Theory of Generalized Canonical Transformations for Birkhoff Systems

Transformation is an important means to study problems in analytical mechanics. It is often difficult to solve dynamic equations, and the use of variable transformation can make the equations easier to solve. The theory of canonical transformations plays an important role in solving Hamilton’s canonical equations. Birkhoffian mechanics is a natural generalization of Hamiltonian mechanics. This paper attempts to extend the canonical transformation theory of Hamilton systems to Birkhoff systems and establish the generalized canonical transformation of Birkhoff systems. First, the definition and criterion of the generalized canonical transformation for the Birkhoff system are established. Secondly, based on the criterion equation and considering the generating functions of different forms, six generalized canonical transformation formulas are derived. As special cases, the canonical transformation formulas of classical Hamilton’s equations are given. At the end of the paper, two examples are given to illustrate the application of the results.


Introduction
Birkhoffian mechanics can be traced back to Birkhoff's monograph Dynamical Systems, which gave a new class of dynamic equations more common than Hamilton's canonical equations and a new class of integral variational principles more common than Hamilton's principle [1]. Santilli [2] studied Birkhoff's equations, the transformation theory of Birkhoff's equations, and the extension of Galileo's relativity and applied Birkhoff's equations to hadron physics. Galiullin et al. [3] studied the inverse problem of Birkhoffian dynamics, the symmetry, and the conformal invariance of Birkhoff systems. Mei et al. have conducted in-depth studies on the dynamics of Birkhoff systems, including Birkhoffian representation of holonomic and nonholonomic systems, integration theory, symmetry theory, inverse problem of dynamics, motion stability, geometric method, and global analysis of Birkhoff systems [4], and extended the results to generalized Birkhoff systems [5]. In recent years, some new advances have been made in the study of dynamics of Birkhoff systems, such as [6][7][8][9][10][11][12][13][14][15][16][17][18][19] and the references therein.
The integration problem of dynamic equations is an important aspect of analytical mechanics. Since it is often difficult to solve the general dynamic equations, the transformation of variables can make the equations easy to solve. The classical Hamilton canonical transformation theory plays an important role in solving dynamic equations. How do we extend Hamilton canonical transformation theory to Birkhoff systems? Santilli first proposed and preliminarily studied the transformation theory of Birkhoff's equations and only gave one kind of generating function and its transformation [2]. In this paper, based on the basic identity of Birkhoff system's generalized canonical transformation, we derive six generalized canonical transformation formulas by selecting different forms of generating functions and give the transformation relations between the old and new variables in each case. The way of selecting the generating function in this paper is different from that in Reference [2]. The canonical transformation formulas of classical Hamilton's equations are the special cases of the generalized canonical transformation formulas of Birkhoff systems. The application of the results is illustrated with two examples. This paper is organized as follows. In Section 2, we give the definition of the generalized canonical transformation for Birkhoff systems and establish the basic identity for constructing the generalized canonical transformation. In Section 3, we present six generating functions of Birkhoff systems, derive the corresponding generalized regular transformation formulas, and discuss their special cases, namely, the canonical transformation of Hamilton systems. Two examples are given in Section 4. We conclude in Section 5.

