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Lie symmetries and conserved quantities of the constraint mechanical systems on time scales
Vol. 79 (2017)
REPORTS ON MATHEMATICAL PHYSICS
No. 3
LIE SYMMETRIES AND CONSERVED QUANTITIES OF THE CONSTRAINT MECHANICAL SYSTEMS ON TIME SCALES P ING -P ING C AI , J ING -L I F U* Institute of Mathematical Physics, Zhejiang Sci-Tech University, Hangzhou 310018, China (e-mails: [email protected], [email protected])
and YONG -X IN G UO Department of Physics, Liaoning University, Shenyang 110036, China (e-mail: [email protected]) (Received February 8, 2016 — Revised January 3, 2017) We introduce a new method to study Lie symmetries and conserved quantities of constraint mechanical systems which include Lagrangian systems, nonconservative systems and nonholonomic systems on time scales T. For the constraint mechanical systems on time scales, based on the transformation Lie group, we get a series of significant results including the variational principle of systems on time scales, the equations of motion, the determining equations, the structure equations, the restriction equations as well as the Lie theorems of the Lie symmetries of the systems on time scales. Furthermore, a set of new conserved quantities of the constraint mechanical systems on time scales are given. More significant is that this work unifies the theories of Lie symmetries of the two cases for the continuous and the discrete constraint mechanical systems by applying the time scales. And then taking the discrete (T = Z) nonholonomic system for example, we derive the corresponding discrete Lie symmetry theory. Finally, two examples are designed to illustrate these results. Keywords: time scale, Lie symmetry, constraint mechanical system.
1.
Introduction The theory of time scales originated in 1988 with the work of Stefan Hilger [1] in order to unify various concepts from the theories of discrete and continuous dynamical systems, and to extend such theories to more general classes of dynamical systems. The time scales calculus theory is applicable to any field in which dynamic processes can be described with discrete or continuous models. The study of the calculus of variations in the context of time scales has its beginning in 2004 with the paper of Martin Bohner [2], with two excellent books dedicated to it [3, 4]. Since the pioneering paper [2], the classical results of the calculus of variations on continuous time (T = R) and discrete time (T = Z) have been unified and generalized to * Corresponding
author. [279]
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a time scale T: Euler–Lagrange equations [5, 6], necessary optimality conditions for variational problems subject to isoperimetric constraints [7, 8], boundary value problems [9, 10], high-order delta derivatives [11–13], weak maximum principle for variable endpoints optimal control problems [14], an invariant group of parameter transformations [15]. The more general theory of the calculus of variations on time scales seems to be useful in applications to economics [16]. It is significant to study the symmetries and conserved quantities of dynamical systems not only in mathematics but also in physics. And the development of modern physics such as quantum mechanics, quantum field theory, nuclear physics, and spatially inhomogeneous nonlinearities shows that the principle of symmetry has become the most important principle of exploring the laws of motion of microparticles. There are two main symmetry methods used to seek the conserved quantities of a dynamical system: the Noether symmetry and the Lie symmetry [17, 18]. The Noether method is making good progress, such as Herglotz variational problems [19], problems with isoperimetric constraints [20], problems with delay arguments [21] and problems with noninteger order derivatives [22]. The Lie method has been approved to be a powerful tool to solve differential equations, to study constraint mechanical systems, to discuss controllable dynamical systems, to investigate mechanico-electrical systems and to establish properties of their solution spaces. The theories of Lie symmetry have been described in many papers [23–27]. The Lie group theory has also been applied to discrete equations, such as differential-difference equations, discrete dynamical systems and discrete mechanico-electrical systems [28, 29]. We have seen that the theories of Lie symmetry of the continuous and the discrete systems have been deeply studied and they are studied in two different directions. Here we introduce the concept of time scale which unifies the theories of Lie symmetries of the two cases for the continuous (T = R) and the discrete (T = Z) constraint mechanical systems. This paper systematically studies the conserved quantities and Lie symmetries for constraint mechanical systems on an arbitrary time scale T which unifies and extends the previous formulations of Lie’s method in the discrete-time and continuous domains (cf. [23–29] and references therein). By defining the isochronous variation on time scales, we study the exchanging relationships between the isochronous variation and the delta derivatives as well as the relationships between the isochronous variation and the total variation on time scales. On the basis of the invariance theory of differential equations of motion under the infinitesimal transformations, we study the Lie symmetry of the Lagrangian system. By virtue of the Hamilton principle, we construct the equations of motion of nonconservative systems on time scales, and then we give the method of solution of the Lie symmetries of nonconservative systems on time scales. Furthermore, using the d’Alembert–Lagrange principle on time scales we derive the equations of motion of nonholonomic systems of Chetaev type. Then, by presenting the Lie symmetrical determining equations, the structure equations and the constrained restriction equations of the constraint mechanical systems on time scales T, we get a new type of Noether conserved quantity which only depends on the variables t, q, q σ , q 1 . For the classical discrete time (T = Z)
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we can reduce previous results on an arbitrary time scale to the classical results of the Lie symmetries of the discrete nonholonomic systems. Finally, in order to illustrate applications of the results, we calculate two examples. 2.
Basics on the time scale calculus A time scale is a nonempty closed subset of real numbers, and we usually denote it by the symbol T. The two most popular examples are (T = R) and (T = Z). We define the forward and backward jump operators σ, ρ : T → T by σ (t) = inf{s ∈ T : s > t} and ρ(t) = sup{s ∈ T : s < t}, for all t ∈ T,
(supplemented by inf Ø = sup T and Ø = inf T). The graininess function µ : T → [0, ∞) is defined by µ(t) = σ (t) − t. (1)
Hence the graininess function is constant and equal to 0 if (T = R) while it is constant and equal to 1 for (T = Z). However, a time scale T could have nonconstant graininess. A point t ∈ T is called right scattered, right dense, left scattered and left dense if σ (t) > t, σ (t) = t, ρ(t) < t, ρ(t) = t hold, respectively. Throughout we let a, b ∈ T with a < b. For an interval [a, b] ∩ T we simply write [a, b] when this is not ambiguous. We also define [a, b]κ := [a, b]\(ρ(b), b]
and
2
[a, b]κ := [a, b]\(ρ(ρ(b)), b].
We say that a function f : T → R is delta differentiable at t ∈ Tκ provided there exists a real number f 1 (t) such that for all ε > 0 there is a neighbourhood U = (t − δ, t + δ) ∩ T of t with f (σ (t)) − f (s) − f 1 (t) (σ (t) − s) ≤ ε |σ (t) − s| for all t ∈ U. For differentiable f , the formula
f σ = f + µf 1
(2)
is very useful and easy to prove. If f and g are both differentiable, then so is f g with (3) (fg)1 = f 1 g + f σ g 1 ,
where we abbreviate f ◦ σ by f σ . Next, a function f : T → R is called rd-continuous if it is continuous in right-dense points and if its left-sided limits exist in left-dense points. Crd denotes 1 the set of all rd-continuous functions, while Crd denotes the set of all differentiable functions with rd-continuous derivative. It is known that rd-continuous functions possess an antiderivative, i.e. there exists a function F : T → R with F 1 (t) = f (t), and in this case an integral of f from a to b (a, b ∈ T) is defined by Z b f (t)1t = F (b) − F (a) . (4) a
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3. Variational relationships on time scales 3.1.
Exchange relationship between the isochronous variation and the delta derivatives
Consider two infinitely closed orbits α and α + dα. We denote the generalized coordinates by q = q(t, α), q = q(t, α + dα), corresponding to the two infinitely closed orbits, respectively, in giving time on time scale T. We define the isochronous variation as δq = q (t, α + dα) − q (t, α) .
(5)
Extending q = q(t, α + dα) to the linear terms of dα, we obtain q (t, α + dα) = q (t, α) +
∂q (t, α) dα. ∂α
(6)
Substituting Eq. (6) into Eq. (5), we get δq = Similarly we have
∂q (t, α) dα. ∂α
(7)
1 ∂q (t, α) 1 δq = dα. 1t 1t ∂α
(8)
According to Eq. (6) we get δq 1 = q 1 (t, α + dα) − q 1 (t, α) =
1 ∂q (t, α) dα, 1t ∂α
(9)
and comparing Eq. (9) with Eq. (8), we obtain Similarly we have
δq 1 = (δq)1 .
(10)
δq σ = (δq)σ .
(11)
We call Eqs. (10) and (11) the exchanging relationships with respect to the delta derivatives and isochronous variation. 3.2.
The isochronous variation on time scales
We continue studying the infinitely closed orbits α and α + dα. The generalized coordinates are given by q = q(t, α), q ∗ = q ∗ (t, α), for any t ∈ T, where t = t (α), so we have q = q [t (α) , α] , taking total variation for q we obtain
∂q [t (α) , α] ∂q [t (α) , α] ∂t dα + dα. ∂α ∂t ∂α Since 1t is the variation of time with respect to α, therefore 1q =
1t =
∂t dα. ∂α
(12)
(13)
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Substituting Eqs. (7) and (13) into Eq. (12), we get 1q = δq + q 1 1t.
(14)
We call Eq. (14) the relationship between the isochronous variation and the total variation on time scales T. Using Eqs. (10) and (14) we have 1q 1 = δq 1 + q 11 · 1t.
(15)
Differentiating both sides of Eq. (14) with respect to t we obtain (1q)1 = δq 1 + q 11 · (1t)σ + q 1 · (1t)1 , and according to Eqs. (15) and
(16), we get
1q 1 = (1q)1 − q 11 (1t)σ − q 1 (1t)1 + q 11 · 1t = (1q)1 − q 1 (1t)1 − µ (t) · (1t)1 q 11 .
Similarly we have
1q 11 = δq 11 + q 111 · 1t, σ2
(1q)11 = δq 11 + q 111 (1t) From Eqs. (18) and
(16)
+ q 11 (1t)σ 1 + q 11 (1t)1σ + q 1 (1t)11 .
(17)
(18) (19)
(19) we obtain 2
1q 11 = q 111 · 1t + (1q)11 − q 111 (1t)σ − q 11 (1t)σ 1 − q 11 (1t)1σ − q 1 (1t)11 . (20) Differentiating both sides of Eq. (17) with respect to t we obtain 1 2 1q 1 =(1q)11 − q 111 (1t)σ − q 11 (1t)σ 1 − q 11 (1t)1σ − q 1 (1t)11 (21) + q 111 (1t)σ + q 11 (1t)1 . By virtue of Eqs. (20) and (21) we get 1 1q 11 = 1q 1 − q 11 (1t)1 − µ (t) · (1t)1 q 111 . 4. 4.1.
(22)
Lie symmetry of the Lagrange systems on time scales The Lagrange equations of a Lagrangian system
The equations of motion of a Lagrangian system with delta derivatives can be written in the form [2] 1 ∂3 L t, q σ (t) , q 1 (t) = ∂2 L t, q σ (t) , q 1 (t) , (23) 1t where L = L t, q σ , q 1 is the Lagrangian of the system on time scales. In general,
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it is supposed that the system (23) is nonsingular, i.e. ∂ 2L ∂ 2L 1 6= 0. 2 + µ 1 + µ ∂q 1 ∂q σ ∂ q1
(24)
Expanding Eqs. (23), we can seek the generalized acceleration with delta derivatives as q 11 = h t, q σ , q 1 . (25)
4.2.
