Issue 
Wuhan Univ. J. Nat. Sci.
Volume 29, Number 2, April 2024



Page(s)  106  116  
DOI  https://doi.org/10.1051/wujns/2024292106  
Published online  14 May 2024 
Mathematics
CLC number: O316
Canonical Transformations and Poisson Theory for Dynamics with NonStandard Lagrangians
^{1}
School of Mathematical Sciences, Suzhou University of Science and Technology, Suzhou 215009, Jiangsu, China
^{2}
School of Civil Engineering, Suzhou University of Science and Technology, Suzhou 215011, Jiangsu, China
^{†} Corresponding author. Email: zhy@mail.usts.edu.cn
Received:
8
June
2023
The canonical transformation and Poisson theory of dynamical systems with exponential, powerlaw, and logarithmic nonstandard Lagrangians are studied, respectively. The criterion equations of canonical transformation are established, and four basic forms of canonical transformations are given. The dynamic equations with nonstandard Lagrangians admit Lie algebraic structure. From this, we establish the Poisson theory, which makes it possible to find new conservation laws through known conserved quantities. Some examples are put forward to demonstrate the use of the theory and verify its effectiveness.
Key words: nonstandard Lagrangians / dynamical equations / canonical transformation / Poisson theory
Cite this article: ZHU Lin, ZHANG Yi. Canonical Transformations and Poisson Theory for Dynamics with NonStandard Lagrangians[J]. Wuhan Univ J of Nat Sci, 2024, 29(2): 106116.
Biography: ZHU Lin, female, Master candidate, research direction: analytical mechanics. Email: 649405840@qq.com
Fundation item: Supported by the National Natural Science Foundation of China (12272248, 11972241)
© Wuhan University 2024
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
0 Introduction
As commonly understood, for a system of particles, the kinetic energy minus the potential energy is defined as a Lagrangian, which can be called the standard Lagrangian or the natural Lagrangian. The socalled "nonstandard Lagrangian", in which, in general, there is neither kinetic energy item nor potential energy item can be traced back to the "nonnatural Lagrangians" mentioned by Arnold in his monograph^{ [1]}. In recent years, it has been found that many nonconservative systems or nonlinear dynamical equations, such as the nonlinear Emden equation, Riccati equation, Lienard nonlinear oscillation equation, Duffingvan der Pol equation, etc., can be derived from variational principles whose action functional takes nonstandard Lagrangians as the integrand. To address dissipative and nonlinear dynamics problems, ElNabulsi^{ [2]} introduced two actions with nonstandard Lagrangians, namely exponential form, and powerlaw form, and derived their corresponding EulerLagrange equations. Saha and Talukdar^{[3,4]} researched the variational inverse problem, and constructed several important nonstandard Lagrangians for dissipative and nonlinear dynamics, such as oscillators moving in viscous media, the nonlinear Emden equation, the LotkaVolterra model, etc. Musielak^{[5]} constructed Lagrangians for variablecoefficient dissipative dynamics systems, including standard and nonstandard Lagrangians. Symmetry and conserved quantity have always been a hot spot in analytical mechanics^{ [68]}. Recently, Zhang and his collaborators have studied Noether theorems for systems with nonstandard Lagrangians ^{[9,10]}, nonstandard Hamiltonians^{[11]}, nonstandard Birkhoffians^{[12]}, and Lie symmetry ^{[13,14]}, Mei symmetry ^{[15]}, first integral and method of reduction^{ [16,17]}. There have been some results about nonlinear dynamical equations and their symmetries with nonstandard Lagrangians^{[1826]}, but canonical transformation and Poisson theory for nonstandard Lagrangian dynamics have not been involved.
