On Noether symmetries and form invariance of mechanico-electrical systems

Physics Letters A 331 (2004) 138–152 www.elsevier.com/locate/pla

On Noether symmetries and form invariance of mechanico-electrical systems Jing-Li Fu a,b , Li-Qun Chen b,c,∗ a Department of Applied Physics, Zhejiang University of Science, Hangzhou 310018, China b Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, Shanghai 200072, China c Department of Mechanics, Shanghai University, Shanghai 200072, China

Received 1 July 2004; received in revised form 31 August 2004; accepted 31 August 2004 Available online 15 September 2004 Communicated by R. Wu

Abstract This Letter focuses on form invariance and Noether symmetries of mechanico-electrical systems. Based on the invariance of Hamiltonian actions for mechanico-electrical systems under the infinitesimal transformation of the coordinates, the electric quantities and the time, the authors present the Noether symmetry transformation, the Noether quasi-symmetry transformation, the generalized Noether quasi-symmetry transformation and the general Killing equations of Lagrange mechanico-electrical systems and Lagrange–Maxwell mechanico-electrical systems. Using the invariance of the differential equations, satisfied by physical quantities, such as Lagrangian, non-potential general forces, under the infinitesimal transformation, the authors propose the definition and criterions of the form invariance for mechanico-electrical systems. The Letter also demonstrates connection between the Noether symmetries and the form invariance of mechanico-electrical systems. An example is designed to illustrate these results.  2004 Elsevier B.V. All rights reserved. PACS: 03.20.+i Keywords: Mechanico-electrical system; Form invariance; Noether symmetry; Infinitesimal transformation

1. Introduction It is well known that symmetric principles are key issues in mathematics and physics. There are close relationships between symmetries and conserved quantities. Many methods have been developed to seek conserved * Corresponding author. Department of Mechanics, Shanghai University, Shanghai 200436, China.

E-mail addresses: [email protected] (J.-L. Fu), [email protected] (L.-Q. Chen). 0375-9601/$ – see front matter  2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2004.08.059

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139

quantities of mechanical systems [1]. Noether first proposed the famous Noether symmetry theory which deals with the invariance of Hamiltonian actions under infinitesimal transformations [2]. Djuki´c and Vujanovi´c [3] further investigated Noether symmetries and conserved quantities of non-conservative holonomic systems by including generalized velocities. Studies of Noether symmetry are very active in many areas such as mathematics, mechanics, and classical fields etc. [4–8]. For example, the Noether theory was recently applied to Birkhoffian systems, to 2D spinning particles and, along with the hyperextended Scaler-tensor theory, to the FLRW models (it is logical to consider an anisotropic and inhomogeneous model) [9–11]. Nevertheless, mechanico-electrical systems have rarely been explored in this way so far. Recently, Mei presented a form invariance theory dealing with the invariance of the differential equation satisfied by physical quantities such as Lagrangian, non-potential generalized forces and generalized constrained forces under the infinitesimal transformation about time and generalized coordinates [12,13]. The authors studied the form invariance of the canonical and the generalized canonical Hamiltonian systems [14]. The approach is significant in seeking conserved quantities. In this Letter, the authors generalize the Noether theory and the form invariance theory to mechanic-electrical systems, and then proposed new methods to locate conserved quantities of mechanic-electrical systems.

2. Lagrange–Maxwell equations of mechanico-electrical systems Mechanico-electrical systems are systems in which a mechanical process and an electromagnetic process are coupled. The mechanical part, constituted by N particles, is described by n generalized coordinates qs (s = 1, 2, . . . , n), and the electrodynamical part, constituted by m electric circuits, is described by the model of electrics. Consider a mechanico-electrical system composed of m circuits consisting of linear conductors and capacitors. Let ik denote the current, uk denote the electronic potential, ek (e˙k = ik ) denote the charge in the capacitor, Rk denote the resistance, and Ck denote the capacitance. Then the Lagrangian of the mechanico-electrical system is L = T (q, q) ˙ − V (q) + Wm (q, e) − We (q, e),

(1)

where T and V are, respectively, the kinetic energy and the potential energy, and the electric field energy and the magnetic field energy of the mth circuit are, respectively, defined by 1 ek2 1 , Wm = Lkr ik ir (k, r = 1, . . . , m), (2) 2 Ck 2 where Lkr (k = r) is the mutual inductance between the kth circuit and the rth circuit, Lkk is self inductance of the kth return circuit, and repeated suffixes stand for the sum. Motion of the system is governed by the Lagrange–Maxwell equation We =

∂L ∂F d ∂L ∂L ∂F d ∂L − + = uk , − + = Qs (k = 1, . . . , m; s = 1, . . . , n), (3) dt ∂ e˙k ∂ek ∂ e˙k dt ∂ q˙s ∂qs ∂ q˙s where Qs is generalized non-potential forces. Eq. (3) consists of n + m 2-order ordinary differential equations with respect to n generalized coordinates qs (s = 1, 2, . . . , n) and m generalized electric quantities ek (k = 1, 2, . . . , m). In Eq. (3), the dissipative function F of the system is ˙ F = Fe (˙e) + Fm (q, q),

(4)

where the electric dissipative function is given by 1 1 Fe = Rk ik2 = Rk e˙k2 (k = 1, . . . , m) (5) 2 2 and Fm is the dissipative function of the viscous frictional damping force, and Qs is non-potential general force.