Infinitesimal transformations and the determining equations
We introduce the infinitesimal transformations in terms of time and generalized coordinate as t ∗ = t + 1t, q ∗ = q(t) + 1q. (26) or their expanded form t ∗ = t + ετ (t, q) + o (ε) ,
q ∗ (t) = q (t) + εξ (t, q) + o (ε) .
(27)
where ε is the infinitesimal parameter and τ (t, q), ξ(t, q) are the generators of infinitesimal transformations. Eqs. (27) are one-parameter Lie-point group of transformations. Taking the infinitesimal generator vector X(0) = τ
∂ ∂ + ξσ σ , ∂t ∂q
(28)
the first extended infinitesimal generator is X(1) = τ
∂ ∂ ∂ + ξ σ σ + ξ 1 − τ 1 q 1 − µ (t) τ 1 q 11 , ∂t ∂q ∂q 1
and the second is h i 1 X(2) = X (1) + ξ 1 − τ 1 q 1 − µ (t) τ 1 q 11 − τ 1 q 11 − µ (t) τ 1 q 111
(29)
∂ , (30) ∂q 11
where
1 ξ (t, q (t)) 1t and τ (t, q), ξ(t, q) are infinitesimal generators. Then based on the invariance of the differential equations (25) under the infinitesimal transformations (27), if and only if [30] X(2) q 11 − h t, q σ , q 1 = 0, (31) ξ σ (t, q (t)) = ξ (σ (t) , q (σ (t))) , ξ 1 (t, q (t)) =
we can have
h
i h i 1 + 2τ 1 + µτ 11 h − µ µτ 1 + 2µτ 1 q 111 = X(1) h. (32) Eqs. (32) are called the determining equations on time scales T which the generators ξ and τ should satisfy. If ξ and τ satisfy the determining equations (32), ξ 11 − τ 11 q 1 −
µτ 1
1
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the corresponding transformations are the Lie symmetry’s transformations of the Lagrangian systems on time scales. 4.3.
The structure equation and conserved quantity
D EFINITION 1. A quantity C t, q, q σ , q 1 is said to be a conserved quantity if 1 C t, q (t) , q σ (t) , q 1 (t) = 0 is preserved along all q(t) that satisfy and only if 1t the Euler–Lagrange equations (23).
Lie symmetries do not always generate Noether conserved quantities. The subsequent propositions give the condition under which Lie symmetries generate conserved quantities and the form of conserved quantities. T HEOREM 1. For the infinitesimal generators ξ, τ satisfying the determining Eqs. (32), if there is a gauge function G = G t, q σ , q 1 satisfying the equation Lτ 1 + X(1) L + ∂3 L · µτ 1 q 11 + G1 = 0,
(33)
then the system possesses a conserved quantity with delta derivatives C t, q, q σ , q 1 = ∂3 L t, q σ , q 1 · ξ (t, q) + L t, q σ , q 1 − ∂3 L t, q σ , q 1 · q 1 − ∂1 L t, q σ , q 1 · µ (t) · τ (t, q) = const . (34) Proof : Using the Euler–Lagrange equations (23) and the structure equation (33), we obtain 1 [∂3 L t, q σ , q 1 · ξ (t, q) 1t + L t, q σ , q 1 − ∂3 L t, q σ , q 1 · q 1 − ∂1 L t, q σ , q 1 · µ (t) · τ (t, q) 1 + G t, q σ , q 1 = ∂3 L t, q σ , q 1 · ξ σ (t, q) + ∂3 L t, q σ , q 1 · ξ 1 (t, q) 1t 1 σ + τ (t, q) · L t, q σ , q 1 − ∂3 L t, q σ , q 1 · q 1 − ∂1 L t, q σ , q 1 · µ (t) 1t 1 + τ (t, q) · L t, q σ , q 1 − ∂3 L t, q σ , q 1 · q 1 − ∂1 L t, q σ , q 1 µ (t) + G1 t, q σ , q 1 = ∂2 L t, q σ , q 1 · ξ σ (t, q) + ∂3 L t, q σ , q 1 · ξ 1 (t, q) + ∂1 L t, q σ , q 1 · τ (t, q) + L t, q σ , q 1 · τ 1 (t, q) − ∂3 L t, q σ , q 1 · τ 1 (t, q) · q 1 + G1 t, q σ , q 1 1 G t, q σ , q 1 = L t, q σ , q 1 τ 1 + X(1) L t, q σ , q 1 + ∂3 L t, q σ , q 1 · µτ 1 q 11 + 1t = 0. Eq. (33) is called the structure equation for the Lagrangian systems on time scales.
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If the transformations (27) satisfy Noether’s identity ∂1 L t, q σ , q 1 · τ (t, q) + ∂2 L t, q σ , q 1 · ξ σ (t, q) + ∂3 L t, q σ , q 1 · ξ 1 (t, q) 1 + L t, q σ , q 1 · τ 1 − ∂3 L t, q σ q 1 · τ 1 · q 1 = − G t, q σ , q 1 , (35) 1t then the transformations (27) are called the Noether symmetrical transformations of the system (25). T HEOREM 2. The structure equation with delta derivatives (33) of the Lie symmetry is equivalent to the Noether identity (35). 5. Lie symmetries of nonconservative systems on time scales 5.1.
Hamilton’s principle and Lagrange equations for nonconservative systems with delta derivatives
L EMMA 1 Z b a
[2]
(Dubois–Reymond). Let g ∈ Crd , g : [a, b] → Rn , then
1 g T (t) · η1 (t) 1t = 0 for all η ∈ Crd with η (a) = η (b) = 0
holds if and only if g (t) ≡ c on [a, b]κ for some c ∈ Rn . Assuming that the kinetic energy function of the system is T = T t, q σ , q 1 , Hamilton’s principle states that the actual pace exists when the Hamiltonian action has determining value. So the Hamilton principle for nonconservative systems with delta derivatives can be written in the form Z b (36) (δT + Qδq σ )1t = 0, a
σ
where Qδq is the virtual work of the generalized force. Taking total variation for the function T , we get δT =
∂T ∂T δq σ + 1 δq 1 , σ ∂q ∂q
(37)
substituting Eq. (37) into Eq. (36) we have Z b Z b ∂T ∂T ∂T ∂T σ 1 σ σ 1 δq + 1 δq + Qδq 1t = Q + σ δq + 1 δq 1t ∂q σ ∂q ∂q ∂q a a Z b ∂T ∂T σ 1 = Q + σ (δq) + 1 (δq) 1t ∂q ∂q a
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#1 ! ∂T τ, q σ (τ ) , q 1 (τ ) 1τ · δq = Q+ ∂q σ (τ ) a a # "Z ! t ∂T τ, q σ (τ ) , q 1 (τ ) ∂T 1 1 1τ (δq) + − Q+ (δq) 1t ∂q σ (τ ) ∂q 1 a ) ! Z b( Z t ∂T τ, q σ (τ ) , q 1 (τ ) ∂T = − Q+ 1τ (δq)1 1t = 0. 1 σ (τ ) ∂q ∂q a a Z
b
("Z
t
Therefore, by Lemma 1, we obtain ! Z b ∂T τ, q σ (τ ) , q 1 (τ ) ∂T − Q+ 1τ ≡ const, t ∈ [a, b] , ∂q 1 ∂q σ (τ ) a hence
1 ∂T ∂T − σ − Q = 0, 1 1t ∂q ∂q
(38)
when Q contains conservative force Q′ and nonconservative force Q′′ , and Q′ satisfies the following conditions: If Q′ is potential, i.e. there exists a function V = V (q σ , t) such that Q′ = −
∂V , ∂q σ
(39)
substituting Eq. (39) into (38) we obtain 1 ∂T ∂T ∂V − σ + σ − Q′′ = 0. 1 1t ∂q ∂q ∂q
(40)
As the function only depends on the generalized coordinates, therefore ∂V = 0, ∂q 1
(41)
using Eq. (41), from Eq. (40) we have 1 ∂ (T − V ) ∂ (T − V ) − = Q′′ . 1t ∂q 1 ∂q σ
(42)
Introducing function L = T − V , we can rewrite Eqs. (42) as 1 ∂L ∂L − σ = Q′′ . 1 1t ∂q ∂q
If Q′ has a generalized potential, i.e. there exists a function U = U t, q σ , q 1 such that ∂U 1 ∂U Q′ = σ − , ∂q 1t ∂q 1
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then Eqs. (38) can be written as 1 ∂ (T + U ) ∂ (T + U ) − = Q′′ , 1t ∂q 1 ∂q σ and introducing the function L = T + U = T − V , we can obtain the Euler–Lagrange equations of nonconservative systems with delta derivatives, ∂L 1 ∂L − σ = Q′′ . 1 1t ∂q ∂q 5.2.
(43)
Lie symmetries of the system
In general, it is supposed that the system (43) is nonsingular, i.e. ∂ 2L ∂ 2L 1 6= 0, + µ 1 + µ 2 ∂q 1 ∂q σ ∂ q1
and from Eqs. (43) we can find all accelerations as q 11 = β t, q σ , q 1 .
(44)
(45)
The necessary and sufficient conditions under which Eqs. (45) remain invariant are h i h i 1 1 ξ 11 − τ 11 q 1 − µτ 1 + 2τ 1 + µτ 11 β − µ µτ 1 + 2µτ 1 q 111 = X(1) β, (46) we hereinafter give the definition of Lie symmetries and conserved quantities of nonconservative systems on time scales. D EFINITION 2. If the generators ξ, τ satisfy the determining equations (46), then the corresponding symmetries are called Lie symmetries of holonomic systems (43) on time scales. T HEOREM 3. For the infinitesimal generators ξ, τ satisfying the determining Eqs. (46), if there is a gauge function G = G t, q σ , q 1 satisfying the structure equation Lτ 1 + X(1) (L) + ∂3 L · µτ 1 q 11 + Q ξ σ − τ σ q 1σ + G1 = 0, (47)
then the holonomic system possesses the following conserved quantity with delta derivative: C t, q, q σ , q 1 = ∂3 L t, q σ , q 1 · ξ (t, q) + L t, q σ , q 1 − ∂3 L t, q σ , q 1 · q 1 − ∂1 L t, q σ , q 1 · µ (t) · τ (t, q) = const . (48)
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Proof : Using Eqs. (43) and the structure equation (47) we obtain 1 [∂3 L t, q σ , q 1 · ξ (t, q) 1t + L t, q σ , q 1 −∂3 L t, q σ , q 1 ·q 1 −∂1 L t, q σ , q 1 ·µ (t) ·τ (t, q) 1 ∂3 L t, q σ , q 1 ·ξ σ (t, q)+∂3 L t, q σ , q 1 ·ξ 1 (t, q) +G t, q σ , q 1 = 1t 1 L t, q σ , q 1 −∂3 L t, q σ , q 1 ·q 1 − ∂1 L t, q σ , q 1 ·µ (t) +τ σ (t, q)· 1t +τ 1 (t, q)· L t, q σ , q 1 −∂3 L t, q σ , q 1 ·q 1 − ∂1 L t, q σ , q 1 µ (t) +G1 t, q σ , q 1 = (∂2 L t, q σ , q 1 +Q t, q σ , q 1 ·ξ σ (t, q)+∂3 L t, q σ , q 1 ·ξ 1 (t, q) +[∂1 L t, q σ , q 1 −Q t, q σ , q 1 q 1σ τ σ +τ 1 L t, q σ , q 1 −∂3 L t, q σ , q 1 ·q 1 −∂1 L t, q σ , q 1 ·µ (t) +G1 t, q σ , q 1 = Lτ 1 +X(1) (L)+Q ξ σ −τ σ q 1σ +G1 = 0.
