_{C j} + Δ p_{C j})] \cdot (1 + \frac{d}{d t} Δ t) d t - \int_{t_{1}}^{t_{2}} (p_{C i} \cdot {}_{t_{1}}^{C}D D_{t}^{α} q_{C i} - H_{C} (t, q_{C}, p_{C})) d t \\ = \int_{t_{1}}^{t_{2}} [p_{C i} {}_{t_{1}}^{C}D D_{t}^{α} q_{C i} + p_{C i} {}_{t_{1}}^{C}D D_{t}^{α} δ q_{C i} + p_{C i} Δ t \frac{d}{d t} {}_{t_{1}}^{C}D D_{t}^{α} q_{C i} + Δ p_{C i} {}_{t_{1}}^{C}D D_{t}^{α} q_{C i} - H_{C} (t, q_{C}, p_{C}) - \frac{p_{C i}}{Γ (1 - α)} {(t - t_{1})}^{- α} {\dot{q}}_{C i} (t_{1}) Δ t_{1} \\ - \frac{\partial H_{C}}{\partial t} Δ t - \frac{\partial H_{C}}{\partial p_{C i}} Δ p_{C i} + (p_{C i} {}_{t_{1}}^{C}D D_{t}^{α} q_{C i} - H_{C}) \frac{d}{d t} Δ t - \frac{\partial H_{C}}{\partial q_{C i}} Δ q_{C i}] d t - \int_{t_{1}}^{t_{2}} (p_{C i} \cdot {}_{t_{1}}^{C}D D_{t}^{α} q_{C i} - H_{C} (t, q_{C}, p_{C})) d t \\ = θ_{C} \int_{t_{1}}^{t_{2}} [p_{C i} {}_{t_{1}}^{C}D D_{t}^{α} (ξ_{C i} - {\dot{q}}_{C i} ξ_{C 0}) + (p_{C i} \frac{d}{d t} {}_{t_{1}}^{C}D D_{t}^{α} q_{C i} - \frac{\partial H_{M}}{\partial t}) ξ_{C 0} - \frac{\partial H_{C}}{\partial q_{C i}} ξ_{C i} \\ - \frac{p_{C i}}{Γ (1 - α)} {(t - t_{1})}^{- α} {\dot{q}}_{C i} (t_{1}) ξ_{C 0} (t_{1}) + (p_{C i} {}_{t_{1}}^{C}D D_{t}^{α} q_{C i} - H_{C}) {\dot{ξ}}_{C 0} + λ_{C a} \frac{\partial ϕ_{C a}}{\partial p_{C i}} η_{C i}] d t \end{array}$ Mathematical equation (36)

where

${}_{{\bar{t}}_{1}}^{C}D D_{\bar{t}}^{α} {\bar{q}}_{C i} = {}_{t_{1}}^{C}D D_{t}^{α} q_{C i} + {}_{t_{1}}^{C}D D_{t}^{α} δ q_{C i} + Δ t \frac{d}{d t} {}_{t_{1}}^{C}D D_{t}^{α} q_{C i} - \frac{1}{Γ (1 - α)} {(t - t_{1})}^{- α} {\dot{q}}_{C i} (t_{1}) Δ t_{1}, ξ_{C 0} (t_{1}) = ξ_{C 0} (t_{1}, q_{C} (t_{1}), p_{C} (t_{1})) .$ Mathematical equation

That fractional Hamilton action (Eq. (33)) remains invariant implies $Δ I_{C} = 0$ Mathematical equation ; therefore, from Eq. (36), we have

$\begin{array}{l} p_{C i} {}_{t_{1}}^{C}D D_{t}^{α} (ξ_{C i} - {\dot{q}}_{C i} ξ_{C 0}) + (p_{C i} \frac{d}{d t} {}_{t_{1}}^{C}D D_{t}^{α} q_{C i} - \frac{\partial H_{C}}{\partial t}) ξ_{C 0} - \frac{\partial H_{C}}{\partial q_{C i}} ξ_{C i} + λ_{C a} \frac{\partial ϕ_{C a}}{\partial p_{C i}} η_{C i} \\ - \frac{p_{C i}}{Γ (1 - α)} {(t - t_{1})}^{- α} {\dot{q}}_{C i} (t_{1}) ξ_{C 0} (t_{1}) + (p_{C i} {}_{t_{1}}^{C}D D_{t}^{α} q_{C i} - H_{C}) {\dot{ξ}}_{C 0} = 0, i = 1,2, \dots, n \end{array}$ Mathematical equation (37)