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2.1. Variation principle of mechanico-electrical systems The Hamiltonian action of a mechanico-electrical system is defined as
t2 L(t, q, q, ˙ e, e˙ ) dt,

S(γ ) =

(6)

t1

where γ is an arbitrary curve. Introduce infinitesimal transformations of the generalized coordinates, the generalized electric quantities and the time t • = t + t,

qs• (t) = qs (t) + qs ,

ek• = ek + ek

(s = 1, . . . , n; k = 1, . . . , m),

(7)

where isochronous variations are used. Their expanded forms are ˙ e, e˙ ), t • = t + εα ξ0α (t, q, q,

ek• (t) = ek (t) + εα ηkα (t, q, q, ˙ e, e˙ ), (8) where εα (α = 1, . . . , r) is small parameter, and ξ0α , ξsα , ηkα are the infinitesimal generators. Hamiltonian action S(γ ) of the system under the infinitesimal transformation (8) is written as 

S γ

•




t2 =

qs• (t) = qs (t) + εα ξsα (t, q, q, ˙ e, e˙ ),

  L t • , q • , q˙ • , e• , e˙ • dt • .

(9)

t1

From Eq. (7), one obtains

  d(t) dt = dt + d(t) = 1 + dt. dt •

(10)

Application of the variational calculus to the Hamiltonian action S(γ ) leads to 

S = S γ

∗




t2 ∂L ∂L ∂L ∂L d(t) ∂L + t + − S(γ ) ≈ qs + q˙s + ek + e˙k dt. L dt ∂t ∂qs ∂ q˙s ∂ek ∂ e˙k

(11)

t1

Here only the linear term of small parameter ε is taken into consideration. That is, all non-linear terms of small parameter ε are neglected in the variation S. The relationships between the isochronous variation and the complete variation are given by qs = δqs + q˙s t, ek = δek + e˙k t, d d δ q˙s = (δqs ), δ e˙k = (δek ). dt dt Substitution of Eq. (12) into (11) yields
t2 S = t1

q˙s = δ q˙s + q¨s t,

e˙k = δ e˙k + e¨k t,

      ∂L d ∂L ∂L d ∂L d ∂L ∂L δqs + δek + − − δqs + δek . Lt + dt ∂ q˙s ∂ e˙k ∂qs dt ∂ q˙s ∂ek dt ∂ e˙k

(12)

(13)

Eq. (13) is a fundamental variation formula of the Hamiltonian action of mechanico-electrical systems. Inserting Eq. (7) into Eq. (13) leads to 


t2 S =

εα t1

      ∂L α ∂L α d ∂L α ∂L d ∂L α d ∂L α ¯ ¯ η¯ + − − ξ + η¯ dt, ξ + Lξ0 + dt ∂ q˙s s ∂ e˙k k ∂qs dt ∂ q˙s s ∂ek dt ∂ e˙k k

(14)

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where ξ¯sα = ξsα − q˙s ξ0α ,

η¯ kα = ηkα − e˙k ξ0α .

(15)

3. Noether theory of the mechanico-electrical systems It is well known that a system has the Noether symmetry when its Hamilton action is invariant under an infinitesimal transformation. The definitions and criterions of Noether symmetry transformations for mechanico-electrical systems are given as follow. 3.1. Noether symmetry transformations of mechanico-electrical systems Definition 1. If the Hamiltonian action of a mechanico-electrical system is invariant under an infinitesimal transformation, and the following condition S = 0

(16)

always holds, then the transformation is called the Noether symmetry transformation associated with the mechanico-electrical system. Definition 1, with Eqs. (13) and (14), leads to: Criterion 1. Infinitesimal transformation (7) is a Noether symmetry transformation of mechanico-electrical system (3), on the condition that ∂L ∂L ∂L ∂L ∂L d t + qs + q˙s + ek + e˙k + L (t) = 0. ∂t ∂qs ∂ q˙s ∂ek ∂ e˙k dt

(17)

Criterion 2. Infinitesimal transformation (8) is a Noether symmetry transformation of mechanico-electrical system (3), if r equations       d ∂L ¯ α ∂L α d ∂L ¯ α ∂L d ∂L α ∂L β (18) η¯ + − − ξ + η¯ = 0 ξ + Lξ0 + dt ∂ q˙s s ∂ e˙k k ∂qs dt ∂ q˙s s ∂ek dt ∂ e˙k k hold. Consider the following relationships t = εα ξ0α (t, q, q, ˙ e, e˙ ),

qs (t) = εα ξsα (t, q, q, ˙ e, e˙ ),

ek (t) = εα ηkα (t, q, q, ˙ e, e˙ ),

(19)

then Eq. (17) is written as r equations

  ∂L α ∂L α ∂L α ∂L α ∂L α ∂L ∂L ξ0 + ξ˙s + ξs + ηk + η˙ k + L − q˙s − e˙k ξ˙0α = 0 ∂t ∂qs ∂ q˙s ∂ek ∂ e˙k ∂ q˙s ∂ e˙k

If let α = 1 in Eq. (20), the Noether equality is obtained   ∂L ∂L ∂L ∂L ∂L ∂L ∂L ˙ ξ0 + ξs + ξs + ηk + η˙ k + L − q˙s − e˙k ξ˙0 = 0. ∂t ∂qs ∂ q˙s ∂ek ∂ e˙k ∂ q˙s ∂ e˙k

(α = 1, . . . , r).

(20)

(21)

The Noether symmetry transformation of the mechanico-electrical system can be determined by Criterion 1 and Criterion 2.