T HEOREM 4. The structure equation of the holonomic system of the Lie symmetry (47) is equivalent to the following Noether’s identity, ∂1 L t, q σ , q 1 · τ (t, q) + ∂2 L t, q σ , q 1 · ξ σ (t, q) + ∂3 L t, q σ , q 1 · ξ 1 (t, q) 1 + L t, q σ , q 1 · τ 1 − ∂3 L t, q σ q 1 · τ 1 · q 1 + Q ξ σ − τ σ q 1σ = − G t, q σ , q 1 . 1t
The method of solution of Lie symmetries of holonomic nonconservative systems on time scales is the following: first, establish the determining equations (46) with the delta derivative, and seek the generator ξ, τ from these equations; secondly, substitute the generators obtained into the structure equation (47) to determine G t, q σ , q 1 ; finally, substitute ξ, τ and G t, q σ , q 1 into the formula (48) to get the conserved quantities of the symmetries of holonomic systems on time scales. 6. 6.1.
Lie symmetries of nonholonomic mechanical systems of Chetaev type on time scales The Lagrange equations for nonholonomic Chetaev systems
Let the configuration of a mechanical system be determined by the generalized coordinate q σ on time scale T. Its motion is subject to the ideal nonholonomic constraints of Chetaev type f β t, q σ , q 1 = 0, (β = 1, . . . , g) . (49)
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The restriction of conditions (49) on the virtual displacements are ∂f β σ δq = 0; ∂q 1
(50)
we call equations (50) the Chetaev condition on time scales T. The equations of motion can be written in the form 1 ∂L ∂L ∂f β ′′ − = Q + λ , (51) β 1t ∂q 1 ∂q σ ∂q 1 where Q′′ = Q′′ t, q σ , q 1 is the nonconservative force and L = L t, q σ , q 1 the Lagrangian of the system. Before integrating the equations of motion, we can determine λβ as functions of t, q σ , q 1 , λβ = λβ (t , q σ , q 1 . (52) Substituting this into Eqs. (51) we obtain
Let
∂L 1 ∂L − σ = Q′′ + 3, 1 1t ∂q ∂q ∂f β 3 = 3 t, q σ , q 1 = λβ 1 . ∂q ∂ 2L ∂ 2L 1 6= 0, 2 + µ 1 + µ ∂q 1 ∂q σ ∂ q1
expanding Eqs. (51) we can get all accelerations as q 11 = α t, q σ , q 1 ,
(53)
(54)
(55)
which are called the equations of motion of holonomic system corresponding to the nonholonomic system (49), (51). If the initial conditions satisfy Eqs. (49), the solution of Eqs. (55) gives the motion of the nonholonomic system. The invariance of Eqs. (55) and (49) under the infinitesimal transformations (27) leads to the fulfilment of the following determining equations with delta derivatives, h i h i 1 1 ξ 11 − τ 11 q 1 − µτ 1 + 2τ 1 + µτ 11 α − µ µτ 1 + 2µτ 1 q 111 = X(1) α, (56) and the inivariance of the constraint equations (49) under the infinitesimal transformations (27) can be made to satisfy the restriction equations X(1) f β t, q σ , q 1 = 0. (57) Considering the Chetaev conditions (50) and
δq σ = 1q σ − q σ 1 · 1t = ε ξ σ − τ σ q 1σ ,
(58)
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we have the following additional restriction equations ∂f β σ σ 1σ ξ − τ q =0 ∂q 1
(β = 1, . . . , g) .
(59)
D EFINITION 3. If the generators τ (t, q), ξ(t, q) satisfy the determining equations (56), then the transformations are called the Lie symmetrical transformations of the corresponding holonomic system (55). D EFINITION 4. If the generators τ (t, q), ξ(t, q) satisfy the determining equations (56) and the restriction equations (57), then the transformations are called the weakly Lie symmetrical transformations of the nonholonomic system (49), (51). D EFINITION 5. If the generators τ (t, q), ξ(t, q) satisfy the determining equations (56), the restriction equations (57) and the additional restriction equations (59), then the transformations are called the strongly Lie symmetrical transformations of the nonholonomic system (49), (51). The Lie symmetry can lead to a Noether-type conserved quantity under certain conditions. T HEOREM 5. For the generators τ (t, q), ξ(t, q) satisfying the determining equations (56) and the restriction equations (57), if there exists a gauge function G = G t, q σ , q 1 satisfying the structure equation Lτ 1 + X(1) (L) + ∂3 L · µτ 1 q 11 + Q′′ + 3 ξ σ − τ σ q 1σ + G1 = 0, (60)
then there exists a Noether-type conserved quantity corresponding to the Lie symmetries for the nonholonomic system of Chetaev type on time scales, i.e. C t, q, q σ , q 1 = ∂3 L t, q σ , q 1 · ξ (t, q) + L t, q σ , q 1 − ∂3 L t, q σ , q 1 · q 1 − ∂1 L t, q σ , q 1 · µ (t)] τ (t, q) + G t, q σ , q 1 = const . (61)
Proof : Using the Euler–Lagrange equations (53) and the structure equation (60) we obtain 1 [∂3 L t, q σ , q 1 · ξ (t, q) 1t + L t, q σ , q 1 − ∂3 L t, q σ , q 1 · q 1 − ∂1 L t, q σ , q 1 · µ (t) · τ (t, q) 1 + G t, q σ , q 1 = ∂3 L t, q σ , q 1 · ξ σ (t, q) + ∂3 L t, q σ , q 1 · ξ 1 (t, q) 1t 1 + τ σ (t, q) · L t, q σ , q 1 − ∂3 L t, q σ , q 1 · q 1 − ∂1 L t, q σ , q 1 · µ (t) 1t 1 + τ (t, q) · L t, q σ , q 1 − ∂3 L t, q σ , q 1 · q 1 − ∂1 L t, q σ , q 1 µ (t) + G1 t, q σ , q 1
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= (∂2 L t, q σ , q 1 +Q t, q σ , q 1 + 3 t, q σ , q 1 · ξ σ (t, q) + ∂3 L t, q σ , q 1 · ξ 1 (t, q) + [∂1 L t, q σ , q 1 − Q t, q σ , q 1 + 3 t, q σ , q 1 q 1σ τ σ + τ 1 L t, q σ , q 1 − ∂3 L t, q σ , q 1 · q 1 − ∂1 L t, q σ , q 1 · µ (t) + G1 t, q σ , q 1 = Lτ 1 + X(1) (L) + ∂3 L · µτ 1 q 11 + Q′′ + 3 ξ σ − τ σ q 1σ + G1 = 0.
T HEOREM 6. For the generators τ (t, q), ξ(t, q) satisfying the determining equations (56) and the restriction equations (57) and the additional restriction equa tions (59), if there exists a gauge function G = G t, q σ , q 1 satisfying the structure equation (60), then the system possesses the strongly Lie symmetrical conserved quantity (61). 7. Discussion We have seen that an invariance of the constraint mechanical systems can be obtained under the infinitesimal transformations with respect to both the time and the state variables on an arbitrary time scale T. To obtain the classical Noether theorems of the two cases, for the continuous and the discrete constraint mechanical systems, we assume T = R, T = Z, respectively. For the classical continuous time (T = R), the previous results are reduced to the classical results of the Lie symmetries [31]. Here let us take a nonholonomic system for example, we will consider the Lie symmetries of discrete nonholonomic systems on discrete time (T = Z). The time t is a discrete variable: t ∈ Z. The horizon consists of N periods, t = M, M + 1, . . . , M + N − 1 where M and N are fixed integers, instead of a continuous interval. The purpose of the following work is to deduce the discrete equations of motion, the discrete structure equations and the discrete conserved quantities of discrete nonholonomic systems on discrete time T = Z from the results we have obtained on an arbitrary time scale T. We refer to the literatures for further reading on Lie symmetries of the mechanical systems for the discrete case [28]. 7.1.
The discrete equations of motion for nonholonomic systems
In discrete mechanics, continuous curve in configuration space is replaced with the discrete sequence qt (t = M, M + 1, · · · , M + N − 1). The discrete versions of L and Q are L (t, qt+1 , 1q) and Q (t, qt+1 , 1q), respectively. So Eqs. (34) can be written in the form of discrete equations: ∂L (t, qt+1 , 1q) ∂L (t − 1, qt , 11 q) ∂L (t, qt+1 , 1q) − − ∂ (1q) ∂ (11 q) ∂qt+1 ′′ = Q (t, qt+1 , 1q) + 3 (t, qt+1 , 1q) , (t = M, M + 1, · · · , M + N − 1), where 3 = λα
∂f (k, qk+1 , 1q) ∂ (1q)
(62)
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are constraint forces corresponding to discrete nonholonimic constraints (and λα is discrete constraint multiplication), 1 the difference operator, 1q = qt+1 − qt . The discrete equations of motion can be put in the form 12 q = γ (t, qt+1 , 1q) ,
(63)
where we abbreviate 1 (1q) as 12 q. 7.2.
First integrals of the discrete nonoholonomic system
The vector field of generators (28) turns out to be ∂ ∂ , (64) X∗(0) = τk + ξk+1 ∂t ∂qk+1 which can be prolonged to the two- and three-point schemes ∂ ∂ ∂ X∗(1) = τk + ξk+1 + 1ξ − 1τ · 1q − 1τ · 12 q , (65) ∂t ∂qk+1 ∂ (1q) ∂ . X∗(2) = X∗(1) + 1 1ξ − 1τ · 1q − 1τ · 12 q − 1τ · 12 q − 1τ · 13 q ∂ 12 q (66)
The invariance of discrete equations (63) under the infinitesimal transformations (27) leads to the fulfilment of the following discrete determining equations 12 ξ − 1q · 12 τ − 2 1τ + 12 τ 12 q − 21τ + 12 τ 13 q = X∗(1) 12 q , (67)
where 1ξ = ξt+1 − ξt , 1τ = τt+1 − τt . The constrained restriction equation (57) becomes X∗ (1) f β (t, qt+1 , 1q) = 0,
(68)
and the discrete restriction of conditions f (t, qt+1 , 1q) = 0 on the discrete virtual displacements is given by ∂f (t, qt+1 , 1q) (69) (ξt+1 − τt+1 1qt+1 ) = 0, ∂ (1q) which are the discrete analogue of Appell–Chetaev conditions with respect to the discrete nonholonomic constraints restricting generators of infinitesimal transformations. We hereinafter give the Lie symmetries of the discrete nonholonomic systems on discrete-time (T = Z). C OROLLARY 1. For the infinitesimal generators τt , ξt satisfying the determining equation (67) and the restriction equation (68), if there exists a discrete gauge function G = G (t, qt+1 , 1q) such that the identity L (k, qk+1 , 1q) · 1τ + X∗(1) L (k, qk+1 , 1q)
+ ∂3 L (k, qk+1 , 1q) · 1τ · 12 q + 1G (k, qk+1 , 1q) = 0
(70)
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holds, then the discrete nonholonomic system has the discrete conserved quantities I (t, qt , qt+1 , 1q) = ∂3 L (t, qt+1 , 1q) · ξt (t, q) + L (t, qt+1 , 1q) − ∂3 L (t, qt+1 , 1q) · 1q − ∂1 L (t, qt+1 , 1q) · τt (t, q) = const . (71) Proof : Using Eqs. (62) and the discrete structure equation (70), we obtain ∂L (t, qt+1 , 1q) +3 (t, qt+1 , 1q)) ξt+1 1I = Q′′ (t, qt+1 , 1q) + ∂qt+1 ∂L (t, qt+1 , 1q) + · 1ξ ∂ (1q) ∂L (t, qt+1 , 1q) + − (Q (t, qt+1 , 1q) + 3 (t, qt+1 , 1q)) 1qt+1 τt+1 ∂t ∂L (t, qt+1 , 1q) ∂L (t, qt+1 , 1q) 1τ + L (t, qt+1 , 1q) − · 1q − ∂ (1q) ∂t + 1G (t, qt+1 , 1q) ∂L (t, qt+1 , 1q) ∂L (t, qt+1 , 1q) ∂L (t, qt+1 , 1q) = ξt+1 + τt + 1ξ + L (t, qt+1 , 1q) ∂t ∂qt+1 ∂ (1q) ∂L (t, qt+1 , 1q) − 1q] 1τ + Q′′ + 3 (ξt+1 − τt+1 · 1qt+1 ) + 1G (t, qt+1 , 1q) = 0. ∂ (1q) Eq. (70) is called the discrete structure equation corresponding to the Lie symmetries of discrete nonholonomic systems, and Eq. (71) is called the discrete conserved quantities associated with the systems. Furthermore, if the transformations (27) satisfy the Noether identity ∂L ∂L ∂L ∂L τt + ξt+1 + · 1ξ + L − 1q 1τ + 1G ∂t ∂qt+1 ∂ (1q) ∂ (1q) + Q′′ + 3 (ξt+1 − τt+1 · 1qt+1 ) = 0, (72)
then the transformations (27) are called the discrete Noether symmetrical transformations of the nonoholonomic systems. We also have the following corollary. C OROLLARY 2. The discrete structure equation (70) of the Lie symmetry is equivalent to the discrete Noether identity (72).