Equation (37) is called the fractional Noether identity with the Caputo fractional derivatives for the fractional constrained Hamiltonian system (Eq. (17)). The infinitesimal transformations in this case are called the Noether symmetric transformations with the Caputo fractional derivatives, which determine the Noether symmetry.

Theorem 3 If the infinitesimal generators $ξ_{C 0}$ Mathematical equation , $ξ_{C i}$ , $η_{C i}$ and $η_{C i}^{α}$ satisfy the fractional Noether identity (Eq. (37)), then a fractional conserved quantity with the Caputo fractional derivatives exists for the fractional constrained Hamiltonian system (Eq. (17)) as follows:

$\begin{array}{l} C_{C} = (p_{C i} {}_{t_{1}}^{C}D D_{t}^{α} q_{C i} - H_{C}) ξ_{C 0} + \int_{t_{1}}^{t} [p_{C i} {}_{t_{1}}^{C}D D_{τ}^{α} (ξ_{C i} - {\dot{q}}_{C i} ξ_{C 0}) - (ξ_{C i} - {\dot{q}}_{C i} ξ_{C 0}) {}_{τ}^{R L}D D_{t_{2}}^{α} p_{C i}] d τ \\ [- t 1 t p C i Γ 1 - α τ - t 1 - α q C i t 1 ξ C 0 t 1 d τ = c o n s t] \end{array}$ Mathematical equation (38)

Proof It is obtained from Eqs. (17), (37) and (38) that

$\begin{array}{l} \frac{d}{d t} C_{C} = ({\dot{p}}_{C i} {}_{t_{1}}^{C}D D_{t}^{α} q_{C i} + p_{C i} \frac{d}{d t} {}_{t_{1}}^{C}D D_{t}^{α} q_{C i} - \frac{\partial H_{C}}{\partial t} - \frac{\partial H_{C}}{\partial q_{C i}} {\dot{q}}_{C i} - \frac{\partial H_{C}}{\partial p_{C i}} {\dot{p}}_{C i}) ξ_{C 0} + (p_{C i} {}_{t_{1}}^{C}D D_{t}^{α} q_{C i} - H_{C}) {\dot{ξ}}_{C 0} \\ + p_{C i} {}_{t_{1}}^{C}D D_{t}^{α} (ξ_{C i} - {\dot{q}}_{C i} ξ_{C 0}) - (ξ_{C i} - {\dot{q}}_{C i} ξ_{C 0}) {}_{t}^{R L}D D_{t_{2}}^{α} p_{C i} - \frac{p_{C i}}{Γ (1 - α)} {(t - t_{1})}^{- α} {\dot{q}}_{C i} (t_{1}) ξ_{C 0} (t_{1}) \\ = \frac{\partial H_{C}}{\partial q_{C i}} ξ_{C i} + ({\dot{p}}_{C i} {}_{t_{1}}^{C}D D_{t}^{α} q_{C i} - \frac{\partial H_{C}}{\partial p_{C i}} {\dot{p}}_{C i} - \frac{\partial H_{C}}{\partial q_{C i}} {\dot{q}}_{C i}) ξ_{C 0} - (ξ_{C i} - {\dot{q}}_{C i} ξ_{C 0}) {}_{t}^{R L}D D_{t_{2}}^{α} p_{C i}^{α} - λ_{C a} \frac{\partial ϕ_{C a}}{\partial p_{C i}} η_{C i} \\ = (\frac{\partial H_{C}}{\partial q_{C i}} - {}_{t}^{R L}D D_{t_{2}}^{α} p_{C i} + λ_{C a} \frac{\partial ϕ_{C a}}{\partial q_{C i}}) (ξ_{C i} - {\dot{q}}_{C i} ξ_{C 0}) - λ_{C a} \frac{\partial ϕ_{C a}}{\partial p_{C i}} η_{C i} - λ_{C a} \frac{\partial ϕ_{C a}}{\partial q_{C i}} (ξ_{C i} - {\dot{q}}_{C i} ξ_{C 0}) + ({\dot{p}}_{C i} {}_{t_{1}}^{C}D D_{t}^{α} q_{C i} - \frac{\partial H_{C}}{\partial p_{C i}} {\dot{p}}_{C i}) ξ_{C 0} \\ = - λ_{C a} \frac{\partial ϕ_{C a}}{\partial p_{C i}} η_{C i} + {\dot{p}}_{C i} ξ_{C 0} λ_{C a} \frac{\partial ϕ_{C a}}{\partial p_{C i}} - λ_{C a} \frac{\partial ϕ_{C a}}{\partial q_{C i}} (ξ_{C i} - {\dot{q}}_{C i} ξ_{C 0}) = - λ_{C a} \frac{\partial ϕ_{C a}}{\partial p_{C i}} (η_{C i} - {\dot{p}}_{C i} ξ_{C 0}) - λ_{C a} \frac{\partial ϕ_{C a}}{\partial q_{C i}} (ξ_{C i} - {\dot{q}}_{C i} ξ_{C 0}) = 0, \end{array}$ Mathematical equation