Transformation is an important technique to study problems in analytical mechanics. Through variable transformation, the original complex differential equations become more accessible to solve. Canonical transformation is an essential kind of transformation, and the Hamilton equation retains its canonical form under canonical transformation, thus laying the foundation for HamiltonJacobi theory^{ [2729]}. In recent years, classical canonical transformation theory has been extended to fractional mechanical systems^{ [30]}, Birkhoff systems^{ [31,32]}, and time scale cases^{ [33]}. However, the transformation theory of the nonstandard Lagrangian system is still an open subject. The first integral of a dynamic system provides excellent convenience for solving the differential equations of motion. Poisson theory is an important tool for many scholars to study dynamic systems. In the classical mechanics, the Poisson theory encompasses the definition of Poisson parentheses, the establishment of Poisson conditions for the first integral, and the derivation of a new first integral based on a known one. Mei^{ [3436]} studied the algebraic structure and Poisson theory for holonomic and nonholonomic systems, specifically Birkhoff systems. Subsequently, research on Poisson theory has yielded a series of significant achievements^{ [3741]}. Since dynamical systems constructed from nonstandard Lagrangians are usually nonlinear and can be simplified using canonical transformations, it is possible to find more first integrals using Poisson's method. In this article, we will explore the canonical transformation and Poisson theory with exponential, powerlaw, and logarithmic nonstandard Lagrangians and use the two tools of canonical transformation and Poisson theory to solve the motion and first integrals of nonlinear system.
1 Dynamic Systems with Exponential Lagrangians
1.1 Canonical Transformations
Let the configuration of the mechanical system be determined by generalized coordinates , the action with exponential Lagrangians is:
Corresponding to action (1), the Hamilton principle is expressed as:
And the dynamic equations are^{ [6]}
Define the generalized momentum corresponding to exponential Lagrangians as:
and the exponential Hamiltonian as:
After canonicalization, the equations (3) read:
We study the transformation from variables and to new variables and , namely:
Let , then, the transformation is invertible. If under transformation (7), the form of equations (6) remains unchanged, i.e.,
where is a new function. The transformation (7) is the canonical transformation.
Since equation (6) is deduced from principle (2), the expression in equation (8) must align with that of (6). Therefore, we have:
Therefore, the integrand functions can be written as the following relation:
where an arbitrary function is called a generating function. Equation (11) can be written as:
Equation (12) is the criterion equation for the canonical transformation. Since canonical variables , and , are related by transformation relations (7), only variables are independent. We can select the generating function as a function of variables and thus obtain the following four basic types of canonical transformations.
The first type of generating function, denoted by , has the form:
Substituting formula (13) into equation (12), we obtain:
Let the coefficients of , , and be zero, respectively, and we get:
The second type of generating function is:
Substituting formula (16) into equation (12), we obtain:
So we have:
The third type of generating function is:
Then we get:
The fourth type of generating function is:
Then we get:
1.2 Poisson Theory
Let , , then equations (6) can be transformed into contravariant algebraic form:
where is a contravariant tensor.
Let be some functions. According to equation (23), we define a product
It is easy to verify that the product (24) satisfies the left distributive law, right distributive law and scalar law. Furthermore, it adheres to antisymmetry and Jacobi identity. Thus, we have:
The system (23), which is determined by exponential Lagrangians, admits not only a compatible algebraic structure but also a Lie algebraic structure.
Given that Eq. (23) possesses a Lie algebraic structure, we can establish Poisson theory as follows:
The sufficient and necessary condition for to be a first integral of the system (23) is that:
Formula (25) is referred to as the generalized Poisson condition with exponential Lagrangians.
If the exponential Hamiltonian does not depend on , then is the first integral of the system (23).
If and are two first integrals of the system (23), which are not involute, then is also a first integral.
If is a first integral of the system (23) which involves , but the exponential Hamiltonian does not depend on , then are all first integrals.
If is a first integral of the system (23), which involves the variable , but the exponential Hamiltonian does not depend on , then are all first integrals.
1.3 Examples
Try to find the canonical transformation of the system and solve its motion.
From action (26), we get . Substituting into equation (3), we get:
According to Eqs. (4) and (5), we get:
Select the generating function:
From Eq. (15), we get:
And the new Hamiltonian is:
The canonical equation with as the new variables is:
By integrating the above equation, we get:
Substituting Eq. (34) into Eq. (31), we get:
On the basis of in Example 1, we try to find canonical transformations of the other three basic forms.
Take the generating function as:
Then the transformation (18) gives:
Take
Then the transformation (20) gives:
Take
Then the transformation (22) gives:
Let
The Hamiltonian (29) can be written as
The equation exhibits a Lie algebraic structure. Utilizing the generalized Poisson condition (25), we can readily verify that:
are the first integrals of the system.
Assuming the initial conditions and , the trajectory of motion , the Hamiltonian , and the conserved quantities and can be easily calculated, as shown in Fig. 1 and Fig. 2.