We can see that this method also easily applies to investigate the Lie symmetries of the discrete Lagrangian and nonconservative Lagrangian systems. 8.
Examples We will give two theoretical examples to illustrate applications of the results. The applications and implications of the study of this paper will be the next research direction.
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E XAMPLE 1. We first consider an example of a conservation law of a nonconservative system on a discrete but nonhomogeneous time scale (graininess is not constant). The time scale and the Lagrangian of the system are and
T = {2n : n ∈ N ∪ {0}}, L t, q σ , q 1 = t + q σ q 1 ,
Q t, q σ , q 1 = 0.
(73) (74)
Eqs. (45) give the equation of motion of the system
q1 . (75) 2t The determining equation of the Lie symmetries of Eq. (75) under the infinitesimal transformation ξ = ξ(t, q), τ = τ (t, q) is q 11 = −
ξ1 q1 τ 11 q 1 τ 1 q 1 =τ· 2 + − . 2t 4t 4 8t We can obtain the following solution of Eq. (76): ξ 11 +
ln t , τ = 0. ln 2 The structure equation with delta derivatives (47) gives ξ=
τ 1 t + τ + q 1 ξ σ + q σ ξ 1 = −G1 .
(76)
(77)
(78)
Substituting the generators (77) into the structure equation (78) yields q ln 2t . (79) ln 2 According to Theorem 3, substituting the generator (77) and the gauge function (79) into the formula (48), we get the following conserved quantity G=−
tq 1 ln t I t, q, q σ , q 1 = − q = const . ln 2 In this example, the transformation is also Noether symmetrical and such fact is easily verified by direct application of Definition 2, 1I = (tξ )1 q 1σ + tξ q 11 − q 1 = tξ q 11 − q 1 + ξ σ + tξ 1 q 1 + tq 11 1t = tξ 1 q 1 − q 1 = 0. √ E XAMPLE 2. Let T = {( 2 − 1)n : n ∈ N ∪ {0}}, and the Lagrangian, the nonpotential generalized force, the constraint equation of a nonholonomic mechanical system on T are, respectively, 2 t2 q1 σ 1 L t, q , q = + (q σ )2 , Q′′ = q 1 − 2q σ , (80) 2
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f = q σ + tq 1 = 0.
(81)
We study the Lie symmetries and the conserved quantities of this nonholonomic system on time scale T. The differential equation of the system can be expressed as follows √ √ 2t − 1 q 1 − λt = 0. (82) 3 − 2 2 t 2 q 11 + By differentiating Eq. (81) with respect to t we have √ 1 2q 11 . q = √ 2 2−3 t
(83)
Substituting Eq. (82) into Eq. (83) we obtain
q1 λ t, q σ , q 1 = − . t The determining equations (56) give h i h i 1 1 ξ 11 − τ 11 q 1 − µτ 1 + 2τ 1 + µτ 11 α − µ µτ 1 + 2µτ 1 q 111 √ √ 1 2q 2 1 1 1 1 11 , √ =τ √ + ξ − τ q − µτ q 3 − 2 2 t2 2 2−3 t
(84)
(85)
and the restriction conditions (57) give
τ q 1 + ξ σ + ξ 1 − τ 1 q 1 − µτ 1 q 11 t = 0.
(86)
We let the infinitesimal transformations be
1 ξ= . (87) t Obviously, the formulae (85) and (86) can both be satisfied, therefore, the formula (87) corresponds to the Lie symmetrical transformation. Substituting Eqs. (87) into the structure equation (60), we get the gauge function q . (88) G= √ 2−1 Substituting Eqs. (87) and (88) into the formula (61), we obtain the conserved quantity of the system q = const. (89) I t, q, q σ , q 1 = tq 1 + √ 2−1 It is easy to verify by Definition 4 that √ √ 1I q1 2 1 1 11 = tq +√ = 2 − 1 tq + √ q 1 = 0. 1t 2−1 2−1 τ = 0,
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Summary
In summary, we have founded the Lie symmetries and the conserved quantities of the constraint mechanical systems on an arbitrary time scale, which is a new method to investigate the constrained mechanical systems. This is a significant work which extends the theories of Lie symmetry of the two cases for the continuous (T = R) and the discrete (T = Z) constraint mechanical systems to the systems on an arbitrary time scale T. Using this approach, it might also be possible to obtain Lie symmetries of the mechanico-electrical systems, the optimal control systems and other constraint mechanical systems on time scales. Acknowledgments This work has been partially supported by the National Natural Science Foundations of China (Grant Nos. 11472247, 11272287 and 11072218), and by Zhejiang Provincial Natural Science Foundation of China (Grant No.Y6110314). REFERENCES [1] B. Aulbach and S. Hilger: A unified approach to continuous and discrete dynamics, Collo. Math. Sci. 53 (1990), 37. [2] M. Bohner: Calculus of variations on time scales, Dyn. Syst. Appl. 13 (2004), 339. [3] M. Bohner and A. Peterson: Dynamic Equations on Time Scales, Birkh¨auser, Boston 2001. [4] M. Bohner and A. Peterson: Advances in Dynamic Equations on Time Scales, Birkh¨auser, Boston 2003. [5] R. Hilscher and V. Zeidan: Calculus of variations on time scales, J. Math. Anal. Appl. 289 (2004), 143. [6] N. Martins and D. F. M. Torres: Noether’s symmetry theorem for nabla problems of the calculus of variations, Appl. Math. Lett. 23 (2010), 1432. [7] R. Almeida and D. F. M. Torres: Isoperimetric problems on time scales with nabla derivatives, J. Vib. Control 6 (2009), 951. [8] A. B. Malinowska and D. F. M. Torres: Necessary and sufficient conditions for local Pareto optimality on time scales, J. Math. Phys. Sci. 6 (2009), 803. [9] R. P. Agarwal, M. Bohner and D. O’Rega: Dynamic equations on time scales, J. Comp. Appl. Math. 141 (2002), 27. [10] F. M. Atici and G. S. Guseinov: On Green’s functions and positive solutions for boundary value problems on time scales, J. Comp. Appl. Math. 141 (2002), 75. [11] R. A. C. Ferreira and D. F. M. Torres: High-order Calculus of Variations on Time Scales, Springer, Berlin 2008. [12] M. Bohner and G. S. Guseinov: Double integral calculus of variations on time scales, Comput. Math. Appl. 54 (2007), 45. [13] Z. Bartosiewicz and E. Pawluszewicz: Realizations of nonlinear control systems on time scales, IEEE Trans. Autom. Control 53 (2008), 571. [14] R. Hilscher and V. Zeidan: Weak maximum principle and accessory problem for control problems on time scales, Nonlinear Anal. 70 (2009), 3209. [15] D. F. M. Torres: Noether’s theorem on time scales, J. Math. Anal. Appl. 342 (2008), 1220. [16] M. Dryl and D. F. M. Torres: A general delta-nabla calculus of variations on time scales with application to economics, Int. J. Dyn. Syst. Diff. Eqns. 5 (2014), 42. [17] A. E. Noether: Invariante Variations Problem, Math. Phys. Kl.2 (1918), 235. [18] P. J. Olver: Applications of Lie Groups to Differential Equations, Springer, New York 1993. [19] S. P. S. Santos, N. Martins and D. F. M. Torres: Variational problems of Herglotz type with time delay: DuBois–Reymond condition and Noether’s first theorem, Discrete Contin. Dyn. Syst. 35 (2015), 4593.
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[20] G. S. F. Frederico and D. F. M. Torres: Fractional isoperimetric Noether’s theorem in the Riemann–Liouville sense, Rep. Math. Phys. 71 (2013), 291. [21] G. S. F. Frederico, T. Odzijewicz and D. F. M. Torres, Noether’s theorem for non-smooth extremals of variational problems with time delay, Appl. Anal. 93 (2014), 153. [22] G. S. F. Frederico and D. F. M. Torres: Fractional Noether’s theorem in the Riesz-Caputo sense, Appl. Math. Comput. 217 (2010), 1023. [23] J. L. Fu, L. Q. Chen and S. Jimnez, Non-Noether symmetries and Lutzky conserved quantities for mechanico-electrical systems, Phys. Lett. A 358 (2006), 5. [24] Y. Y. Zhao: Conservative quantities and Lie symmetries of nonconservative dynamical systems, Acta Mech. Sin. 26 (1994), 380. [25] J. Belmonte-Beitia, V. M. P´erez-Garc´ıa and V. Vekslerchik: Lie symmetries and solitons in nonlinear systems with spatially inhomogeneous nonlinearities, Phys. Rev. Lett. 98 (2007), 064102. [26] P. E. Hydon: Symmetries and first integrals of ordinary difference equations, Proc. R. Soc., Ser. A 456 (2000), 2835. [27] P. E. Hydon: Symmetry Methods for Ordinary Differential Equations, Cambridge University Press, Cambridge 1999. [28] J. L. Fu, B. Y. Chen and H. Fu: Velocity-dependent symmetries and non-Noether conserved quantities of electromechanical systems, Sci. China 54 (2011), 288. [29] J. L. Fu, H. Fu and R. W. Liu: Hojman conserved quantities of discrete mechanicoCelectrical systems constructed by continuous symmetries, Phys. Lett. A 374 (2010), 1812. [30] G. W. Bluman and S. Kumei: Symmetries and differential equations, Springer, New York 1989. [31] F. X. Mei: Application of Lie groups and Lie Algebra to constraint mechanical systems, Science Press, Beijing 1999.