where

$\begin{array}{l} δ ϕ_{C a} (t, q_{C j}, p_{C j}) = \frac{\partial ϕ_{C a}}{\partial q_{C i}} δ q_{C i} + \frac{\partial ϕ_{C a}}{\partial p_{C i}} δ p_{C i} = \frac{\partial ϕ_{C a}}{\partial q_{C i}} (Δ q_{C i} - {\dot{q}}_{C i} Δ t) + \frac{\partial ϕ_{C a}}{\partial p_{C i}} (Δ p_{C i} - {\dot{p}}_{C i} Δ t) \\ = θ_{C} [\frac{\partial ϕ_{C a}}{\partial q_{C i}} (ξ_{C i} - {\dot{q}}_{C i} ξ_{C 0}) + \frac{\partial ϕ_{C a}}{\partial p_{C i}} (η_{C i} - {\dot{p}}_{C i} ξ_{C 0})] = 0 . \end{array}$ Mathematical equation

The proof is completed.

Let $Δ I_{C} = {\bar{I}}_{C} - I_{C} = - \int_{t_{1}}^{t_{2}} \frac{d}{d t} (Δ G_{C}) d t$ Mathematical equation , where $Δ G_{C} = θ_{C} G_{C}$ and $G_{C} = G_{C} (t, q_{C j}, p_{C j})$ is a gauge function with the Caputo fractional derivatives; then, from Eq. (37), we obtain

$\begin{array}{l} p_{C i} {}_{t_{1}}^{C}D D_{t}^{α} (ξ_{C i} - {\dot{q}}_{C i} ξ_{C 0}) + (p_{C i} \frac{d}{d t} {}_{t_{1}}^{C}D D_{t}^{α} q_{C i} - \frac{\partial H_{C}}{\partial t}) ξ_{C 0} - \frac{\partial H_{C}}{\partial q_{C i}} ξ_{C i} + λ_{C a} \frac{\partial ϕ_{C a}}{\partial p_{C i}} η_{C i} \\ - \frac{p_{C i}}{Γ (1 - α)} {(t - t_{1})}^{- α} {\dot{q}}_{C i} (t_{1}) ξ_{C 0} (t_{1}) + (p_{C i} {}_{t_{1}}^{C}D D_{t}^{α} q_{C i} - H_{C}) {\dot{ξ}}_{C 0} + {\dot{G}}_{C} = 0, i = 1,2, \dots, n \end{array}$ Mathematical equation (39)

Equation (39) is called the fractional Noether-quasi identity with the Caputo fractional derivatives for the fractional constrained Hamiltonian system (Eq. (17)). The infinitesimal transformations in this case are called Noether-quasi symmetric transformations, which determine the Noether-quasi symmetry. Then, a fractional conserved quantity with the Caputo fractional derivatives can be obtained.