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3.2. Noether quasi-symmetry transformations of mechanico-electrical systems Bessel-Hegen gave the definition of the gauge transformation of Lagrangian systems [15]. Based on the idea, the Noether quasi-symmetry transformation can be defined for mechanico-electrical systems. For a mechanico-electrical system with its Lagrangian L(t, q, q, ˙ e, e˙ ) and a gauge function G(t, q, q, ˙ e, e˙ ), let d G(t, q, q, ˙ e, e˙ ). (22) dt Then the Noether quasi-symmetry transformation of the system can be obtained through substituting Eq. (22) into Eq. (11). ˙ e, e˙ ) = L(t, q, q, ˙ e, e˙ ) + L1 (t, q, q,

Definition 2. If the Hamiltonian action of a mechanico-electrical system is quasi-invariant under an infinitesimal transformation (7), that is, the following condition
t2 S = −

d (G) dt dt

(23)

t1

always holds for gauge function G, then the transformation is defined as Noether quasi-symmetry transformation. According to Definition 2, Eqs. (13) and (14) leads to the follows propositions. Criterion 3. Infinitesimal transformation (7) is the Noether quasi-symmetry transformation of a mechanicoelectrical system, if the transformation satisfies the following equation ∂L ∂L d ∂L ∂L ∂L d t + qs + q˙s + ek + e˙k + L (t) = − (G). ∂t ∂qs ∂ q˙s ∂ek ∂ e˙k dt dt

(24)

Criterion 4. Infinitesimal transformation (8) is the Noether quasi-symmetry transformation of a mechanicoelectrical system, if the r equations       d ∂L ¯ α ∂L α d ∂L ¯ α ∂L d ∂L α dGα ∂L α (25) η¯ k + − − ξs + η¯ k = − ξs + Lξ0 + dt ∂ q˙s ∂ e˙k ∂qs dt ∂ q˙s ∂ek dt ∂ e˙k dt hold. Based on the independence of the parameter εα , Eq. (19) and the following relationship G = εα Gα ,

(26)

Eq. (24) yields r equations   ∂L α ∂L α ∂L ˙ α ∂L α ∂L α ∂L ∂L ˙ α. ξ0 + ξs + ηk + η˙ k + L − q˙s − e˙k ξ0α = −G ξs + ∂t ∂qs ∂ q˙s ∂ek ∂ e˙k ∂ q˙s ∂ e˙k

(27)

When α = 1, Eq. (27) gives the Noether equality

  ∂L ∂L ˙ ∂L ∂L ∂L ∂L ∂L ˙ ξ0 + ξs + ηk + η˙ k + L − q˙s − e˙k ξ0 = −G. ξs + ∂t ∂qs ∂ q˙s ∂ek ∂ e˙k ∂ q˙s ∂ e˙k

(28)

The Noether quasi-symmetry transformation of mechanico-electrical systems can be determined via Criterion 3 and Criterion 4 when the Lagrangian L of the system is given.

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3.3. Generalized Noether quasi-symmetry transformation of the mechanico-electrical systems Notice that Qs − ∂Fs /∂ q˙s is the generalized force, and uk − ∂Fe /∂ e˙k is the generalized electrokinetic potential of the return circuit. If the Lagrangian of a mechanico-electrical system satisfies •


t2


t2 L(t, q, q, ˙ e, e˙ ) dt = t1•

t1

 
t2
t2 ∂F ∂F L1 (t, q, q, ˙ e, e˙ ) dt + Qs − δqs + uk − δek , ∂ q˙s ∂ e˙k t1

(29)

t1

then the Hamiltonian action of the mechanico-electrical system is generalized quasi-invariant under the infinitesimal transformation (7). Definition 3. An infinitesimal transformation (7) is a generalized Noether quasi-symmetry transformation, if the Hamiltonian action of a mechanico-electrical systems is generalized quasi-invariant under the infinitesimal transformation (7). That is, the following condition
t2 S = − t1

    ∂F ∂F d (G) + Qs − δqs + uk − δek dt dt ∂ q˙s ∂ e˙k

(30)

holds, where (Qs − ∂F /∂ q˙s )δqs is the sum of virtual work done by generalized non-potential forces, and (uk − ∂F /∂ e˙k )δek is the sum of virtual work done by generalized electromotive forces. Definition 3, Eqs. (13) and (14) lead to the follows propositions. Criterion 5. There exists a generalized Noether quasi-symmetry transformation in a mechanico-electrical system, if the infinitesimal transformation (7) satisfies the following equation ∂L ∂L ∂L ∂L ∂L d t + qs + q˙s + ek + e˙k + L (t) = ∂t ∂qs ∂ q˙s ∂ek ∂ e˙k dt     ∂F ∂F d + Qs − (qs − q˙s t) + uk − (ek − e˙k t) = − (G). ∂ q˙s ∂ e˙k dt

(31)

Criterion 6. There exists a generalized Noether quasi-symmetry transformation in a mechanico-electrical system, if the infinitesimal transformation (8) satisfies the following equation       d ∂L ¯ α ∂L α d ∂L ∂F ¯ α ∂L d ∂L ∂F ∂L α η¯ + − + Qs − − + uk − ξ + η¯ α ξ + Lξ0 + dt ∂ q˙s s ∂ e˙k k ∂qs dt ∂ q˙s ∂ q˙s s ∂ek dt ∂ e˙k ∂ e˙k k dGα (α = 1, . . . , r). =− (32) dt Due to the independence of the parameter εα , Eqs. (19), (27) and (31) lead to   ∂L α ∂L α ∂L α ∂L α ∂L α ∂L ∂L ˙ ξ + ξ + ξ + η + η˙ + L − q˙s − e˙k ξ˙0α = ∂t 0 ∂qs s ∂ q˙s s ∂ek k ∂ e˙k k ∂ q˙s ∂ e˙k       ∂F  α ∂F  α ˙ α. + Qs − ξs − q˙s ξ0α + uk − ηk − e˙k ξ0α = −G ∂ q˙s ∂ e˙k

(33)