REPORTS ON MATHEMATICAL PHYSICS
No. 3
LIE SYMMETRIES AND CONSERVED QUANTITIES OF THE CONSTRAINT MECHANICAL SYSTEMS ON TIME SCALES P ING -P ING C AI , J ING -L I F U* Institute of Mathematical Physics, Zhejiang Sci-Tech University, Hangzhou 310018, China (e-mails: [email protected], [email protected])
and YONG -X IN G UO Department of Physics, Liaoning University, Shenyang 110036, China (e-mail: [email protected]) (Received February 8, 2016 — Revised January 3, 2017) We introduce a new method to study Lie symmetries and conserved quantities of constraint mechanical systems which include Lagrangian systems, nonconservative systems and nonholonomic systems on time scales T. For the constraint mechanical systems on time scales, based on the transformation Lie group, we get a series of significant results including the variational principle of systems on time scales, the equations of motion, the determining equations, the structure equations, the restriction equations as well as the Lie theorems of the Lie symmetries of the systems on time scales. Furthermore, a set of new conserved quantities of the constraint mechanical systems on time scales are given. More significant is that this work unifies the theories of Lie symmetries of the two cases for the continuous and the discrete constraint mechanical systems by applying the time scales. And then taking the discrete (T = Z) nonholonomic system for example, we derive the corresponding discrete Lie symmetry theory. Finally, two examples are designed to illustrate these results. Keywords: time scale, Lie symmetry, constraint mechanical system.
1.
Introduction The theory of time scales originated in 1988 with the work of Stefan Hilger [1] in order to unify various concepts from the theories of discrete and continuous dynamical systems, and to extend such theories to more general classes of dynamical systems. The time scales calculus theory is applicable to any field in which dynamic processes can be described with discrete or continuous models. The study of the calculus of variations in the context of time scales has its beginning in 2004 with the paper of Martin Bohner [2], with two excellent books dedicated to it [3, 4]. Since the pioneering paper [2], the classical results of the calculus of variations on continuous time (T = R) and discrete time (T = Z) have been unified and generalized to * Corresponding
author. [279]
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a time scale T: Euler–Lagrange equations [5, 6], necessary optimality conditions for variational problems subject to isoperimetric constraints [7, 8], boundary value problems [9, 10], high-order delta derivatives [11–13], weak maximum principle for variable endpoints optimal control problems [14], an invariant group of parameter transformations [15]. The more general theory of the calculus of variations on time scales seems to be useful in applications to economics [16]. It is significant to study the symmetries and conserved quantities of dynamical systems not only in mathematics but also in physics. And the development of modern physics such as quantum mechanics, quantum field theory, nuclear physics, and spatially inhomogeneous nonlinearities shows that the principle of symmetry has become the most important principle of exploring the laws of motion of microparticles. There are two main symmetry methods used to seek the conserved quantities of a dynamical system: the Noether symmetry and the Lie symmetry [17, 18]. The Noether method is making good progress, such as Herglotz variational problems [19], problems with isoperimetric constraints [20], problems with delay arguments [21] and problems with noninteger order derivatives [22]. The Lie method has been approved to be a powerful tool to solve differential equations, to study constraint mechanical systems, to discuss controllable dynamical systems, to investigate mechanico-electrical systems and to establish properties of their solution spaces. The theories of Lie symmetry have been described in many papers [23–27]. The Lie group theory has also been applied to discrete equations, such as differential-difference equations, discrete dynamical systems and discrete mechanico-electrical systems [28, 29]. We have seen that the theories of Lie symmetry of the continuous and the discrete systems have been deeply studied and they are studied in two different directions. Here we introduce the concept of time scale which unifies the theories of Lie symmetries of the two cases for the continuous (T = R) and the discrete (T = Z) constraint mechanical systems. This paper systematically studies the conserved quantities and Lie symmetries for constraint mechanical systems on an arbitrary time scale T which unifies and extends the previous formulations of Lie’s method in the discrete-time and continuous domains (cf. [23–29] and references therein). By defining the isochronous variation on time scales, we study the exchanging relationships between the isochronous variation and the delta derivatives as well as the relationships between the isochronous variation and the total variation on time scales. On the basis of the invariance theory of differential equations of motion under the infinitesimal transformations, we study the Lie symmetry of the Lagrangian system. By virtue of the Hamilton principle, we construct the equations of motion of nonconservative systems on time scales, and then we give the method of solution of the Lie symmetries of nonconservative systems on time scales. Furthermore, using the d’Alembert–Lagrange principle on time scales we derive the equations of motion of nonholonomic systems of Chetaev type. Then, by presenting the Lie symmetrical determining equations, the structure equations and the constrained restriction equations of the constraint mechanical systems on time scales T, we get a new type of Noether conserved quantity which only depends on the variables t, q, q σ , q 1 . For the classical discrete time (T = Z)
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we can reduce previous results on an arbitrary time scale to the classical results of the Lie symmetries of the discrete nonholonomic systems. Finally, in order to illustrate applications of the results, we calculate two examples. 2.
Basics on the time scale calculus A time scale is a nonempty closed subset of real numbers, and we usually denote it by the symbol T. The two most popular examples are (T = R) and (T = Z). We define the forward and backward jump operators σ, ρ : T → T by σ (t) = inf{s ∈ T : s > t} and ρ(t) = sup{s ∈ T : s < t}, for all t ∈ T,
(supplemented by inf Ø = sup T and Ø = inf T). The graininess function µ : T → [0, ∞) is defined by µ(t) = σ (t) − t. (1)
Hence the graininess function is constant and equal to 0 if (T = R) while it is constant and equal to 1 for (T = Z). However, a time scale T could have nonconstant graininess. A point t ∈ T is called right scattered, right dense, left scattered and left dense if σ (t) > t, σ (t) = t, ρ(t) < t, ρ(t) = t hold, respectively. Throughout we let a, b ∈ T with a < b. For an interval [a, b] ∩ T we simply write [a, b] when this is not ambiguous. We also define [a, b]κ := [a, b]\(ρ(b), b]
and
2
[a, b]κ := [a, b]\(ρ(ρ(b)), b].
We say that a function f : T → R is delta differentiable at t ∈ Tκ provided there exists a real number f 1 (t) such that for all ε > 0 there is a neighbourhood U = (t − δ, t + δ) ∩ T of t with f (σ (t)) − f (s) − f 1 (t) (σ (t) − s) ≤ ε |σ (t) − s| for all t ∈ U. For differentiable f , the formula
f σ = f + µf 1
(2)
is very useful and easy to prove. If f and g are both differentiable, then so is f g with (3) (fg)1 = f 1 g + f σ g 1 ,
where we abbreviate f ◦ σ by f σ . Next, a function f : T → R is called rd-continuous if it is continuous in right-dense points and if its left-sided limits exist in left-dense points. Crd denotes 1 the set of all rd-continuous functions, while Crd denotes the set of all differentiable functions with rd-continuous derivative. It is known that rd-continuous functions possess an antiderivative, i.e. there exists a function F : T → R with F 1 (t) = f (t), and in this case an integral of f from a to b (a, b ∈ T) is defined by Z b f (t)1t = F (b) − F (a) . (4) a
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3. Variational relationships on time scales 3.1.
Exchange relationship between the isochronous variation and the delta derivatives
Consider two infinitely closed orbits α and α + dα. We denote the generalized coordinates by q = q(t, α), q = q(t, α + dα), corresponding to the two infinitely closed orbits, respectively, in giving time on time scale T. We define the isochronous variation as δq = q (t, α + dα) − q (t, α) .
(5)
Extending q = q(t, α + dα) to the linear terms of dα, we obtain q (t, α + dα) = q (t, α) +
∂q (t, α) dα. ∂α
(6)
Substituting Eq. (6) into Eq. (5), we get δq = Similarly we have
∂q (t, α) dα. ∂α
(7)
1 ∂q (t, α) 1 δq = dα. 1t 1t ∂α
(8)
According to Eq. (6) we get δq 1 = q 1 (t, α + dα) − q 1 (t, α) =
1 ∂q (t, α) dα, 1t ∂α
(9)
and comparing Eq. (9) with Eq. (8), we obtain Similarly we have
δq 1 = (δq)1 .
(10)
δq σ = (δq)σ .
(11)
We call Eqs. (10) and (11) the exchanging relationships with respect to the delta derivatives and isochronous variation. 3.2.
The isochronous variation on time scales
We continue studying the infinitely closed orbits α and α + dα. The generalized coordinates are given by q = q(t, α), q ∗ = q ∗ (t, α), for any t ∈ T, where t = t (α), so we have q = q [t (α) , α] , taking total variation for q we obtain
∂q [t (α) , α] ∂q [t (α) , α] ∂t dα + dα. ∂α ∂t ∂α Since 1t is the variation of time with respect to α, therefore 1q =
1t =
∂t dα. ∂α
(12)
(13)
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Substituting Eqs. (7) and (13) into Eq. (12), we get 1q = δq + q 1 1t.
(14)
We call Eq. (14) the relationship between the isochronous variation and the total variation on time scales T. Using Eqs. (10) and (14) we have 1q 1 = δq 1 + q 11 · 1t.
(15)
Differentiating both sides of Eq. (14) with respect to t we obtain (1q)1 = δq 1 + q 11 · (1t)σ + q 1 · (1t)1 , and according to Eqs. (15) and
(16), we get
1q 1 = (1q)1 − q 11 (1t)σ − q 1 (1t)1 + q 11 · 1t = (1q)1 − q 1 (1t)1 − µ (t) · (1t)1 q 11 .
Similarly we have
1q 11 = δq 11 + q 111 · 1t, σ2
(1q)11 = δq 11 + q 111 (1t) From Eqs. (18) and
(16)
+ q 11 (1t)σ 1 + q 11 (1t)1σ + q 1 (1t)11 .
(17)
(18) (19)
(19) we obtain 2
1q 11 = q 111 · 1t + (1q)11 − q 111 (1t)σ − q 11 (1t)σ 1 − q 11 (1t)1σ − q 1 (1t)11 . (20) Differentiating both sides of Eq. (17) with respect to t we obtain 1 2 1q 1 =(1q)11 − q 111 (1t)σ − q 11 (1t)σ 1 − q 11 (1t)1σ − q 1 (1t)11 (21) + q 111 (1t)σ + q 11 (1t)1 . By virtue of Eqs. (20) and (21) we get 1 1q 11 = 1q 1 − q 11 (1t)1 − µ (t) · (1t)1 q 111 . 4. 4.1.
(22)
Lie symmetry of the Lagrange systems on time scales The Lagrange equations of a Lagrangian system
The equations of motion of a Lagrangian system with delta derivatives can be written in the form [2] 1 ∂3 L t, q σ (t) , q 1 (t) = ∂2 L t, q σ (t) , q 1 (t) , (23) 1t where L = L t, q σ , q 1 is the Lagrangian of the system on time scales. In general,
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it is supposed that the system (23) is nonsingular, i.e. ∂ 2L ∂ 2L 1 6= 0. 2 + µ 1 + µ ∂q 1 ∂q σ ∂ q1
(24)
Expanding Eqs. (23), we can seek the generalized acceleration with delta derivatives as q 11 = h t, q σ , q 1 . (25)
4.2.
