1 Introduction

The symmetric theory has been widely used in mathematics,physics,mechanics,and dynamical systems. In early years,Lie group theory method has been a popular and useful method in the study of ordinary differential equations (ODEs) and partial differential equations (PDEs)^{[ 1, 2, 3, 4, 5, 6 ]}. Bluman^{[ 1 ]} and Olver^{[ 2, 3 ]} have showed that one can use canonical coordinates or differentials invariant’s method to reduce the order or to solve the differential equation with Lie group. In other words,given ODEs or PDEs with known Lie symmetries,one can obtain the corresponding canonical coordinates or differentials invariants,and then get the equations after reduction of order. Furthermore,the solutions of the equations can be obtained.

In the late 20th century,Noether symmetry and Lie symmetry were widely introduced to solve the problems of constraint mechanical system^{[ 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28 ]}. Noether,the famous German mathematician, established the Noether theorem that can be described as that each successive symmetry of the mechanical system action has the corresponding conserved quantity. The Noether symmetry is that the Hamilton action is an invariant under infinitesimal transformations of group in phase space. Calculating the killing equations,we can obtain the infinitesimal transformations which can be substituted to the Noether theorem to get the Noether conserved quantities. In addition,the basic idea of Lie symmetry is to keep the equation of motion invariant under the infinitesimal transformations. We can obtain generational functions and normal functions from determining equations and structural equations. Then,the conserved quantities can be obtained from Lie theorem with the infinitesimal transformation and structural equations. Nowadays,many good and useful results have been obtained using the symmetric theory to constrained mechanical system in the papers by Feng-xiang MEI et al.

The main point of using the canonical coordinates method to constraint mechanical system is that the corresponding canonical coordinates can be calculated with the infinitesimal transformations which can be obtained through determining equations,and then the first integrals of the equation of motion can be derived. In recent years,canonical coordinates method has been widely used to solve many problems in many fields^{[ 29, 30, 31, 32, 33, 34, 35, 36, 37 ]}. However,applying canonical coordinates method to dynamic system and obtaining the corresponding first integral have not been studied. In the paper,this problem will be solved. 2 Canonical coordinates

In this section,we review the definition and properties of the canonical coordinates which have been given in Ref. ^{[ 1 ]},where some good results have been established about canonical coordinates. Now,let us consider a one-parameter Lie group transformation

in which x = (x₁,x₂,· · · ,x_n) are variables,ξ(x) = ∂X/∂ε(x;ε) and X = are infinitesimal and infinitesimal generator of (1). Let us introduce the coordinate transformation, Thus,the corresponding infinitesimal generators become Y =.

Definition 1 The change of coordinates (2) defines a set of canonical coordinates for the one-parameter Lie group of transformations (1). Then,the group (1) turns into

Theorem 1 Assume that the canonical coordinates of equation are (s(x,y),r(x,y)),non- trivial one-parameter Lie group of transformations should satisfy

where infinitesimal generators are X = ξ(x,y)∂/∂x +η(x,y)∂/∂y . Then,solving the nth-order ODE yⁿ = f(x,y,y′,· · · ,y^n-1) reduces to solving the (n − 1)th-order one where z = ds/dt ,and the nontrivial one-parameter Lie group transformations are

Theorem 2 If z = φ(r;C₁,C₂,· · · ,C_n−1) is the general solution of (5),then the general solution of ODE is

where C₁,C₂,· · · ,C_n are constants. 3 Lie point symmetry of constraint homonomic system

In this part,Lie group theory is used to determine Lie point symmetries of the motion of equation of the single-freedom constraint homonomic system. The system is non-singular and can be described by the Euler-Lagrange equation in the form as

Case 1 If Q(t,q,q˙) = 0,the system is a conservative homonomic system,the equation of motion can be expressed in the form as

Case 2 If Q(t,q,q˙) 6= 0,the system is a non-conservative homonomic system. After decoupling, the equation of motion still has the same form as (9).

We assume that the system has one-parameter Lie point symmetry. Thus,the infinitesimal transformations with respect to time and generalized coordinates can be introduced as follows:

where ε is a parameter,and ξ and η are infinitesimals. The group is determined by its infinitesimal generator, The generator of the first extended group is given by Its second extension has the form Thus,the basic idea of Lie symmetry is to keep (9) invariant under the infinitesimal transformations (10) that must satisfy the determining equation Then,we have Obviously,we can get the infinitesimal transformations from (15). 4 Canonical coordinates and first integral of conservative homonomic system Now,we discuss a homonomic system with the method of canonical coordinates. We had worked out the infinitesimal transformation (ξ(t,q),η(t,q)) with (15). Then,using (4) and (11), we obtain (r,s) = (f(t,q),g(t,q)). Finally,the derivations of canonical coordinates are given by If s_qr_t − s_tr_q 6≠ 0,we have Then,we can obtain q˙ and from (17), Substituting (18) into (9) leads to Canonical coordinate s is free from function ',where z = ds/dt . If (19) has the general solution, then the general solution of (9) can be obtained. Unless,we can obtain the first integral of the equation of the motion.