Definition and Criterion of Generalized Canonical Transformations
We consider a mechanical system described by 2n Birkhoff's variables a μ ðμ = 1, 2, ⋯, 2nÞ. The differential equations of motion of the system can be expressed as the following Birkhoff's equations: where B = Bðt, a ν Þ is the Birkhoffian and R μ = R μ ðt, a ν Þ are Birkhoff's functions. If there is a contemporaneous transformation where the equations of motion expressed by the new variables a μ remain in the form of Birkhoff's equations, i.e., where B = Bðt, a ν Þ and R μ = R μ ðt, a ν Þ are the new Birkhoffian and new Birkhoff's functions, then the transformation (2) is the generalized canonical transformation of the Birkhoff system.
Considering that Birkhoff's equations are directly derived from the Pfaff-Birkhoff principle, we give a general definition of the generalized canonical transformation of the Birkhoff system, as follows: Definition 1. For the Birkhoff system (1), if the contemporaneous transformation (2) preserves the Pfaff-Birkhoff principle in the transition from the old variables to the new variables where the notation δð * Þ represents the isochronous variation of ð * Þ, then the transformation is called a generalized canonical transformation of the system.
According to Definition 1, equations (4) and (5) need to be satisfied simultaneously for the generalized canonical transformation, but this does not mean that their integrand functions are exactly the same. In general, they can differ from each other by the total derivative of any function Fðt, a ν , a ν Þ with respect to time t. Due to Considering δa μ ðt 1 Þ = δa μ ðt 0 Þ = δ a μ ðt 1 Þ = δ a μ ðt 0 Þ = 0 ðμ = 1, 2, ⋯, 2nÞ, so the variation of equation (6) is zero, we have Hence, we obtain the following. holds.
Formula (8) is called the criterion equation to judge whether the given transformation of the Birkhoff system is a generalized canonical transformation. The function F is called the generating function.
Since the old variables a μ , the new variables a μ , and time t are connected by 2n transformation equation (2), only 2n variables are independent except for the variable . Selecting independent variables usually can have different schemes. Thus, the generating function can also have different forms.

Generating Functions and Generalized Canonical Transformations
For the convenience of interpretation, we express Birkhoff's variables as a = fa s , a s g and Birkhoff's functions as R = fR s , R s g, where s = 1, 2, ⋯, n. Then, the criterion equation (8) can be expressed as where R s , R s are functions of the old variables a k , a k and R s , Theorem 3. If the old variables a s and the new variables a s ðs = 1, 2, ⋯, nÞ are regarded as 2n independent variables, namely, the generating function is taken as F 1 ðt, a s , a s Þ, then the transformation determined by the following equations is the generalized canonical transformation of Birkhoff system (1), where F 1 ðt, a s , a s Þ is called the first kind of generating function.
Proof. Let We have Since a s , a s ðs = 1, 2, ⋯, nÞ are independent variables, we have Substituting formula (12) into the criterion equation (9), and considering the relation (13), we get By the independence of da s and d a s , we get the results easily. The theorem is proved.
Hamilton's principle is a special case of the Pfaff-Birkhoff principle, and Hamilton's canonical equation is a special case of Birkhoff's equation. Therefore, the generalized canonical transformations of the Birkhoff system are naturally suitable for the Hamilton system. In fact, if we take a s = q s , a s = p s , R s = p s , R s = 0, B = H, then equation (1) is reduced to the following Hamilton canonical equations Equation (8) becomes the basic identity for constructing the canonical transformation of the Hamilton system, i.e., The transformation (10) gives So, from Theorem 3, we have the following corollary.

Corollary 4.
If we take the old variables q s and the new variables q s ðs = 1, 2, ⋯, nÞ as 2n independent variables, the transformation determined by equation (17) is the canonical transformation of Hamilton system (15), where F 1 ðt, q s , q s Þ is called the first kind of generating function.

Theorem 5.
If the old variables a s and the new variables a s ðs = 1, 2, ⋯, nÞ are regarded as 2n independent variables, namely, the generating function is taken as F 2 ðt, a s , a s Þ, then the transformation determined by the following equations

Advances in Mathematical Physics
is the generalized canonical transformation of Birkhoff system (1), where F 2 ðt, a s , a s Þ is called the second kind of generating function.
Proof. Let We have Since a s , a s ðs = 1, 2, ⋯, nÞ are independent variables, we have Substituting formula (20) into equation (9), and considering the relations (21), we get By the independence of da s and d a s , we get the results easily. The theorem is proved. For the Hamilton system (15), equation (18) gives So, from Theorem 5, we have the following corollary.
Corollary 6. If we take the old variables q s and the new variables p s ðs = 1, 2, ⋯, nÞ as 2n independent variables, the transformation determined by equation (23) is the canonical transformation of Hamilton system (15), where F 2 ðt, q s , p s Þ is called the second kind of generating function.