Theorem 4 If the infinitesimal generators $ξ_{C 0}$ Mathematical equation , $ξ_{C i}$ , $η_{C i}$ , and a gauge function $G_{C}$ satisfy the fractional Noether-quasi identity (Eq. (39)), then a fractional conserved quantity with the Caputo fractional derivatives exists for the fractional constrained Hamiltonian system (Eq. (17))

$\begin{array}{l} C_{G C} = (p_{C i} {}_{t_{1}}^{C}D D_{t}^{α} q_{C i} - H_{C}) ξ_{C 0} + \int_{t_{1}}^{t} [p_{C i} {}_{t_{1}}^{C}D D_{τ}^{α} (ξ_{C i} - {\dot{q}}_{C i} ξ_{C 0}) - (ξ_{C i} - {\dot{q}}_{C i} ξ_{C 0}) {}_{τ}^{R L}D D_{t_{2}}^{α} p_{C i}] d τ \\ - \int_{t_{1}}^{t} \frac{p_{C i}}{Γ (1 - α)} {(τ - t_{1})}^{- α} {\dot{q}}_{C i} (t_{1}) ξ_{C 0} (t_{1}) d τ + G_{C} = c o n s t \end{array}$ Mathematical equation (40)

Proof The intended result can be obtained from Eqs. (17), (39) and (40).

Remark 2 Based on the Caputo fractional derivatives, the Noether-quasi symmetry is more general than the Noether symmetry. In fact, by setting $G_{C} = 0$ Mathematical equation , Theorem 4 reduces to Theorem 3.

Remark 3 Based on the Caputo fractional derivatives, if let $α \to 1$ Mathematical equation , then the fractional primary constraint (Eq. (16)), the fractional constrained Hamilton equation (Eq. (17)) and the Noether theorem (Theorem 3) reduce to the corresponding classical integer-order cases, which are consistent with the results in Ref. [6].

An example is presented to illustrate the results and methods above.

Example 2 For the Lagrangian

$L_{C} = q_{C 2} \cdot {}_{t_{1}}^{C}D D_{t}^{α} q_{C 1} - q_{C 1} \cdot {}_{t_{1}}^{C}D D_{t}^{α} q_{C 2} + {(q_{C 1})}^{2} + {(q_{C 2})}^{2}$ Mathematical equation (41)

find its conserved quantity.

For this Lagrangian $L_{C}$ Mathematical equation , there exist two fractional primary constraints ^[30]:

$ϕ_{C 1} = p_{C 1} - q_{C 2} = 0, ϕ_{C 2} = p_{C 2} + q_{C 1} = 0 .$ Mathematical equation (42)

The fractional constrained Hamilton equations can also be established ^[30]:

${}_{t}^{R L} D_{t_{2}}^{α} p_{C 1} = - 2 q_{C 1} + {}_{t_{1}}^{C}D D_{t}^{α} q_{C 2}, {}_{t}^{R L}D D_{t_{2}}^{α} p_{C 2} = - 2 q_{C 2} - {}_{t_{1}}^{C}D D_{t}^{α} q_{C 1}$ Mathematical equation (43)

The fractional Noether-quasi identity (Eq. (39)) gives

$\begin{array}{l} p_{C 1} {}_{t_{1}}^{C}D D_{t}^{α} (ξ_{C 1} - {\dot{q}}_{C 1} ξ_{C 0}) + p_{C 2} {}_{t_{1}}^{C}D D_{t}^{α} (ξ_{C 2} - {\dot{q}}_{C 2} ξ_{C 0}) + 2 q_{C 1} ξ_{C 1} + 2 q_{C 2} ξ_{C 2} + λ_{C 1} η_{C 1} + λ_{C 2} η_{C 2} + (p_{C 1} \frac{d}{d t} {}_{t_{1}}^{C}D D_{t}^{α} q_{C 1} + p_{C 2} \frac{d}{d t} {}_{t_{1}}^{C}D D_{t}^{α} q_{C 2}) ξ_{C 0} \\ - \frac{p_{C 1}}{Γ (1 - α)} {(t - t_{1})}^{- α} {\dot{q}}_{C 1} (t_{1}) ξ_{C 0} (t_{1}) - \frac{p_{C 2}}{Γ (1 - α)} {(t - t_{1})}^{- α} {\dot{q}}_{C 2} (t_{1}) ξ_{C 0} (t_{1}) + (p_{C 1} {}_{t_{1}}^{C}D D_{t}^{α} q_{C 1} + p_{C 2} {}_{t_{1}}^{C}D D_{t}^{α} q_{C 2} - H_{C}) {\dot{ξ}}_{C 0} + {\dot{G}}_{C} = 0 \end{array}$ Mathematical equation (44)

