Open Access
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

© Wuhan University 2024

Licence Creative CommonsThis 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 so-called "non-standard Lagrangian", in which, in general, there is neither kinetic energy item nor potential energy item can be traced back to the "non-natural 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, Duffing-van der Pol equation, etc., can be derived from variational principles whose action functional takes non-standard Lagrangians as the integrand. To address dissipative and nonlinear dynamics problems, El-Nabulsi [2] introduced two actions with non-standard Lagrangians, namely exponential form, and power-law form, and derived their corresponding Euler-Lagrange equations. Saha and Talukdar[3,4] researched the variational inverse problem, and constructed several important non-standard Lagrangians for dissipative and nonlinear dynamics, such as oscillators moving in viscous media, the nonlinear Emden equation, the Lotka-Volterra model, etc. Musielak[5] constructed Lagrangians for variable-coefficient dissipative dynamics systems, including standard and non-standard Lagrangians. Symmetry and conserved quantity have always been a hot spot in analytical mechanics [6-8]. Recently, Zhang and his collaborators have studied Noether theorems for systems with non-standard Lagrangians [9,10], non-standard Hamiltonians[11], non-standard 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 non-standard Lagrangians[18-26], but canonical transformation and Poisson theory for non-standard 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 Hamilton-Jacobi theory [27-29]. 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 non-standard 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 [34-36] 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 [37-41]. Since dynamical systems constructed from non-standard 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, power-law, and logarithmic non-standard 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:

(1)

Corresponding to action (1), the Hamilton principle is expressed as:

(2)

And the dynamic equations are [6]

(3)

Define the generalized momentum corresponding to exponential Lagrangians as:

(4)

and the exponential Hamiltonian as:

(5)

After canonicalization, the equations (3) read:

(6)

We study the transformation from variables and to new variables and , namely:

(7)

Let , then, the transformation is invertible. If under transformation (7), the form of equations (6) remains unchanged, i.e.,

(8)

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:

(9)

(10)

Therefore, the integrand functions can be written as the following relation:

(11)

where an arbitrary function is called a generating function. Equation (11) can be written as:

(12)

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:

(13)

Substituting formula (13) into equation (12), we obtain:

(14)

Let the coefficients of , , and be zero, respectively, and we get:

(15)

The second type of generating function is:

(16)

Substituting formula (16) into equation (12), we obtain:

(17)

So we have:

(18)

The third type of generating function is:

(19)

Then we get:

(20)

The fourth type of generating function is:

(21)

Then we get:

(22)

1.2 Poisson Theory

Let , , then equations (6) can be transformed into contravariant algebraic form:

(23)

where is a contravariant tensor.

Let be some functions. According to equation (23), we define a product

(24)

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:

Theorem 1

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:

Theorem 2

The sufficient and necessary condition for to be a first integral of the system (23) is that:

(25)

Formula (25) is referred to as the generalized Poisson condition with exponential Lagrangians.

Theorem 3

If the exponential Hamiltonian does not depend on , then is the first integral of the system (23).

Theorem 4

If and are two first integrals of the system (23), which are not involute, then is also a first integral.

Theorem 5

If is a first integral of the system (23) which involves , but the exponential Hamiltonian does not depend on , then are all first integrals.

Theorem 6

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

Example 1

Consider a nonlinear dynamical system with exponential Lagrangians. The action is [2]

(26)

Try to find the canonical transformation of the system and solve its motion.

From action (26), we get . Substituting into equation (3), we get:

(27)

According to Eqs. (4) and (5), we get:

(28)

(29)

Select the generating function:

(30)

From Eq. (15), we get:

(31)

And the new Hamiltonian is:

(32)

The canonical equation with as the new variables is:

(33)

By integrating the above equation, we get:

(34)

Substituting Eq. (34) into Eq. (31), we get:

(35)

Example 2

On the basis of in Example 1, we try to find canonical transformations of the other three basic forms.

Take the generating function as:

(36)

Then the transformation (18) gives:

(37)

Take

(38)

Then the transformation (20) gives:

(39)

Take

(40)

Then the transformation (22) gives:

(41)

Example 3

Try to study the first integral of the system in Example 1.