Theorem 7.
If the old variables a s and the new variables a s ðs = 1, 2, ⋯, nÞ are regarded as 2n independent variables, namely, the generating function is taken as F 3 ðt, a s , a s Þ, then the transformation determined by the following equations is the generalized canonical transformation of Birkhoff system (1), where F 3 ðt, a s , a s Þ is called the third kind of generating function.
Proof. Let

Advances in Mathematical Physics
Since a s , a s ðs = 1, 2, ⋯, nÞ are independent variables, we have Substituting formula (26) into equation (9), and considering the relation (27), we get By the independence of da s and d a s , we get the results easily. The theorem is proved. For the Hamilton system (15), equation (24) give So, from Theorem 7, we have the following corollary.

Corollary 8.
If we take the old variables p s and the new variables q s ðs = 1, 2, ⋯, nÞ as 2n independent variables, the transformation determined by equation (29) is the canonical transformation of Hamilton system (15), where F 3 ðt, p s , q s Þ is called the third kind of generating function.

Theorem 9.
If the old variables a s and the new variables a s ðs = 1, 2, ⋯, nÞ are regarded as 2n independent variables, namely, the generating function is taken as F 4 ðt, a s , a s Þ, then the transformation determined by the following equations is the generalized canonical transformation of Birkhoff system (1), where F 4 ðt, a s , a s Þ is called the fourth kind of generating function.
Proof. Let We have Since a s , a s ðs = 1, 2, ⋯, nÞ are independent variables, we have Substituting formula (32) into equation (9), and considering the relation (33), we get By the independence of da s and d a s , we get the results easily. The theorem is proved.

Advances in Mathematical Physics
For the Hamilton system (15), equation (30) gives So, from Theorem 9, we have the following corollary.

Corollary 10.
If we take the old variables p s and the new variables p s ðs = 1, 2, ⋯, nÞ as 2n independent variables, the transformation determined by equation (35) is the canonical transformation of Hamilton system (15), where F 4 ðt, p s , p s Þ is called the fourth kind of generating function.

Theorem 11.
If the old variables a s and a s ðs = 1, 2, ⋯, nÞ are regarded as 2n independent variables, namely, the generating function is taken as F 5 ðt, a s , a s Þ, then the transformation determined by the following equations is the generalized canonical transformation of Birkhoff system (1), where F 5 ðt, a s , a s Þ is called the fifth kind of generating function.
Proof. Let We have Since a s , a s ðs = 1, 2, ⋯, nÞ are independent variables, we have Substituting formula (38) into equation (9), and considering the relation (39), we get By the independence of da s and da s , we get the results easily. The theorem is proved. For the Hamilton system (15), equation (36) gives So, from Theorem 11, we have the following corollary.

Corollary 12.
If we take the old variables q s and p s ðs = 1, 2, ⋯, nÞ as 2n independent variables, the transformation determined by equation (41) is the canonical transformation of Hamilton system (15), where F 5 ðt, q s , p s Þ is called the fifth kind of generating function.

Advances in Mathematical Physics
Theorem 13. If the new variables a s and a s ðs = 1, 2, ⋯, nÞ are regarded as 2n independent variables, namely, the generating function is taken as F 6 ðt, a s , a s Þ, then the transformation determined by the following equations is the generalized canonical transformation of Birkhoff system (1), where F 6 ðt, a s , a s Þ is called the sixth kind of generating function.
Proof. Let Since a s , a s ðs = 1, 2, ⋯, nÞ are independent variables, we have Substituting formula (44) into equation (9), and considering the relations (45), we get By the independence of d a s and d a s , we get the results easily. The theorem is proved. For the Hamilton system (15), equation (42) gives So, from Theorem 13, we have the following corollary.