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The generalized Noether quasi-symmetry transformation can be determined by Criterion 5 and Criterion 6. When α = 1, Eq. (33) gives the generalized Noether equality   ∂L ∂L ∂L ∂L ∂L ∂L ∂L ˙ ξs + ξs + ηk + η˙ k + L − q˙s − e˙k ξ˙0 = ξ0 + ∂t ∂qs ∂ q˙s ∂ek ∂ e˙k ∂ q˙s ∂ e˙k     ∂F ∂F ˙ + Qs − (34) (ξs − q˙s ξ0 ) + uk − (ηk − e˙k ξ0 ) = −G. ∂ q˙s ∂ e˙k 3.4. Killing equations of mechanico-electrical systems Eqs. (17), (24) and (31) can be, respectively, cast into the Killing equations or the generalized Killing equations, one-order differential equations with respect to infinitesimal generators ξ0 , ξs , ηk and G. Expanding Eq. (17), one obtains    ∂L ∂ξ0 ∂ξ0 ∂L ∂L ∂L ∂L ∂ξ0 ξ0 + + ξs + ηk + L − q˙s − e˙k q˙l + e˙j ∂t ∂qs ∂ek ∂ q˙s ∂ e˙k ∂t ∂ql ∂ej     ∂ξs ∂L ∂ξs ∂ξs ∂ηk ∂L ∂ηk ∂ηk + + + (35) q˙l + e˙k + q˙s + e˙j = 0, ∂ q˙s ∂t ∂ql ∂ek ∂ e˙k ∂t ∂qs ∂ej   ∂L ∂L ∂ξs ∂L ∂ηk ∂L ∂ξ0 ∂ξ0 ∂L q˙s − e˙k + + − e˙k = 0, L− (36) ∂ q˙s ∂ e˙k ∂ q˙l ∂ q˙s ∂ q˙l ∂ e˙k ∂ q˙l ∂ e˙k ∂ q˙l   ∂ξ0 ∂L ∂L ∂L ∂ξs ∂L ∂ηk ∂L ∂ηk L− (37) q˙s − e˙k + + − q˙s = 0. ∂ q˙s ∂ e˙k ∂ e˙j ∂ q˙s ∂ e˙j ∂ e˙k ∂ e˙j ∂ q˙s ∂ e˙j Eqs. (35), (36) and (37) are (n + m + 1)r one-order partial differential equations with respect to (n + m + 1)r unknown functions ξ0 , ξs , ηk . They are the Killing equations of a Lagrange mechanico-electrical system. The generators can be solved from Eqs. (35), (36) and (37), for given Lagrangian of the system. From Eq. (24) one can analogously get    ∂L ∂ξ0 ∂ξ0 ∂L ∂L ∂L ∂L ∂ξ0 ξ0 + + ξs + ηk + L − q˙s − e˙k q˙l + e˙j ∂t ∂qs ∂ek ∂ q˙s ∂ e˙k ∂t ∂ql ∂ej     ∂ξs ∂L ∂ξs ∂ξs ∂ηk ∂G ∂L ∂ηk ∂ηk ∂G ∂G + + − + (38) q˙l + e˙k + q˙s + e˙j = − q˙l − e˙j , ∂ q˙s ∂t ∂ql ∂ek ∂ e˙k ∂t ∂qs ∂ej ∂t ∂ql ∂ e˙j   ∂L ∂L ∂ξs ∂L ∂ηk ∂L ∂ξ0 ∂G ∂L ∂ξ0 q˙s − e˙k + + − e˙k = − , L− (39) ∂ q˙s ∂ e˙k ∂ q˙l ∂ q˙s ∂ q˙l ∂ e˙k ∂ q˙l ∂ e˙k ∂ q˙l ∂ q˙l   ∂ξ0 ∂L ∂L ∂L ∂ξs ∂L ∂ηk ∂L ∂ηk ∂G L− (40) q˙s − e˙k + + − q˙s = − . ∂ q˙s ∂ e˙k ∂ e˙j ∂ q˙s ∂ e˙j ∂ e˙k ∂ e˙j ∂ q˙s ∂ e˙j ∂ e˙j Eqs. (38), (39) and (40) are the Killing equations with a gauge function of the mechanico-electrical system, which are (n + m + 2)r one-order partial differential equations with respect to (n + m + 2)r unknown functions ξ0 , ξs , ηk and G. Eq. (31) yields    ∂L ∂ξ0 ∂ξ0 ∂L ∂L ∂L ∂L ∂ξ0 ξ0 + + ξs + ηk + L − q˙s − e˙k q˙l + e˙j ∂t ∂qs ∂ek ∂ q˙s ∂ e˙k ∂t ∂ql ∂ej       ∂ξs ∂L ∂ηk ∂ηk ∂L ∂ξs ∂ξs ∂ηk ∂F + + (ξs − q˙s ξ0 ) + q˙l + e˙k + q˙s + e˙j + Qs − ∂ q˙s ∂t ∂ql ∂ek ∂ e˙k ∂t ∂qs ∂ej ∂ q˙s

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  ∂F ∂G ∂G ∂G − (ηk − e˙k ξ0 ) = − + uk − q˙l − e˙j , ∂ e˙k ∂t ∂ql ∂ e˙j   ∂L ∂L ∂ξs ∂L ∂ηk ∂L ∂ξ0 ∂G ∂L ∂ξ0 q˙s − e˙k + + − e˙k = − , L− ∂ q˙s ∂ e˙k ∂ q˙l ∂ q˙s ∂ q˙l ∂ e˙k ∂ q˙l ∂ e˙k ∂ q˙l ∂ q˙l   ∂L ∂ξ0 ∂L ∂L ∂ξs ∂L ∂ηk ∂L ∂ηk ∂G L− q˙s − e˙k + + − q˙s = − . ∂ q˙s ∂ e˙k ∂ e˙j ∂ q˙s ∂ e˙j ∂ e˙k ∂ e˙j ∂ q˙s ∂ e˙j ∂ e˙j

145

(41) (42) (43)

Eqs. (41), (42) and (43) are the generalized Killing equations of Lagrange–Maxwell mechanico-electrical system, which are (n + m + 2)r one-order partial differential equations with respect to (n + m + 2)r unknown functions ξ0 , ξs , ηk and G. For given the Lagrangian, the generalized force (Qs − ∂F /∂ q˙s ), the generalized electromotive force (uk − ∂F /∂ e˙k ) of the system, the generators and gauge function can be solved from Eqs. (41), (42) and (43). 3.5. Noether theorems of Lagrange mechanico-electrical systems Consider a Lagrangian mechanico-electrical system satisfying conditions Qs − ∂F /∂ q˙s = 0 and uk − ∂F /∂ e˙k = 0. Eq. (3) becomes ∂L d ∂L − = 0, dt ∂ q˙s ∂qs

d ∂L ∂L − = 0. dt ∂ e˙k ∂ek

(44)