Through computation, we can verify that

$ξ_{C 0} = - 1, ξ_{C 1} = ξ_{C 2} = 0, η_{C 1} = η_{C 2} = 0, G_{C} = 0$ Mathematical equation (45)

is a solution to Eq. (44). Finally, Theorem 4 gives the fractional conserved quantity

$C_{G C} = \int_{t_{1}}^{t} (p_{C 1} \frac{d}{d τ} {}_{t_{1}}^{C}D D_{τ}^{α} q_{C 1} + p_{C 2} \frac{d}{d τ} {}_{t_{1}}^{C}D D_{τ}^{α} q_{C 2}) d τ - [p_{C 1} {}_{t_{1}}^{C}D D_{t}^{α} q_{C 1} + p_{C 2} {}_{t_{1}}^{C}D D_{t}^{α} q_{C 2} + q_{C 1}^{2} + q_{C 2}^{2}]$ Mathematical equation (46)

4 Conclusion

Noether theorems for the singular systems with the mixed derivatives and with only Caputo fractional derivatives are studied for the first time. Theorems 1-4 are all new work. Besides, the constrained Hamiltonian system on time scales is another topic deserved to be done.

References

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[R1] Parra J C. On singular lagrangians and Dirac's method[J]. Revista Mexicana De Física, 2012, 58(1): 61-68. [MathSciNet] [Google Scholar]

[R2] Deriglazov A. Classical Mechanics Hamiltonian and Lagrangian Formalism [M]. Berlin: Springer-Verlag, 2010. [Google Scholar]

[R3] Cawley R. Determination of the Hamiltonian in the presence of constraints[J]. Physical Review Letters, 1979, 42(7): 413-416. [NASA ADS] [CrossRef] [Google Scholar]

[R4] Mittelstaedt P. Klassische Mechanik [M]. Germany: Hochschultaschenbücher Verlag, 1970. [Google Scholar]

[R5] Li Z. Symmetries in Constrained Canonical Systems [M]. Beijing: Science Press, 2002(Ch). [Google Scholar]

[R6] Li Z P. Contrained Hamiltonian Systems and Their Symmetrical Properties [M]. Beijing: Beijing Polytechnic University Press, 1999(Ch). [Google Scholar]

[R7] Li Z P. Classical and Quantal Dynamics of Contrained Systems and Their Symmetrical Properties [M]. Beijing: Beijing Polytechnic University Press, 1993(Ch). [Google Scholar]

[R8] Dirac P A M. Generalized Hamiltonian dynamics[J]. Canadian Journal of Mathematics, 1950, 2: 129-148. [CrossRef] [Google Scholar]

[R9] Noether A E. Invariante variationsprobleme [J]. Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, 1918, KI: 235-257. [Google Scholar]

[R10] Mei F X, Wu H B. Dynamics of Constrained Mechanical Systems [M]. Beijing: Beijing Institute of Technology Press, 2009. [Google Scholar]

[R11] Mei F X. Aanlytical Mechanics [M]. Beijing: Beijing Institute of Technology Press, 2013. [Google Scholar]

[R12] 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]

[R13] Frederico G S F, Torres D F M. A formulation of Noether's theorem for fractional problems of the calculus of variations[J]. Journal of Mathematical Analysis and Applications, 2007, 334(2): 834-846. [NASA ADS] [CrossRef] [MathSciNet] [Google Scholar]

[R14] Frederico G S F, Torres D F M. Fractional conservation laws in optimal control theory[J]. Nonlinear Dynamics, 2008, 53(3): 215-222. [CrossRef] [MathSciNet] [Google Scholar]