In this part,two different examples about conservative homonomic system are considered. The new conserved quantities have been constructed from the different canonical coordinates.

Example 1 We now consider an example of the harmonic oscillator^{[ 11 ]},the equation of motion is

and the determining equation of the system under the infinitesimal transformation (10) is (21) is a polynomial equation of degree three in q˙. Let the coefficients of q˙n euqal to zero (n = 0, 1,2,3),we obtain a system of partial differential equations as To find the corresponding infinitesimal generators,(22) should be solved. Then,we have After some calculations,(4) and (23)-(25) yield the canonical coordinates of the system, If the canonical coordinates do not satisfy the condition of the coordinates of (26),then by use of (20) and (27)-(28),(19) can be written as where z₂ = ds₂/dr₂ ,and z₃ = ds₃/dr₃ . Thus,the general solutions of (26) and (27) can be given as According to Theorem 2,we have the general solutions of (20) where C_i (i = 1,2,3,4) are constants.

Example 2 Now,we use the method of canonical coordinates to seek the conserved quantity of the variable mass homonomic systems^{[ 15 ]}. The spherical raindrops are falling straight down in the atmosphere,and are supplied with water vapor. Due to condensation,the masses of the raindrops increase with a rate that is proportional to the surface area. α is a proportionality coefficient. The initial speed of the raindrops is v0.

For dm/dt = 4πv²,v = v0 + αt,we have m = 4πv³/3 . When the absolute velocity of particles into the atmosphere is zero and m only depends on t,we can get that the reaction thrust p is zero from

The Lagrangian of the system is L = 1 2mq˙² − mgq,where q is the vertical coordinate. The equation of motion is

Then,we have the determining equation of the system under the infinitesimal transformation (10), (36) yields a set of equations that can be separated with respect to q˙. The resulting equations are The solution of (37) is Using (4) and (38),we get the canonical coordinates of the system From (16)-(17),the original equation can be rewritten in the form as Setting z = ds/dr ,one has In order to obtain the conserved quantity of the system,we should choose the canonical coordinates r appropriately. Through calculation,we get the function !(t) as follows. (i) Set !(t) = kt,where k is a nonzero arbitrary constant. (41) can be expressed as Therefore,the system possesses the conserved quantity. According to Theorem 2,we can also get the general solution of the system as follows: where c is a constant.

(ii) Let !(t) = ekt,where k is a nonzero arbitrary constant. One has

Obviously,one conserved quantity of the system can be deduced from (44).

(iii) Set !(t) = sin t. (41) can be expressed as

Three conserved quantities (42),(44),and (45) of the system have been found finally. We should point out that the different first integral of the system would be obtained by taking different !(t). 5 Canonical coordinates and first integral of non-conservative homonomic system

In the section,the canonical coordinates method used to non-conservative system will be discussed. The canonical coordinates (r,s) are determined by (4) and (11). Then,we take the differential of canonical coordinates

If the basic condition s_qr_t − s_tr_q 6≠0 can be satisfied,we have Substituting (47) into (9) yields an ordinary differential equation The first integral of the non-conservative system has been given.

Example 3 The Euler-Lagrange equation and the generalized force of the system are in the form of L = 1/2 q˙² and Q = q˙. Thus,the equation of the motion can be written as

The determining equation of the system (49) under the infinitesimal transformation (10) is equating to zero the coefficients of powers of q˙ leads to a system of four partial differential equations, and the solution is

Different solutions of (51) can be obtained,if parameters c1 and c2 are given different values in (52). We discuss two possibilities in the following.

(i) Set c₁ = 1 and c₂ = 0. The infinitesimal transformation becomes

Group (10) is completely defined by One can obtain the canonical coordinates with (4) and (54) like We can easily get From (56),we have Set z = ds/dr . Substituting (57) into (49) leads to Obviously,(48) is the first integral of the system.

(ii) Set c₁ = 1 and c₂ = 1,then we get the infinitesimal transformation (1,q + 1),whose corresponding generator is

The derivation of canonical coordinates can be written as where canonical coordinates are (r,s) = (e^−t(q + 1),t). For s_qr_t − s_tr_q 6≠ 0,q˙ and q¨ can be written in the forms of Set z = ds/dr . Substituting (61) into (49) leads to

Obviously,the system has the conserved quantity (62) and the general solution as s = cer, where c is a constant. 6 Conclusions

In this paper,the first integral of the homonomic system with the canonical coordinates method has been obtained. The method to construct canonical coordinates for the equations of motion in multi-degree of freedom for a constraint system remains to be studied.