Infinitesimal transformations and the determining equations
We introduce the infinitesimal transformations in terms of time and generalized coordinate as t ∗ = t + 1t, q ∗ = q(t) + 1q. (26) or their expanded form t ∗ = t + ετ (t, q) + o (ε) ,
q ∗ (t) = q (t) + εξ (t, q) + o (ε) .
(27)
where ε is the infinitesimal parameter and τ (t, q), ξ(t, q) are the generators of infinitesimal transformations. Eqs. (27) are one-parameter Lie-point group of transformations. Taking the infinitesimal generator vector X(0) = τ
∂ ∂ + ξσ σ , ∂t ∂q
(28)
the first extended infinitesimal generator is X(1) = τ
∂ ∂ ∂ + ξ σ σ + ξ 1 − τ 1 q 1 − µ (t) τ 1 q 11 , ∂t ∂q ∂q 1
and the second is h i 1 X(2) = X (1) + ξ 1 − τ 1 q 1 − µ (t) τ 1 q 11 − τ 1 q 11 − µ (t) τ 1 q 111
(29)
∂ , (30) ∂q 11
where
1 ξ (t, q (t)) 1t and τ (t, q), ξ(t, q) are infinitesimal generators. Then based on the invariance of the differential equations (25) under the infinitesimal transformations (27), if and only if [30] X(2) q 11 − h t, q σ , q 1 = 0, (31) ξ σ (t, q (t)) = ξ (σ (t) , q (σ (t))) , ξ 1 (t, q (t)) =
we can have
h
i h i 1 + 2τ 1 + µτ 11 h − µ µτ 1 + 2µτ 1 q 111 = X(1) h. (32) Eqs. (32) are called the determining equations on time scales T which the generators ξ and τ should satisfy. If ξ and τ satisfy the determining equations (32), ξ 11 − τ 11 q 1 −
µτ 1
1
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the corresponding transformations are the Lie symmetry’s transformations of the Lagrangian systems on time scales. 4.3.
The structure equation and conserved quantity
D EFINITION 1. A quantity C t, q, q σ , q 1 is said to be a conserved quantity if 1 C t, q (t) , q σ (t) , q 1 (t) = 0 is preserved along all q(t) that satisfy and only if 1t the Euler–Lagrange equations (23).
Lie symmetries do not always generate Noether conserved quantities. The subsequent propositions give the condition under which Lie symmetries generate conserved quantities and the form of conserved quantities. T HEOREM 1. For the infinitesimal generators ξ, τ satisfying the determining Eqs. (32), if there is a gauge function G = G t, q σ , q 1 satisfying the equation Lτ 1 + X(1) L + ∂3 L · µτ 1 q 11 + G1 = 0,
(33)
then the system possesses a conserved quantity with delta derivatives C t, q, q σ , q 1 = ∂3 L t, q σ , q 1 · ξ (t, q) + L t, q σ , q 1 − ∂3 L t, q σ , q 1 · q 1 − ∂1 L t, q σ , q 1 · µ (t) · τ (t, q) = const . (34) Proof : Using the Euler–Lagrange equations (23) and the structure equation (33), we obtain 1 [∂3 L t, q σ , q 1 · ξ (t, q) 1t + L t, q σ , q 1 − ∂3 L t, q σ , q 1 · q 1 − ∂1 L t, q σ , q 1 · µ (t) · τ (t, q) 1 + G t, q σ , q 1 = ∂3 L t, q σ , q 1 · ξ σ (t, q) + ∂3 L t, q σ , q 1 · ξ 1 (t, q) 1t 1 σ + τ (t, q) · L t, q σ , q 1 − ∂3 L t, q σ , q 1 · q 1 − ∂1 L t, q σ , q 1 · µ (t) 1t 1 + τ (t, q) · L t, q σ , q 1 − ∂3 L t, q σ , q 1 · q 1 − ∂1 L t, q σ , q 1 µ (t) + G1 t, q σ , q 1 = ∂2 L t, q σ , q 1 · ξ σ (t, q) + ∂3 L t, q σ , q 1 · ξ 1 (t, q) + ∂1 L t, q σ , q 1 · τ (t, q) + L t, q σ , q 1 · τ 1 (t, q) − ∂3 L t, q σ , q 1 · τ 1 (t, q) · q 1 + G1 t, q σ , q 1 1 G t, q σ , q 1 = L t, q σ , q 1 τ 1 + X(1) L t, q σ , q 1 + ∂3 L t, q σ , q 1 · µτ 1 q 11 + 1t = 0. Eq. (33) is called the structure equation for the Lagrangian systems on time scales.
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If the transformations (27) satisfy Noether’s identity ∂1 L t, q σ , q 1 · τ (t, q) + ∂2 L t, q σ , q 1 · ξ σ (t, q) + ∂3 L t, q σ , q 1 · ξ 1 (t, q) 1 + L t, q σ , q 1 · τ 1 − ∂3 L t, q σ q 1 · τ 1 · q 1 = − G t, q σ , q 1 , (35) 1t then the transformations (27) are called the Noether symmetrical transformations of the system (25). T HEOREM 2. The structure equation with delta derivatives (33) of the Lie symmetry is equivalent to the Noether identity (35). 5. Lie symmetries of nonconservative systems on time scales 5.1.
Hamilton’s principle and Lagrange equations for nonconservative systems with delta derivatives
L EMMA 1 Z b a
[2]
(Dubois–Reymond). Let g ∈ Crd , g : [a, b] → Rn , then
1 g T (t) · η1 (t) 1t = 0 for all η ∈ Crd with η (a) = η (b) = 0
holds if and only if g (t) ≡ c on [a, b]κ for some c ∈ Rn . Assuming that the kinetic energy function of the system is T = T t, q σ , q 1 , Hamilton’s principle states that the actual pace exists when the Hamiltonian action has determining value. So the Hamilton principle for nonconservative systems with delta derivatives can be written in the form Z b (36) (δT + Qδq σ )1t = 0, a
σ
where Qδq is the virtual work of the generalized force. Taking total variation for the function T , we get δT =
∂T ∂T δq σ + 1 δq 1 , σ ∂q ∂q
(37)
substituting Eq. (37) into Eq. (36) we have Z b Z b ∂T ∂T ∂T ∂T σ 1 σ σ 1 δq + 1 δq + Qδq 1t = Q + σ δq + 1 δq 1t ∂q σ ∂q ∂q ∂q a a Z b ∂T ∂T σ 1 = Q + σ (δq) + 1 (δq) 1t ∂q ∂q a
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#1 ! ∂T τ, q σ (τ ) , q 1 (τ ) 1τ · δq = Q+ ∂q σ (τ ) a a # "Z ! t ∂T τ, q σ (τ ) , q 1 (τ ) ∂T 1 1 1τ (δq) + − Q+ (δq) 1t ∂q σ (τ ) ∂q 1 a ) ! Z b( Z t ∂T τ, q σ (τ ) , q 1 (τ ) ∂T = − Q+ 1τ (δq)1 1t = 0. 1 σ (τ ) ∂q ∂q a a Z
b
("Z
t
Therefore, by Lemma 1, we obtain ! Z b ∂T τ, q σ (τ ) , q 1 (τ ) ∂T − Q+ 1τ ≡ const, t ∈ [a, b] , ∂q 1 ∂q σ (τ ) a hence
1 ∂T ∂T − σ − Q = 0, 1 1t ∂q ∂q
(38)
when Q contains conservative force Q′ and nonconservative force Q′′ , and Q′ satisfies the following conditions: If Q′ is potential, i.e. there exists a function V = V (q σ , t) such that Q′ = −
∂V , ∂q σ
(39)
substituting Eq. (39) into (38) we obtain 1 ∂T ∂T ∂V − σ + σ − Q′′ = 0. 1 1t ∂q ∂q ∂q
(40)
As the function only depends on the generalized coordinates, therefore ∂V = 0, ∂q 1
(41)
using Eq. (41), from Eq. (40) we have 1 ∂ (T − V ) ∂ (T − V ) − = Q′′ . 1t ∂q 1 ∂q σ
(42)
Introducing function L = T − V , we can rewrite Eqs. (42) as 1 ∂L ∂L − σ = Q′′ . 1 1t ∂q ∂q
If Q′ has a generalized potential, i.e. there exists a function U = U t, q σ , q 1 such that ∂U 1 ∂U Q′ = σ − , ∂q 1t ∂q 1
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then Eqs. (38) can be written as 1 ∂ (T + U ) ∂ (T + U ) − = Q′′ , 1t ∂q 1 ∂q σ and introducing the function L = T + U = T − V , we can obtain the Euler–Lagrange equations of nonconservative systems with delta derivatives, ∂L 1 ∂L − σ = Q′′ . 1 1t ∂q ∂q 5.2.
(43)
Lie symmetries of the system
In general, it is supposed that the system (43) is nonsingular, i.e. ∂ 2L ∂ 2L 1 6= 0, + µ 1 + µ 2 ∂q 1 ∂q σ ∂ q1
and from Eqs. (43) we can find all accelerations as q 11 = β t, q σ , q 1 .
(44)
(45)
The necessary and sufficient conditions under which Eqs. (45) remain invariant are h i h i 1 1 ξ 11 − τ 11 q 1 − µτ 1 + 2τ 1 + µτ 11 β − µ µτ 1 + 2µτ 1 q 111 = X(1) β, (46) we hereinafter give the definition of Lie symmetries and conserved quantities of nonconservative systems on time scales. D EFINITION 2. If the generators ξ, τ satisfy the determining equations (46), then the corresponding symmetries are called Lie symmetries of holonomic systems (43) on time scales. T HEOREM 3. For the infinitesimal generators ξ, τ satisfying the determining Eqs. (46), if there is a gauge function G = G t, q σ , q 1 satisfying the structure equation Lτ 1 + X(1) (L) + ∂3 L · µτ 1 q 11 + Q ξ σ − τ σ q 1σ + G1 = 0, (47)
then the holonomic system possesses the following conserved quantity with delta derivative: C t, q, q σ , q 1 = ∂3 L t, q σ , q 1 · ξ (t, q) + L t, q σ , q 1 − ∂3 L t, q σ , q 1 · q 1 − ∂1 L t, q σ , q 1 · µ (t) · τ (t, q) = const . (48)
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Proof : Using Eqs. (43) and the structure equation (47) we obtain 1 [∂3 L t, q σ , q 1 · ξ (t, q) 1t + L t, q σ , q 1 −∂3 L t, q σ , q 1 ·q 1 −∂1 L t, q σ , q 1 ·µ (t) ·τ (t, q) 1 ∂3 L t, q σ , q 1 ·ξ σ (t, q)+∂3 L t, q σ , q 1 ·ξ 1 (t, q) +G t, q σ , q 1 = 1t 1 L t, q σ , q 1 −∂3 L t, q σ , q 1 ·q 1 − ∂1 L t, q σ , q 1 ·µ (t) +τ σ (t, q)· 1t +τ 1 (t, q)· L t, q σ , q 1 −∂3 L t, q σ , q 1 ·q 1 − ∂1 L t, q σ , q 1 µ (t) +G1 t, q σ , q 1 = (∂2 L t, q σ , q 1 +Q t, q σ , q 1 ·ξ σ (t, q)+∂3 L t, q σ , q 1 ·ξ 1 (t, q) +[∂1 L t, q σ , q 1 −Q t, q σ , q 1 q 1σ τ σ +τ 1 L t, q σ , q 1 −∂3 L t, q σ , q 1 ·q 1 −∂1 L t, q σ , q 1 ·µ (t) +G1 t, q σ , q 1 = Lτ 1 +X(1) (L)+Q ξ σ −τ σ q 1σ +G1 = 0.