Fig. 1 Simulation of the trajectory of motion in Eq. (35) 
Fig. 2 The values of the Hamiltonian in Eq. (43) and the conserved quantities in Eqs. (44)(45) 
2 Dynamic Systems with PowerLaw Lagrangians
2.1 Canonical Transformations
Let the configuration of the mechanical system be determined by generalized coordinates , the action with powerlaw Lagrangians is:
Corresponding to action (46), the Hamilton principle is expressed as:
And the dynamic equations are^{ [6]}
where .
Define the generalized momentum corresponding to powerlaw Lagrangians as:
And the powerlaw Hamiltonian as:
After canonicalization, the equations (48) read:
We study the transformation from variables and to new variables and , namely:
Let , then, the transformation is invertible. If under transformation (52), the form of equation (51) remains unchanged, i.e.,
where is a new function. The transformation (52) is the canonical transformation.
Since equation (51) is derived from principle (47), the expression in equation (53) must align with (51). Thus, we have:
Therefore, the criterion equation for the canonical transformation is:
According to the case that the generating function contains old and new variables, we get the following four basic types of canonical transformation.
The first type of generating function, denoted by , has the form
Substituting formula (57) into equation (56), we obtain
Let the coefficients of , , and be zero, respectively, we have:
The second type of generating function is:
Substituting formula (60) into equation (56), we obtain:
So we have:
The third type of generating function is:
Then we have:
The fourth type of generating function is:
Then we have:
2.2 Poisson Theory
Let ,, then equation (51) can be transformed into a contravariant algebraic form
where is a contravariant tensor.
Let be some function. According to equation (67), we define a product
The product (68) conforms to Lie algebra axioms and thus has Theorem 7.
The system (67), which is determined by powerlaw Lagrangians, admits not only a compatible algebraic structure but also a Lie algebraic structure.
Since Eq. (67) has a Lie algebraic structure, we can establish Poisson theory as follows:
The sufficient and necessary condition for to be a first integral of the system (67) is that:
Formula (69) is termed the generalized Poisson condition for powerlaw Lagrangians.
If and are two first integrals of the system (67), which are not involute, then is a first integral.
If is a first integral of the system (67) which involves , but the powerlaw Hamiltonian does not depend on , then are all first integrals.
If is a first integral of the system (67) which involves the variable , but the powerlaw Hamiltonian does not depend on the variable , then all are first integrals.
2.3 Examples
Try to find the canonical transformation of the system and solve its motion.
In this problem, ,. Let , then the dynamical equation reads:
This is an overdamped system. According to Eqs. (49) and (50), we get:
Select the generating function:
From Eq. (59), we get:
and
The canonical equation with as the new variables is:
By integrating the above equation, we get:
By substituting Eqs. (78) into Eqs. (75), it can be concluded that the damping harmonic vibration law of the overdamped system is:
On the basis of in Example 4, try to find the generating functions and canonical transformation of the other three basic forms.
Take the generating function as:
Then the transformation (62) gives:
In this case, the generating function is:
Take
Then the transformation (66) gives:
Let
The Hamiltonian (73) can be written as:
The equation has a Lie algebraic structure.
Since does not explicitly contain , the system has an integral by Theorem 9.
By using the generalized Poisson condition (69), we can easily verify that:
are the first integrals of the system.
Calculate the Poisson parentheses:
According to Theorem 10, is also a first integral.
Since involves time , while the powerlaw Hamiltonian does not explicitly involve time , according to Theorem 11, we get
Formula (91) is the first integral.
Assuming the initial conditions and , the trajectory of motion , the Hamiltonian , and the conserved quantities and can be easily calculated, as shown in Fig. 3 and Fig. 4.
Fig. 3 Simulation of the trajectory of motion in Eq. (79) 
Fig. 4 The values of the Hamiltonian in Eq. (86) and the conserved quantities in Eqs. (87)(91) 
3 Dynamic Systems with Logarithmic Lagrangians
3.1 Canonical Transformations
Let the configuration of the mechanical system be determined by generalized coordinates , the action with logarithmic Lagrangians is:
Corresponding to action (92), the Hamilton principle is expressed as:
And the dynamic equations are:
Define the generalized momentum corresponding to logarithmic Lagrangians as:
And the logarithmic Hamiltonian as:
Then equation (94) can be reduced to canonical equations as:
We study the transformation from variables and to new variables and , namely:
Let , then, the transformation is invertible. If under transformation (98), the form of equations (97) remains unchanged, i.e.,
where is a new function. The transformation (98) is the canonical transformation.