Theorem 1. If the infinitesimal transformation (7) is a Noether quasi-symmetry transformation, then the Lagrange mechanico-electrical system possesses the conserved quantities Lξ0 +

∂L ¯ ∂L η¯ k + G = C ξs + ∂ q˙s ∂ e˙k

(s = 1, . . . , n; k = 1, . . . , m),

(45)

where the gauge function is determined by Eq. (28) and C is a constant. Proof. A Noether quasi-symmetry transformation (7) makes Eq. (23) hold. The arbitrariness of the integral region [t1, t2 ] in Eqs. (14) and (23) yields       d ∂L ¯ ∂L ∂L d ∂L ¯ ∂L d ∂L η¯ k + G + − − ξs + η¯ k = 0. ξs + Lξ0 + dt ∂ q˙s ∂ e˙k ∂qs dt ∂ q˙s ∂ek dt ∂ e˙k Substituting Eq. (44) into Eq. (46) leads to   d ∂L ¯ ∂L η¯ k + G = 0. ξs + Lξ0 + dt ∂ q˙s ∂ e˙k

2

(46)

(47)

Theorem 2. If the infinitesimal transformation (8) is Noether symmetry transformation, the Lagrange mechanicoelectrical system possesses linear independent first integral as Lξ0 +

∂L ¯ ∂L η¯ k = C ξs + ∂ q˙s ∂ e˙k

(s = 1, . . . , n; k = 1, . . . , m).

(48)

The proof of Theorem 2 is similar to that of the Theorem 1. Theorem 1 and Theorem 2 are, respectively, the generalized Noether theorem and the Noether theorem for the Lagrange mechanico-electrical system.

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3.6. Noether theorems of Lagrange–Maxwell mechanico-electrical systems For a Lagrange–Maxwell mechanico-electrical system defined by Eq. (3), the conserved quantity can be obtained from the following generalized Noether theorem. Theorem 3. If the infinitesimal transformation (8) is a generalized quasi-symmetry transformation, the Lagrange– Maxwell mechanico-electrical system possesses the conserved quantities Lξ0 +

∂L ∂L η¯ k + G = C ξ¯s + ∂ q˙s ∂ e˙k

(s = 1, . . . , n; k = 1, . . . , m),

(49)

where the gauge function G is given by Eq. (34) and C is a constant. Proof. The generalized quasi-symmetry transformation makes Eq. (30) hold. Due to Eq. (14), Eq. (30) can be expressed as     ∂L ¯ ∂L ∂L d ∂L ∂F ¯ d η¯ k + G + − + Qs − ξs ξs + Lξ0 + dt ∂ q˙s ∂ e˙k ∂qs dt ∂ q˙s ∂ q˙s   ∂L d ∂L ∂F + − + uk − η¯ k dt = 0. ∂ek dt ∂ e˙k ∂ e˙k


t2  t1

Substituting Eq. (8) into Eq. (50), and using arbitrariness of the integral region [t1, t2 ], one obtains   d ∂L ∂L ¯ η¯ k + G = 0. 2 ξs + Lξ0 + dt ∂ q˙s ∂ e˙k

(50)

(51)

Theorem 3 is the generalized Noether theorem of the Lagrange–Maxwell mechanico-electrical systems. Theorem 1 and Theorem 2 of the Lagrange mechanico-electrical systems are special cases of the Theorem 3.

4. Definitions and criterions of form invariance for mechanico-electrical systems Introduce a point transformation with respect to the general coordinates and the general electric quantities and time t • = t + εξ0 (t, q, q, ˙ e, e˙ ),

qs• = qs + εξs (t, q, q, ˙ e, e˙ ),

ek• = ek + εη(t, q, q, ˙ e, e˙ ),

(52)

where ε is a small parameter, ξ0 , ξs , ηk are infinitesimal generators. Under the infinitesimal transformation (52), the physical quantities such as the Lagrangian L = L(t, q, q, ˙ e, e˙ ), the non-potential force Qs = Qs (t, q, q), ˙ the electric dissipative function Fe = F (˙e) and the dissipation ˙ are, respectively, transformed into L• = functions of the viscous frictional damping force Fm = Fm (q, q) • • • • • • • • • • • • L(t , q , q˙ , e , e˙ ), Qs = Qs (t , q , q˙ ), Fe = F (e˙ ), and Fm = Fm (q • , q˙ • ). Expanding L• , Q•s , Fe• , Fm• , one obtains   L• = L t • , q • , q˙ • , e, e˙     ∂L ∂L ∂L ∂L ∂L ˙ ˙ ˙ (53) ξs + (ξs − q˙s ξ0 ) + (η˙ k − e˙k ξ0 ) + O ε2 , = L(t, q, q, ˙ e, e˙ ) + ε ξ0 + ηk + ∂t ∂qs ∂ q˙s ∂e ∂ e˙k       ∂Qs ∂Qs ∂Qs ξ0 + Q•s = Qs t, q • , q˙ • = Qs (t, q, q) (54) ˙ +ε ξs + (ξ˙s − q˙s ξ˙0 ) + O ε2 , ∂t ∂qs ∂ q˙s

J.-L. Fu, L.-Q. Chen / Physics Letters A 331 (2004) 138–152

    ∂Fe Fe• = Fe e˙• = Fe (e) ˙ +ε (η˙ k − e˙ξ˙0 ) + O ε2 , ∂ e˙       ∂Fm ∂Fm • • • ˙ ˙ Fm = Fm q , q˙ = Fm (q, q) ˙ +ε ξs + (ξs − q˙s ξ0 ) + O ε2 . ∂qs ∂ q˙s