T HEOREM 4. The structure equation of the holonomic system of the Lie symmetry (47) is equivalent to the following Noether’s identity, ∂1 L t, q σ , q 1 · τ (t, q) + ∂2 L t, q σ , q 1 · ξ σ (t, q) + ∂3 L t, q σ , q 1 · ξ 1 (t, q) 1 + L t, q σ , q 1 · τ 1 − ∂3 L t, q σ q 1 · τ 1 · q 1 + Q ξ σ − τ σ q 1σ = − G t, q σ , q 1 . 1t
The method of solution of Lie symmetries of holonomic nonconservative systems on time scales is the following: first, establish the determining equations (46) with the delta derivative, and seek the generator ξ, τ from these equations; secondly, substitute the generators obtained into the structure equation (47) to determine G t, q σ , q 1 ; finally, substitute ξ, τ and G t, q σ , q 1 into the formula (48) to get the conserved quantities of the symmetries of holonomic systems on time scales. 6. 6.1.
Lie symmetries of nonholonomic mechanical systems of Chetaev type on time scales The Lagrange equations for nonholonomic Chetaev systems
Let the configuration of a mechanical system be determined by the generalized coordinate q σ on time scale T. Its motion is subject to the ideal nonholonomic constraints of Chetaev type f β t, q σ , q 1 = 0, (β = 1, . . . , g) . (49)
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The restriction of conditions (49) on the virtual displacements are ∂f β σ δq = 0; ∂q 1
(50)
we call equations (50) the Chetaev condition on time scales T. The equations of motion can be written in the form 1 ∂L ∂L ∂f β ′′ − = Q + λ , (51) β 1t ∂q 1 ∂q σ ∂q 1 where Q′′ = Q′′ t, q σ , q 1 is the nonconservative force and L = L t, q σ , q 1 the Lagrangian of the system. Before integrating the equations of motion, we can determine λβ as functions of t, q σ , q 1 , λβ = λβ (t , q σ , q 1 . (52) Substituting this into Eqs. (51) we obtain
Let
∂L 1 ∂L − σ = Q′′ + 3, 1 1t ∂q ∂q ∂f β 3 = 3 t, q σ , q 1 = λβ 1 . ∂q ∂ 2L ∂ 2L 1 6= 0, 2 + µ 1 + µ ∂q 1 ∂q σ ∂ q1
expanding Eqs. (51) we can get all accelerations as q 11 = α t, q σ , q 1 ,
(53)
(54)
(55)
which are called the equations of motion of holonomic system corresponding to the nonholonomic system (49), (51). If the initial conditions satisfy Eqs. (49), the solution of Eqs. (55) gives the motion of the nonholonomic system. The invariance of Eqs. (55) and (49) under the infinitesimal transformations (27) leads to the fulfilment of the following determining equations with delta derivatives, h i h i 1 1 ξ 11 − τ 11 q 1 − µτ 1 + 2τ 1 + µτ 11 α − µ µτ 1 + 2µτ 1 q 111 = X(1) α, (56) and the inivariance of the constraint equations (49) under the infinitesimal transformations (27) can be made to satisfy the restriction equations X(1) f β t, q σ , q 1 = 0. (57) Considering the Chetaev conditions (50) and
δq σ = 1q σ − q σ 1 · 1t = ε ξ σ − τ σ q 1σ ,
(58)
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we have the following additional restriction equations ∂f β σ σ 1σ ξ − τ q =0 ∂q 1
(β = 1, . . . , g) .
(59)
D EFINITION 3. If the generators τ (t, q), ξ(t, q) satisfy the determining equations (56), then the transformations are called the Lie symmetrical transformations of the corresponding holonomic system (55). D EFINITION 4. If the generators τ (t, q), ξ(t, q) satisfy the determining equations (56) and the restriction equations (57), then the transformations are called the weakly Lie symmetrical transformations of the nonholonomic system (49), (51). D EFINITION 5. If the generators τ (t, q), ξ(t, q) satisfy the determining equations (56), the restriction equations (57) and the additional restriction equations (59), then the transformations are called the strongly Lie symmetrical transformations of the nonholonomic system (49), (51). The Lie symmetry can lead to a Noether-type conserved quantity under certain conditions. T HEOREM 5. For the generators τ (t, q), ξ(t, q) satisfying the determining equations (56) and the restriction equations (57), if there exists a gauge function G = G t, q σ , q 1 satisfying the structure equation Lτ 1 + X(1) (L) + ∂3 L · µτ 1 q 11 + Q′′ + 3 ξ σ − τ σ q 1σ + G1 = 0, (60)
then there exists a Noether-type conserved quantity corresponding to the Lie symmetries for the nonholonomic system of Chetaev type on time scales, i.e. C t, q, q σ , q 1 = ∂3 L t, q σ , q 1 · ξ (t, q) + L t, q σ , q 1 − ∂3 L t, q σ , q 1 · q 1 − ∂1 L t, q σ , q 1 · µ (t)] τ (t, q) + G t, q σ , q 1 = const . (61)
Proof : Using the Euler–Lagrange equations (53) and the structure equation (60) we obtain 1 [∂3 L t, q σ , q 1 · ξ (t, q) 1t + L t, q σ , q 1 − ∂3 L t, q σ , q 1 · q 1 − ∂1 L t, q σ , q 1 · µ (t) · τ (t, q) 1 + G t, q σ , q 1 = ∂3 L t, q σ , q 1 · ξ σ (t, q) + ∂3 L t, q σ , q 1 · ξ 1 (t, q) 1t 1 + τ σ (t, q) · L t, q σ , q 1 − ∂3 L t, q σ , q 1 · q 1 − ∂1 L t, q σ , q 1 · µ (t) 1t 1 + τ (t, q) · L t, q σ , q 1 − ∂3 L t, q σ , q 1 · q 1 − ∂1 L t, q σ , q 1 µ (t) + G1 t, q σ , q 1
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= (∂2 L t, q σ , q 1 +Q t, q σ , q 1 + 3 t, q σ , q 1 · ξ σ (t, q) + ∂3 L t, q σ , q 1 · ξ 1 (t, q) + [∂1 L t, q σ , q 1 − Q t, q σ , q 1 + 3 t, q σ , q 1 q 1σ τ σ + τ 1 L t, q σ , q 1 − ∂3 L t, q σ , q 1 · q 1 − ∂1 L t, q σ , q 1 · µ (t) + G1 t, q σ , q 1 = Lτ 1 + X(1) (L) + ∂3 L · µτ 1 q 11 + Q′′ + 3 ξ σ − τ σ q 1σ + G1 = 0.
T HEOREM 6. For the generators τ (t, q), ξ(t, q) satisfying the determining equations (56) and the restriction equations (57) and the additional restriction equa tions (59), if there exists a gauge function G = G t, q σ , q 1 satisfying the structure equation (60), then the system possesses the strongly Lie symmetrical conserved quantity (61). 7. Discussion We have seen that an invariance of the constraint mechanical systems can be obtained under the infinitesimal transformations with respect to both the time and the state variables on an arbitrary time scale T. To obtain the classical Noether theorems of the two cases, for the continuous and the discrete constraint mechanical systems, we assume T = R, T = Z, respectively. For the classical continuous time (T = R), the previous results are reduced to the classical results of the Lie symmetries [31]. Here let us take a nonholonomic system for example, we will consider the Lie symmetries of discrete nonholonomic systems on discrete time (T = Z). The time t is a discrete variable: t ∈ Z. The horizon consists of N periods, t = M, M + 1, . . . , M + N − 1 where M and N are fixed integers, instead of a continuous interval. The purpose of the following work is to deduce the discrete equations of motion, the discrete structure equations and the discrete conserved quantities of discrete nonholonomic systems on discrete time T = Z from the results we have obtained on an arbitrary time scale T. We refer to the literatures for further reading on Lie symmetries of the mechanical systems for the discrete case [28]. 7.1.
The discrete equations of motion for nonholonomic systems
In discrete mechanics, continuous curve in configuration space is replaced with the discrete sequence qt (t = M, M + 1, · · · , M + N − 1). The discrete versions of L and Q are L (t, qt+1 , 1q) and Q (t, qt+1 , 1q), respectively. So Eqs. (34) can be written in the form of discrete equations: ∂L (t, qt+1 , 1q) ∂L (t − 1, qt , 11 q) ∂L (t, qt+1 , 1q) − − ∂ (1q) ∂ (11 q) ∂qt+1 ′′ = Q (t, qt+1 , 1q) + 3 (t, qt+1 , 1q) , (t = M, M + 1, · · · , M + N − 1), where 3 = λα
∂f (k, qk+1 , 1q) ∂ (1q)
(62)
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are constraint forces corresponding to discrete nonholonimic constraints (and λα is discrete constraint multiplication), 1 the difference operator, 1q = qt+1 − qt . The discrete equations of motion can be put in the form 12 q = γ (t, qt+1 , 1q) ,
(63)
where we abbreviate 1 (1q) as 12 q. 7.2.
First integrals of the discrete nonoholonomic system
The vector field of generators (28) turns out to be ∂ ∂ , (64) X∗(0) = τk + ξk+1 ∂t ∂qk+1 which can be prolonged to the two- and three-point schemes ∂ ∂ ∂ X∗(1) = τk + ξk+1 + 1ξ − 1τ · 1q − 1τ · 12 q , (65) ∂t ∂qk+1 ∂ (1q) ∂ . X∗(2) = X∗(1) + 1 1ξ − 1τ · 1q − 1τ · 12 q − 1τ · 12 q − 1τ · 13 q ∂ 12 q (66)
The invariance of discrete equations (63) under the infinitesimal transformations (27) leads to the fulfilment of the following discrete determining equations 12 ξ − 1q · 12 τ − 2 1τ + 12 τ 12 q − 21τ + 12 τ 13 q = X∗(1) 12 q , (67)
where 1ξ = ξt+1 − ξt , 1τ = τt+1 − τt . The constrained restriction equation (57) becomes X∗ (1) f β (t, qt+1 , 1q) = 0,
(68)
and the discrete restriction of conditions f (t, qt+1 , 1q) = 0 on the discrete virtual displacements is given by ∂f (t, qt+1 , 1q) (69) (ξt+1 − τt+1 1qt+1 ) = 0, ∂ (1q) which are the discrete analogue of Appell–Chetaev conditions with respect to the discrete nonholonomic constraints restricting generators of infinitesimal transformations. We hereinafter give the Lie symmetries of the discrete nonholonomic systems on discrete-time (T = Z). C OROLLARY 1. For the infinitesimal generators τt , ξt satisfying the determining equation (67) and the restriction equation (68), if there exists a discrete gauge function G = G (t, qt+1 , 1q) such that the identity L (k, qk+1 , 1q) · 1τ + X∗(1) L (k, qk+1 , 1q)
+ ∂3 L (k, qk+1 , 1q) · 1τ · 12 q + 1G (k, qk+1 , 1q) = 0
(70)
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holds, then the discrete nonholonomic system has the discrete conserved quantities I (t, qt , qt+1 , 1q) = ∂3 L (t, qt+1 , 1q) · ξt (t, q) + L (t, qt+1 , 1q) − ∂3 L (t, qt+1 , 1q) · 1q − ∂1 L (t, qt+1 , 1q) · τt (t, q) = const . (71) Proof : Using Eqs. (62) and the discrete structure equation (70), we obtain ∂L (t, qt+1 , 1q) +3 (t, qt+1 , 1q)) ξt+1 1I = Q′′ (t, qt+1 , 1q) + ∂qt+1 ∂L (t, qt+1 , 1q) + · 1ξ ∂ (1q) ∂L (t, qt+1 , 1q) + − (Q (t, qt+1 , 1q) + 3 (t, qt+1 , 1q)) 1qt+1 τt+1 ∂t ∂L (t, qt+1 , 1q) ∂L (t, qt+1 , 1q) 1τ + L (t, qt+1 , 1q) − · 1q − ∂ (1q) ∂t + 1G (t, qt+1 , 1q) ∂L (t, qt+1 , 1q) ∂L (t, qt+1 , 1q) ∂L (t, qt+1 , 1q) = ξt+1 + τt + 1ξ + L (t, qt+1 , 1q) ∂t ∂qt+1 ∂ (1q) ∂L (t, qt+1 , 1q) − 1q] 1τ + Q′′ + 3 (ξt+1 − τt+1 · 1qt+1 ) + 1G (t, qt+1 , 1q) = 0. ∂ (1q) Eq. (70) is called the discrete structure equation corresponding to the Lie symmetries of discrete nonholonomic systems, and Eq. (71) is called the discrete conserved quantities associated with the systems. Furthermore, if the transformations (27) satisfy the Noether identity ∂L ∂L ∂L ∂L τt + ξt+1 + · 1ξ + L − 1q 1τ + 1G ∂t ∂qt+1 ∂ (1q) ∂ (1q) + Q′′ + 3 (ξt+1 − τt+1 · 1qt+1 ) = 0, (72)
then the transformations (27) are called the discrete Noether symmetrical transformations of the nonoholonomic systems. We also have the following corollary. C OROLLARY 2. The discrete structure equation (70) of the Lie symmetry is equivalent to the discrete Noether identity (72).