Since equation (97) is derived from principle (93), the form of equation (99) must agree with (97), so we have:
Therefore, the integrand functions should satisfy the following criterion equation:
Four basic forms of canonical transformation are given below.
The first type of generating function, denoted as , has the form:
By substituting Eq. (103) into Eq. (102), we get:
Let the coefficients of , , and be zero, respectively, we have:
The second type of generating function is:
By substituting equation (106) into equation (102), we get:
So we have:
The third type of generating function is:
Then we have:
The fourth type of generating function is:
Then we have:
3.2 Poisson Theory
Let ,, then equation (97) can be transformed into a contravariant algebraic form:
where is a contravariant tensor.
Let be some function. According to equation (113), we define a product
The product (114) conforms to Lie algebra axioms and thus has Theorem 13.
The system (113), which is determined by logarithmic Lagrangians, admits not only a compatible algebraic structure but also a Lie algebraic structure.
Since Eq. (113) has a Lie algebraic structure, we can establish Poisson theory as follows:
The sufficient and necessary condition for to be a first integral of the system (113) is that:
Formula (115) is called the generalized Poisson condition with logarithmic Lagrangians.
If the logarithmic Hamiltonian does not depend on , is a first integral of the system (113).
If and are two first integrals of the system (113), which are not in involution, then is a first integral.
If is a first integral of the system (113) which involves , but the logarithmic Hamiltonian does not depend on , then are all first integrals.
If is a first integral of the system (113) which involves the variable , but the logarithmic Hamiltonian does not depend on the variable , then are all first integrals.
3.3 Examples
In this problem, we have . The differential equation of motion reads
According to Eqs. (95) and (96), we get:
Select the generating function:
From Eq. (105), we get:
and
The canonical equation with as the new variables is:
From equation (123), we get:
Solving the differential equation (124) gives:
From Eqs. (121) and (125), we get:
It can be concluded that the damped harmonic vibration law of the critical damped system is:
where,
On the basis of in Example 7, try to find the generating function and canonical transformation of the other three basic forms.
Take the generating function as
Then, the transformation (108) gives
In this case, the generating function is:
Take
Then the transformation (112) gives:
Let
The Hamiltonian (119) can be expressed as
The equation has a Lie algebraic structure.
Since does not explicitly contain , the system has the generalized energy integral by Theorem 15.
By using the generalized Poisson condition (115), we can easily verify that
is a first integral.
Assuming the initial conditions and , the trajectory of motion , the Hamiltonian , and the conserved quantity can be easily calculated, as shown in Fig. 5 and Fig. 6.
Fig. 5 Simulation of the trajectory of motion in Eq. (127) 
Fig. 6 The values of the Hamiltonian in Eq. (135) and the conserved quantity in Eq. (136) 
4 Conclusion
In summary, we investigated the canonical transformation and Poisson theory of three kinds of dynamical equations featuring nonstandard Lagrangians. We presented the criterion equations for the canonical transformation of the systems under consideration. According to different generating function choices, we gave four basic forms of canonical transformation. We verified that dynamic equations with nonstandard Lagrangians have the Lie algebraic structure. Building upon this foundation, we formulated Poisson's theory, comprising five theorems that offer an approach for identifying new conservation laws from known conserved quantities. Several examples were presented to illustrate the application of the results and verify their effectiveness.
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All Figures
Fig. 1 Simulation of the trajectory of motion in Eq. (35)  
In the text 
Fig. 2 The values of the Hamiltonian in Eq. (43) and the conserved quantities in Eqs. (44)(45)  
In the text 
Fig. 3 Simulation of the trajectory of motion in Eq. (79)  
In the text 
Fig. 4 The values of the Hamiltonian in Eq. (86) and the conserved quantities in Eqs. (87)(91)  
In the text 
Fig. 5 Simulation of the trajectory of motion in Eq. (127)  
In the text 
Fig. 6 The values of the Hamiltonian in Eq. (135) and the conserved quantity in Eq. (136)  
In the text 
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