147

(55) (56)

Definition 4. For a Lagrange mechanico-electrical systems, if physical quantities L(t, q, q, ˙ e, e˙ ), Qs (t, q, q), ˙ Fe (˙e), Fm (q, q) ˙ under the infinitesimal transformations (52) satisfy ∂F • d ∂L• ∂L• − = uk − = 0, dt ∂ e˙k ∂ek ∂ e˙k

d ∂L• ∂L• ∂F • − = Q•s − =0 dt ∂ q˙s ∂qs ∂ q˙s

(k = 1, . . . , m; s = 1, . . . , n), (57) the symmetry is defined as the form invariance of the systems under the infinitesimal transformation (52). Criterion 7. If there exists a constant k and a gauge function G = G(t, q, e), and the infinitesimal generators ξ0 , ξs , ηk satisfy ∂L ∂L ˙ ∂L ∂L ∂L ˙ ξ0 + ξs + (ξs − q˙s ξ˙0 ) + ηk + (η˙ k − e˙k ξ˙0 ) = kL − G, ∂t ∂qs ∂ q˙s ∂ek ∂ e˙k

(58)

˙ ˙ ˙ ˙ d ∂G ∂G d ∂G ∂G − + − = 0, dt ∂ q˙s ∂qs dt ∂ e˙k ∂ek

(59)

and

the symmetry is the form invariance of the Lagrange mechanico-electrical systems under the infinitesimal transformation (52). Proof. Define the Euler operators of the Lagrange mechanico-electrical system as Es =

d ∂ ∂ − , dt ∂ q˙s ∂qs

Ek =

d ∂ ∂ − . dt ∂ e˙k ∂ek

(60)

Using Eqs. (53) and (58), one obtains       ˙ − Ek (G) ˙ . Es L• + Ek L• = Es (L) + Ek (L) + ε kEs (L) + kEk (L) − Es (G)

(61)

Substituting Eqs. (44) and (60) into Eq. (61), one knows that the right-hand side of the Eq. (61) equals naught. Therefore     2 Es L• + Ek L• = 0. (62) Definition 5. For the Lagrange–Maxwell mechanico-electrical systems, if physical quantities L(t, q, q, ˙ e, e˙ ), ˙ Fe (e) ˙ and Fm (q, q) ˙ under the infinitesimal transformations (52) satisfy Qs (t, q, q), d ∂L• ∂L• ∂F • − + = uk , dt ∂ e˙k ∂ek ∂ e˙k

d ∂L• ∂L• ∂F • − + = Q•s dt ∂ q˙s ∂qs ∂ q˙s

(k = 1, . . . , m; s = 1, . . . , n),

(63)

then the symmetry is defined as the form invariance of the systems under the infinitesimal transformations (52). Criterion 8. If there exists a constant k and a gauge function G = G(t, q, e) = G1 (t, q, e) + G2 (t, q, e) + G3 (t, q, e) + G4 (t, q, e), and the infinitesimal generators ξ0 , ξs , ηk satisfy the following conditions ∂L ∂L ∂L ∂L ∂L ˙ 1, ξ0 + ξs + (ξ˙s − q˙s ξ˙0 ) + ηk + (η˙ k − e˙k ξ˙0 ) = kL − G ∂t ∂qs ∂ q˙s ∂ek ∂ e˙k

(64a)

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∂Qs ∂Qs ∂Qs ˙ 2, ξs + (ξ˙s − q˙s ξ˙0 ) = kQs − G ξ0 + ∂t ∂qs ∂ q˙s ∂uk ∂uk ∂uk ˙ 3, ξ0 + ηk + (η˙ k − e˙k ξ˙0 ) = kF − G ∂t ∂ek ∂ e˙k ∂F ˙ ∂F ˙ (ξs − q˙s ξ˙0 ) + (η˙ k − e˙k ξ˙0 ) = kF − G, ∂ q˙s ∂ e˙k

(64b) (64c) (64d)

and ˙ 1) − G ˙2+ E s (G

˙4 ∂G = 0, ∂ q˙s

˙ 1) − G ˙3+ E k (G

˙4 ∂G = 0, ∂ e˙k

(65)

then the symmetry is the form invariance of the Lagrange–Maxwell mechanico-electrical systems under the infinitesimal transformation (52). Proof. Substituting Eqs. (53)–(58) into Eq. (63), one has     ∂F • ∂F • + − u•k − Q•s Es L• + Ek L• + ∂ q˙s ∂ e˙k   ∂L ∂L ˙ ∂L ∂L ∂L ξ0 + = Es (L) + εEs ξs + (ξs − q˙s ξ˙0 ) + ηk + (η˙ k − e˙k ξ˙0 ) ∂t ∂qs ∂ q˙s ∂ek ∂ e˙k   ∂F ∂ ∂F ∂F + +ε + (ξ˙l − q˙l ξ˙0 ) + Ek (L) ∂ q˙s ∂ q˙s ∂ql ∂ q˙l   ∂L ∂L ∂L ∂L ∂F ∂L ξ0 + ξs + (ξ˙s − q˙s ξ˙0 ) + ηk + (η˙ k − e˙k ξ˙0 ) + + εEk ∂t ∂qs ∂ q˙s ∂ek ∂ e˙k ∂ e˙k       ∂ ∂F ∂uk ∂Qs ∂Qs ∂Qs − Qs − ε ξ0 + +ε (η˙ j − e˙j ξ˙0 ) − uk − ε ξs + (ξ˙s − q˙s ξ˙0 ) + O ε2 ∂ e˙k ∂ e˙j ∂t ∂t ∂qs ∂ q˙s ˙ ˙ ˙2 = Es (L) + εkEs (L) − Es (G1 ) + Ek (L) + εkEk (L) − Ek (G1 ) − uk − kuk + G ˙ 4 ∂G ˙4 ∂F ∂F ∂F ∂F ∂G ˙ 3. + + εk + + εk − − − Qs − εkQs + G (66) ∂ q˙s ∂ q˙s ∂ e˙k ∂ e˙k ∂ q˙s ∂ e˙k Thus the Criterion 8 is derived from Eqs. (3), (65) and (66). 2