We can see that this method also easily applies to investigate the Lie symmetries of the discrete Lagrangian and nonconservative Lagrangian systems. 8.
Examples We will give two theoretical examples to illustrate applications of the results. The applications and implications of the study of this paper will be the next research direction.
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E XAMPLE 1. We first consider an example of a conservation law of a nonconservative system on a discrete but nonhomogeneous time scale (graininess is not constant). The time scale and the Lagrangian of the system are and
T = {2n : n ∈ N ∪ {0}}, L t, q σ , q 1 = t + q σ q 1 ,
Q t, q σ , q 1 = 0.
(73) (74)
Eqs. (45) give the equation of motion of the system
q1 . (75) 2t The determining equation of the Lie symmetries of Eq. (75) under the infinitesimal transformation ξ = ξ(t, q), τ = τ (t, q) is q 11 = −
ξ1 q1 τ 11 q 1 τ 1 q 1 =τ· 2 + − . 2t 4t 4 8t We can obtain the following solution of Eq. (76): ξ 11 +
ln t , τ = 0. ln 2 The structure equation with delta derivatives (47) gives ξ=
τ 1 t + τ + q 1 ξ σ + q σ ξ 1 = −G1 .
(76)
(77)
(78)
Substituting the generators (77) into the structure equation (78) yields q ln 2t . (79) ln 2 According to Theorem 3, substituting the generator (77) and the gauge function (79) into the formula (48), we get the following conserved quantity G=−
tq 1 ln t I t, q, q σ , q 1 = − q = const . ln 2 In this example, the transformation is also Noether symmetrical and such fact is easily verified by direct application of Definition 2, 1I = (tξ )1 q 1σ + tξ q 11 − q 1 = tξ q 11 − q 1 + ξ σ + tξ 1 q 1 + tq 11 1t = tξ 1 q 1 − q 1 = 0. √ E XAMPLE 2. Let T = {( 2 − 1)n : n ∈ N ∪ {0}}, and the Lagrangian, the nonpotential generalized force, the constraint equation of a nonholonomic mechanical system on T are, respectively, 2 t2 q1 σ 1 L t, q , q = + (q σ )2 , Q′′ = q 1 − 2q σ , (80) 2
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f = q σ + tq 1 = 0.
(81)
We study the Lie symmetries and the conserved quantities of this nonholonomic system on time scale T. The differential equation of the system can be expressed as follows √ √ 2t − 1 q 1 − λt = 0. (82) 3 − 2 2 t 2 q 11 + By differentiating Eq. (81) with respect to t we have √ 1 2q 11 . q = √ 2 2−3 t
(83)
Substituting Eq. (82) into Eq. (83) we obtain
q1 λ t, q σ , q 1 = − . t The determining equations (56) give h i h i 1 1 ξ 11 − τ 11 q 1 − µτ 1 + 2τ 1 + µτ 11 α − µ µτ 1 + 2µτ 1 q 111 √ √ 1 2q 2 1 1 1 1 11 , √ =τ √ + ξ − τ q − µτ q 3 − 2 2 t2 2 2−3 t
(84)
(85)
and the restriction conditions (57) give
τ q 1 + ξ σ + ξ 1 − τ 1 q 1 − µτ 1 q 11 t = 0.
(86)
We let the infinitesimal transformations be
1 ξ= . (87) t Obviously, the formulae (85) and (86) can both be satisfied, therefore, the formula (87) corresponds to the Lie symmetrical transformation. Substituting Eqs. (87) into the structure equation (60), we get the gauge function q . (88) G= √ 2−1 Substituting Eqs. (87) and (88) into the formula (61), we obtain the conserved quantity of the system q = const. (89) I t, q, q σ , q 1 = tq 1 + √ 2−1 It is easy to verify by Definition 4 that √ √ 1I q1 2 1 1 11 = tq +√ = 2 − 1 tq + √ q 1 = 0. 1t 2−1 2−1 τ = 0,
LIE SYMMETRIES AND CONSERVED QUANTITIES. . .
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Summary
In summary, we have founded the Lie symmetries and the conserved quantities of the constraint mechanical systems on an arbitrary time scale, which is a new method to investigate the constrained mechanical systems. This is a significant work which extends the theories of Lie symmetry of the two cases for the continuous (T = R) and the discrete (T = Z) constraint mechanical systems to the systems on an arbitrary time scale T. Using this approach, it might also be possible to obtain Lie symmetries of the mechanico-electrical systems, the optimal control systems and other constraint mechanical systems on time scales. Acknowledgments This work has been partially supported by the National Natural Science Foundations of China (Grant Nos. 11472247, 11272287 and 11072218), and by Zhejiang Provincial Natural Science Foundation of China (Grant No.Y6110314). REFERENCES [1] B. Aulbach and S. Hilger: A unified approach to continuous and discrete dynamics, Collo. Math. Sci. 53 (1990), 37. [2] M. Bohner: Calculus of variations on time scales, Dyn. Syst. Appl. 13 (2004), 339. [3] M. Bohner and A. Peterson: Dynamic Equations on Time Scales, Birkh¨auser, Boston 2001. [4] M. Bohner and A. Peterson: Advances in Dynamic Equations on Time Scales, Birkh¨auser, Boston 2003. [5] R. Hilscher and V. Zeidan: Calculus of variations on time scales, J. Math. Anal. Appl. 289 (2004), 143. [6] N. Martins and D. F. M. Torres: Noether’s symmetry theorem for nabla problems of the calculus of variations, Appl. Math. Lett. 23 (2010), 1432. [7] R. Almeida and D. F. M. Torres: Isoperimetric problems on time scales with nabla derivatives, J. Vib. Control 6 (2009), 951. [8] A. B. Malinowska and D. F. M. Torres: Necessary and sufficient conditions for local Pareto optimality on time scales, J. Math. Phys. Sci. 6 (2009), 803. [9] R. P. Agarwal, M. Bohner and D. O’Rega: Dynamic equations on time scales, J. Comp. Appl. Math. 141 (2002), 27. [10] F. M. Atici and G. S. Guseinov: On Green’s functions and positive solutions for boundary value problems on time scales, J. Comp. Appl. Math. 141 (2002), 75. [11] R. A. C. Ferreira and D. F. M. Torres: High-order Calculus of Variations on Time Scales, Springer, Berlin 2008. [12] M. Bohner and G. S. Guseinov: Double integral calculus of variations on time scales, Comput. Math. Appl. 54 (2007), 45. [13] Z. Bartosiewicz and E. Pawluszewicz: Realizations of nonlinear control systems on time scales, IEEE Trans. Autom. Control 53 (2008), 571. [14] R. Hilscher and V. Zeidan: Weak maximum principle and accessory problem for control problems on time scales, Nonlinear Anal. 70 (2009), 3209. [15] D. F. M. Torres: Noether’s theorem on time scales, J. Math. Anal. Appl. 342 (2008), 1220. [16] M. Dryl and D. F. M. Torres: A general delta-nabla calculus of variations on time scales with application to economics, Int. J. Dyn. Syst. Diff. Eqns. 5 (2014), 42. [17] A. E. Noether: Invariante Variations Problem, Math. Phys. Kl.2 (1918), 235. [18] P. J. Olver: Applications of Lie Groups to Differential Equations, Springer, New York 1993. [19] S. P. S. Santos, N. Martins and D. F. M. Torres: Variational problems of Herglotz type with time delay: DuBois–Reymond condition and Noether’s first theorem, Discrete Contin. Dyn. Syst. 35 (2015), 4593.
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[20] G. S. F. Frederico and D. F. M. Torres: Fractional isoperimetric Noether’s theorem in the Riemann–Liouville sense, Rep. Math. Phys. 71 (2013), 291. [21] G. S. F. Frederico, T. Odzijewicz and D. F. M. Torres, Noether’s theorem for non-smooth extremals of variational problems with time delay, Appl. Anal. 93 (2014), 153. [22] G. S. F. Frederico and D. F. M. Torres: Fractional Noether’s theorem in the Riesz-Caputo sense, Appl. Math. Comput. 217 (2010), 1023. [23] J. L. Fu, L. Q. Chen and S. Jimnez, Non-Noether symmetries and Lutzky conserved quantities for mechanico-electrical systems, Phys. Lett. A 358 (2006), 5. [24] Y. Y. Zhao: Conservative quantities and Lie symmetries of nonconservative dynamical systems, Acta Mech. Sin. 26 (1994), 380. [25] J. Belmonte-Beitia, V. M. P´erez-Garc´ıa and V. Vekslerchik: Lie symmetries and solitons in nonlinear systems with spatially inhomogeneous nonlinearities, Phys. Rev. Lett. 98 (2007), 064102. [26] P. E. Hydon: Symmetries and first integrals of ordinary difference equations, Proc. R. Soc., Ser. A 456 (2000), 2835. [27] P. E. Hydon: Symmetry Methods for Ordinary Differential Equations, Cambridge University Press, Cambridge 1999. [28] J. L. Fu, B. Y. Chen and H. Fu: Velocity-dependent symmetries and non-Noether conserved quantities of electromechanical systems, Sci. China 54 (2011), 288. [29] J. L. Fu, H. Fu and R. W. Liu: Hojman conserved quantities of discrete mechanicoCelectrical systems constructed by continuous symmetries, Phys. Lett. A 374 (2010), 1812. [30] G. W. Bluman and S. Kumei: Symmetries and differential equations, Springer, New York 1989. [31] F. X. Mei: Application of Lie groups and Lie Algebra to constraint mechanical systems, Science Press, Beijing 1999.