[R15] Frederico G S F, Torres D F M. Fractional isoperimetric Noether's theorem in the Riemann-Liouville sense[J]. Reports on Mathematical Physics, 2013, 71(3): 291-304. [NASA ADS] [CrossRef] [MathSciNet] [Google Scholar]

[R16] Frederico G S F, Torres D F M. Fractional Noether's theorem in the Riesz-Caputo sense[J]. Applied Mathematics and Computation, 2010, 217(3): 1023-1033. [CrossRef] [MathSciNet] [Google Scholar]

[R17] Frederico G S F. Fractional optimal control in the sense of Caputo and the fractional Noether's theorem[J]. International Mathematical Forum, 2008, 3(10): 479-493. [MathSciNet] [Google Scholar]

[R18] Jia Q L, Wu H B, Mei F X. Noether symmetries and conserved quantities for fractional forced Birkhoffian systems[J]. Journal of Mathematical Analysis and Applications, 2016, 442(2): 782-795. [CrossRef] [MathSciNet] [Google Scholar]

[R19] Zhou Y. The Fractional Pfaff-Birkhoff Variational Problem and Its Symmetries [D]. Suzhou: Suzhou University of Science and Technology, 2013(Ch). [Google Scholar]

[R20] Zhou S, Fu H, Fu J L. Symmetry theories of Hamiltonian systems with fractional derivatives[J]. Science China Physics, Mechanics and Astronomy, 2011, 54(10): 1847. [CrossRef] [Google Scholar]

[R21] Malinowska A B. A formulation of the fractional Noether-type theorem for multidimensional Lagrangians[J]. Applied Mathematics Letters, 2012, 25(11): 1941-1946. [CrossRef] [MathSciNet] [Google Scholar]

[R22] Atanacković T M, Konjik S, Pilipović S, et al. Variational problems with fractional derivatives: Invariance conditions and Noether's theorem [J]. Nonlinear Analysis Theory Methods & Applications, 2009, 71: 1504-1517. [CrossRef] [Google Scholar]

[R23] Zhai X H, Zhang Y. Noether symmetries and conserved quantities for fractional Birkhoffian systems with time delay [J]. Commun Nonlinear Sci Numer Simulat, 2016, 36: 81-97. [NASA ADS] [CrossRef] [MathSciNet] [Google Scholar]

[R24] Zhang Y, Zhai X H. Noether symmetries and conserved quantities for fractional Birkhoffian systems[J]. Nonlinear Dynamics, 2015, 81: 469-480. [CrossRef] [MathSciNet] [Google Scholar]

[R25] Song C J, Zhang Y. Noether symmetry and conserved quantity for fractional Birkhoffian mechanics and its applications [J]. Fract Calc Appl Anal, 2018, 21: 509-526. [CrossRef] [MathSciNet] [Google Scholar]

[R26] Zhang S H, Chen B Y, Fu J L. Hamilton formalism and Noether symmetry for mechanico electrical systems with fractional derivatives [J]. Chin Phys B, 2012, 21: 100202. [CrossRef] [Google Scholar]

[R27] Song C J. Noether symmetry for fractional Hamiltonian system [J]. Phys Lett A, 2019, 29: 125914. [NASA ADS] [CrossRef] [MathSciNet] [Google Scholar]

[R28] Jin S X, Zhang Y. Noether theorem for non-conservative systems with time delay in phase space based on fractional model[J]. Nonlinear Dynamics, 2015, 82: 663-676. [CrossRef] [MathSciNet] [Google Scholar]

[R29] Jin S X, Zhang Y. Noether theorem for non-conservative Lagrange systems with time delay based on fractional model[J]. Nonlinear Dynamics, 2015, 79: 1169-1183. [CrossRef] [MathSciNet] [Google Scholar]

[R30] Song C J, Agrawal O P. Hamiltonian formulation of systems described using fractional singular Lagrangian[J]. Acta Applicandae Mathematicae, 2021, 172(1): 9. [CrossRef] [MathSciNet] [Google Scholar]

[R31] Podlubny I. Fractional Differential Equations [M]. San Diego: Academic Press, 1999. [Google Scholar]