Corollary 14.
If we take the new variables q s and p s ðs = 1, 2, ⋯, nÞ as 2n independent variables, the transformation determined by equation (47) is the canonical transformation of Hamilton system (15), where F 6 ðt, q s , p s Þ is called the sixth kind of generating function.
The generating functions F 1 ðt, q s , q s Þ, F 2 ðt, q s , p s Þ, F 3 ðt, p s , q s Þ, and F 4 ðt, p s , p s Þ for Hamilton system (15) are consistent with the classical results [20,21], while the fifth kind of generating function F 5 ðt, q s , p s Þ and the sixth kind of generating function F 6 ðt, q s , p s Þ and their corresponding canonical transformations (41) and (47) are generally not reported in the classic textbooks, for example, [20,21].
The application of the theorems given above has two aspects. One is that you can specify the explicit form of any kind of the generating functions. The corresponding generalized canonical transformation can be calculated according to the generating function by using the theorem on implicit functions. Second, if a generalized canonical transformation is specified, the corresponding generating function can be obtained by applying the above transformation formulas.
Example 1. The famous Lane-Emden equation [4,22] arising in the field of mathematical physics and astrophysics can be expressed as the following Birkhoff system: Let us study the generalized canonical transformation of the system.
According to (48), Birkhoff's equations of the system are If we take F 5 = F 5 ðt, a 1 , a 2 Þ as the generating function, by using the transformation formula (36), we have Suppose the transformed Birkhoff's functions are R 1 = a 2 , Substituting (51) into equation (50), we have If we take F 5 = −ð1/2Þða 1 Þ 2 a 2 , from the second equation of (52), we get where f ða 1 Þ represents any differentiable function of a 1 . Substituting equation (53) and F 5 = −ð1/2Þða 1 Þ 2 a 2 into the first equation of (52), we get If we take f ða 1 Þ = a 1 , then we the following generalized canonical transformation According to the third equation of (52), the new Birkhoffian is obtained as From formulas (51) and (56), we get the new Birkhoff's equation as follows: where a 1 and a 2 are the new variables. Here, the new equation (57) is simpler than the original equation (49).
Example 2. We now study a nonconservative system [4], whose Birkhoffian and Birkhoff's functions are Birkhoff's equations of the system can be written as

Advances in Mathematical Physics
According to Birkhoff's equation (59), the second-order differential equation of motion of the system is If we take F 2 ðt, a 1 , a 2 Þ as the generating function, by using the transformation formula (18), we have Suppose new Birkhoff's functions are R 1 = a 2 , Then equation (61) is reduced to If we take the transformation as follows, Substitute formula (64) into the first two equations of formula (63), we get Substituting formula (65) into the third equation of (63), we get B = 1 2 e t a 1 À Á 2 + 1 2 e −t a 2 À Á 2 : From formulas (62) and (66), we obtain the new Birkhoff's equations as follows: − _ a 2 − a 1 e t = 0, where a 1 and a 2 are the new variables. Similarly, other forms of generalized canonical transformations given in this paper can be obtained.

Conclusions
Birkhoffian mechanics is a natural generalization of Hamiltonian mechanics. It is because Birkhoff systems have many good properties, such as autonomous and semiautonomous Birkhoff systems have a Lie algebraic structure and proper symplectic form and Birkhoff's equations have self-adjoint form, that Birkhoff systems are widely used in physics, mechanics, engineering, and other fields. In addition, the generalized canonical transformation has the property of preserving algebraic and geometric structures, which lays a foundation for the Hamilton-Jacobi method, so it is an important aspect of the integral theory of analytical mechanics. In this paper, we studied the generalized canonical transformations of Birkhoff systems. Our main work consists of three aspects. The first is that we derived the criterion equation (8) or (9) of the generalized canonical transformations of Birkhoff systems. The second is that, according to the selection of 2n independent variables, we presented six different forms of generating functions, and the method we constructed generating functions is different from the existing methods, as shown in formula (11)

Data Availability
This article has no additional data.

Conflicts of Interest
There are no conflicts of interest regarding this research work.