5. The connections between the Noether symmetry and the form invariance of mechanico-electrical systems The form invariance of mechanico-electrical systems may lead to the Noether symmetry. Actually, the following proposition can be easily proved. Proposition 1. For a Lagrange–Maxwell mechanico-electrical system (3), if there is a form invariance under the infinitesimal transformation (52) and there exists a gauge function GN = GN (t, q, e) satisfying Eq. (34), the form invariance will lead to a generalized Noether symmetry, and the system possesses a conserved quantity (49). As a special case, there is the connections between the form invariance and the Noether symmetry of Lagrangian mechanico-electrical systems. Proposition 2. For a Lagrange mechanico-electrical system (44), if there is the form invariance under the infinitesimal transformation (52), and there exists a gauge function GN = GN (t, q, e) satisfying Eq. (28), then the form invariance will lead to a Noether symmetry, and the system possesses a conserved quantity (48).

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149

6. An example An electrocircuit of capacitor is presented in Fig. 1. The electrode plank (mass m = 1) of capacitor is connected to a spring with elastic coefficient k. Free vibration along vertical direction is considered between electrode planks, under action of gravity, elastic and electric field force. C denotes capacitance, A denotes a constant, S denotes distance, L1 (L1 = 1) denotes unit inductance, and R denotes resistance. Power of the system is supplied by electromotive force E = E0 sin wt. Study the Noether symmetry of the system. Choose the equilibrium position (O point) of active plank of capacitor as the coordinate origin, and take coordinates q and electric quantity e as generalized coordinates. The kinetic energy of the system is T = 12 q˙ 2 , the spring potential energy is V = 12 kq 2 , the magnetic energy is Wm = 12 e˙2 , and the electric energy of capacitor is 2 We = 12 s−q A e . Then Lagrangian L of the system is 1 1 1 1s −q 2 e , L = q˙ 2 + e˙2 − kq 2 − 2 2 2 2 A dissipative function as

(67)

1 F = Fe = R e˙2 , 2 and non-conservative generalized force is

(68)

ur = E0 sin wt.

(69)

Substituting Lagrange function L into generalized Killing equations (41), (42) and (43), one obtains      1 ∂ξ0 1 2 s −q ∂ξ0 ∂ξ0 1 1 1s −q 2 e ξ− eη + − q˙ 2 − e˙2 − kq 2 − e + q˙ + e˙ −kq + 2A A 2 2 2 2 A ∂t ∂q ∂e     ∂ξ ∂ξ ∂η ∂η ∂η ∂ξ + q˙ + e˙ + e˙ + q˙ + e˙ + (E0 sin wt − R e)(η + q˙ ˙ − eξ ˙ 0) ∂t ∂q ∂e ∂t ∂q ∂e ∂G ∂G ∂G − q˙ − e, ˙ =− ∂t ∂q ∂e   1 2 1 2 1 2 1 s − q 2 ∂ξ0 ∂ξ ∂η ∂ξ0 2 ∂G − q˙ − e˙ − kq − e + q˙ + e˙ − e˙ = − , 2 2 2 2 A ∂ q˙ ∂ q˙ ∂ q˙ ∂ q˙ ∂ q˙   1 ∂η ∂ξ0 2 1 1 1 s − q 2 ∂ξ0 ∂ξ ∂G − q˙ 2 − e˙2 − kq 2 − e + q˙ + e˙ − q˙ = − . 2 2 2 2 A ∂ e˙ ∂ e˙ ∂ e˙ ∂ e˙ ∂ e˙

(70) (71) (72)

Suppose the form of infinitesimal generators as follow

ξ = C , 0

0

˙ ξ = g1 (t, g, e) + f1 (g, e)q˙ + f2 (g, e)e, η = g2 (t, g, e) + f3 (g, e)q˙ + f4 (g, e)e. ˙

(73)

Fig. 1.

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Substitution of Eq. (73) into Eq. (71) and Eq. (72) leads to qf ˙ 1 + ef ˙ 3=−

∂G , ∂ q˙

qf ˙ 2 + ef ˙ 4=−

∂G . ∂ e˙

(74)

From Eq. (74), one gets 1 1 G = − q˙ 2 f1 − e˙2 f4 − q˙ e˙f2 + f5 (q, e), f2 = f3 . 2 2 Substitution of Eqs. (73) and (75) into Eq. (70) yields 1 2 s −q −kq(g1 + f1 q˙ + f2 e) e (g1 + f1 q˙ + f2 e˙) − e(g2 + f2 q˙ + f4 e) ˙ + ˙ 2A A   ∂f1 2 ∂f2 ∂g1 ∂f1 ∂f2 2 ∂g1 ∂g1 + q˙ + q˙ + q˙ e˙ + e˙ + q˙ e˙ + e˙ + q˙ ∂t ∂q ∂q ∂q ∂e ∂e ∂e   ∂g2 ∂g2 ∂f2 2 ∂f4 ∂g2 ∂f2 ∂f4 2 + e˙ + q˙ + q˙ + q˙ e˙ + e˙ + q˙ e˙ + e˙ ∂t ∂q ∂q ∂q ∂e ∂e ∂e + E0 sin wt (g2 + f2 q˙ + f4 e) ˙ − R e(g ˙ 2 + f2 q˙ + f4 e) ˙ − E0 C0 sin wt e˙ + RC0 e˙ ∂f5 1 ∂f5 1 3 ∂f1 1 2 ∂f4 ∂f1 1 3 ∂f4 ∂f2 ∂f5 2 ∂f2 + q˙ + e˙ q˙ + q˙ e˙ − q˙ + q˙ 2 e˙ + e˙ + q˙ e˙ 2 − e. ˙ =− ∂t 2 ∂q 2 ∂q ∂q ∂q 2 ∂e 2 ∂e ∂e ∂e

(75)

(76)

Choosing those terms containing q˙ 3 , q˙ 2 e, ˙ q˙ e˙2 , e˙3 , q˙ 2 , q˙ e, ˙ e˙2 , q, ˙ e, ˙ one obtains the following equations ∂f1 1 ∂f1 = , ∂q 2 ∂q ∂f2 ∂f1 ∂f2 ∂f2 1 ∂f1 + + = + , ∂q ∂e ∂q ∂q 2 ∂e ∂f2 ∂f4 ∂f2 ∂f2 1 ∂f4 + + = + , ∂e ∂q ∂e ∂e 2 ∂q ∂f4 1 ∂f4 = , ∂e 2 ∂e ∂g1 = 0, ∂q ∂g2 − Rf4 = 0, ∂e ∂g1 ∂g2 + − Rf2 = 0, ∂e ∂q ∂f5 1 2 s −q ∂g1 e f1 − ef2 + E0 sin wtf2 + =− , −kqf1 + 2A A ∂t ∂q 1 2 s −q ∂g2 ∂f5 e f2 − ef4 + E0 sin wtf4 − Rg2 − E0 C0 sin wt + + RC0 = − , −kqf2 + 2A A ∂t ∂e 1 2 s −q −kqg1 + e g1 − eg2 + E0 sin wtg2 = 0. 2A A From Eqs. (77)–(80), one has   f1 = −c1 e2 − 2c2 e + c4 , f = c1 qe + c2 q + c3 e + c5 ,  2 f4 = −c1 q 2 − 2c2 q + c6 .

(77) (78) (79) (80) (81) (82) (83) (84) (85) (86)

(87)

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151

Solving Eqs. (81), (82), (83) and (73), one gets g1 = 0,

g2 = 0,

ξ0 = c0 ,

ξ = c4 q, ˙

c1 = 0,

c2 = c3 = c5 = 0,

(88)

and η = c6 e. ˙

(89)

Substituting Eqs. (87) and (88) into Eqs. (84) and (85), and integrating the resulting equation, one has 1 1 s−q c4 e 2 − c6 e2 + E0 c6 e sin wt − E0 c0 e sin wt + Rc0 e. f5 = − kc4 q 2 + 2 2A 2A Based on Eq. (75), the gauge function can be obtained 1 1 1 1 s−q c4 e 2 − c6 e2 + E0 c6 e sin wt − E0 c0 e sin wt + Rc0 e. G = − c4 q˙ 2 − c6 e˙2 − kc4 q 2 + 2 2 2 2A 2A When choosing c0 , c4 , c6 as some special value, one can obtain the following symmetries  c0 = 1, c4 = c6 = 0, (1) ξ0 = 1, ξ = η = 0, G = −E0 e sin wt + Re;  c0 = 1, c4 = 1, c6 = 0, (2) 1 2 ξ0 = 1, ξ = q, ˙ η = 0, G = − 12 q˙ 2 − 12 kq 2 + 2A e − E0 e sin wt + Re;  c0 = 1, c4 = 0, c6 = 1, (3) 2 ξ0 = 1, ξ = 0, η = e˙, G = − 12 e˙ 2 − s−q 2A e + Re;  c0 = 1, c4 = 1, c6 = 1, (4) 1 2 2 ξ0 = 1, ξ = q, ˙ η = e, ˙ G = − 12 q˙ 2 − 12 e˙2 − 12 kq 2 + 2A e − s−q 2A e + Re;  c0 = 0, c4 = 1, c6 = 1, (5) 1 2 2 ξ0 = 0, ξ = q, ˙ η = e, ˙ G = − 12 q˙ 2 − 12 e˙2 − 12 kq 2 + 2A e − s−q 2A e + E0 e sin wt;  c0 = 0, c4 = 1, c6 = 0, (6) 1 2 ˙ η = 0, G = − 12 q˙ 2 − 12 kq 2 + 2A e ; ξ0 = 0, ξ = q,  c0 = 0, c4 = 0, c6 = 1, (7) 2 ξ0 = 0, ξ = 0, η = e˙, G = − 12 e˙ 2 − s−q 2A e + E0 e sin wt.

(90)

(91)

(92) (93) (94) (95) (96) (97) (98)

Substituting the generalized quasi-symmetries (92)–(98) into Eq. (49), one obtains, respectively, the following conserved quantities 1 1 1 1s−q 2 I1 = − q˙ 2 − e˙2 − kq 2 − e − E0 e sin wt + Re = const, 2 2 2 2 A 1 s−q 2 1 2 e + e − E0 e sin wt + Re = const, I2 = − e˙2 − 2 2A 2A 1 1 s−q 2 e + Re = const, I3 = − q˙ 2 − kq 2 − 2 2 A 1 2 s −q 2 e − e + Re = const, I4 = 2A A 1 1 1 1 2 s−q 2 e − e + E0 e sin wt = const, I5 = q˙ 2 + e˙2 − kq 2 + 2 2 2 2A 2A 1 1 1 2 e = const, I6 = q˙ 2 − kq 2 + 2 2 2A 1 s−q 2 e + E0 sin wt = const. I7 = e˙ 2 − 2 2A

(99) (100) (101) (102) (103) (104) (105)

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7. Conclusion The Noether symmetries and the form invariance are extended to mechanico-electrical systems in this Letter. The results here present significant approaches to seek conserved quantities in mechanico-electrical systems.

Acknowledgements The research is supported by the National Science Foundation of China (Grant No. 10372053) and the National Science Foundation of Henan Province Government, China (Grant No. 03011011400).

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