Let

(42)

The Hamiltonian (29) can be written as

(43)

The equation exhibits a Lie algebraic structure. Utilizing the generalized Poisson condition (25), we can readily verify that:

(44)

(45)

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.

thumbnail Fig. 1 Simulation of the trajectory of motion in Eq. (35)

thumbnail Fig. 2 The values of the Hamiltonian in Eq. (43) and the conserved quantities in Eqs. (44)-(45)

2 Dynamic Systems with Power-Law Lagrangians

2.1 Canonical Transformations

Let the configuration of the mechanical system be determined by generalized coordinates , the action with power-law Lagrangians is:

(46)

Corresponding to action (46), the Hamilton principle is expressed as:

(47)

And the dynamic equations are [6]

(48)

where .

Define the generalized momentum corresponding to power-law Lagrangians as:

(49)

And the power-law Hamiltonian as:

(50)

After canonicalization, the equations (48) read:

(51)

We study the transformation from variables and to new variables and , namely:

(52)

Let , then, the transformation is invertible. If under transformation (52), the form of equation (51) remains unchanged, i.e.,

(53)

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:

(54)

(55)

Therefore, the criterion equation for the canonical transformation is:

(56)

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

(57)

Substituting formula (57) into equation (56), we obtain

(58)

Let the coefficients of , , and be zero, respectively, we have:

(59)

The second type of generating function is:

(60)

Substituting formula (60) into equation (56), we obtain:

(61)

So we have:

(62)

The third type of generating function is:

(63)

Then we have:

(64)

The fourth type of generating function is:

(65)

Then we have:

(66)

2.2 Poisson Theory

Let ,, then equation (51) can be transformed into a contravariant algebraic form

(67)

where is a contravariant tensor.

Let be some function. According to equation (67), we define a product

(68)

The product (68) conforms to Lie algebra axioms and thus has Theorem 7.

Theorem 7

The system (67), which is determined by power-law 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:

Theorem 8

The sufficient and necessary condition for to be a first integral of the system (67) is that:

(69)

Formula (69) is termed the generalized Poisson condition for power-law Lagrangians.

Theorem 9

If the power-law Hamiltonian does not depend on , is a first integral of the system (67).

Theorem 10

If and are two first integrals of the system (67), which are not involute, then is a first integral.

Theorem 11

If is a first integral of the system (67) which involves , but the power-law Hamiltonian does not depend on , then are all first integrals.

Theorem 12

If is a first integral of the system (67) which involves the variable , but the power-law Hamiltonian does not depend on the variable , then all are first integrals.

2.3 Examples

Example 4

The action with power-law Lagrangians is [3]

(70)

Try to find the canonical transformation of the system and solve its motion.

In this problem, ,. Let , then the dynamical equation reads:

(71)

This is an over-damped system. According to Eqs. (49) and (50), we get:

(72)

(73)

Select the generating function:

(74)

From Eq. (59), we get:

(75)

and

(76)

The canonical equation with as the new variables is:

(77)

By integrating the above equation, we get:

(78)

By substituting Eqs. (78) into Eqs. (75), it can be concluded that the damping harmonic vibration law of the overdamped system is:

(79)

Example 5

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:

(80)

Then the transformation (62) gives:

(81)

In this case, the generating function is:

(82)

Take

(83)

Then the transformation (66) gives:

(84)

Example 6

Try to study the first integral of the system in Example 1.

Let

(85)

The Hamiltonian (73) can be written as:

(86)

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:

(87)

(88)

(89)

are the first integrals of the system.

Calculate the Poisson parentheses:

(90)

According to Theorem 10, is also a first integral.

Since involves time , while the power-law Hamiltonian does not explicitly involve time , according to Theorem 11, we get

(91)

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.

thumbnail Fig. 3 Simulation of the trajectory of motion in Eq. (79)

thumbnail 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:

(92)

Corresponding to action (92), the Hamilton principle is expressed as:

(93)

And the dynamic equations are:

(94)

Define the generalized momentum corresponding to logarithmic Lagrangians as:

(95)

And the logarithmic Hamiltonian as:

(96)

Then equation (94) can be reduced to canonical equations as:

(97)

We study the transformation from variables and to new variables and , namely:

(98)

Let , then, the transformation is invertible. If under transformation (98), the form of equations (97) remains unchanged, i.e.,

(99)

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:

(100)

(101)

Therefore, the integrand functions should satisfy the following criterion equation:

(102)

Four basic forms of canonical transformation are given below.

The first type of generating function, denoted as , has the form:

(103)

By substituting Eq. (103) into Eq. (102), we get:

(104)

Let the coefficients of , , and be zero, respectively, we have:

(105)

The second type of generating function is:

(106)

By substituting equation (106) into equation (102), we get:

(107)

So we have:

(108)

The third type of generating function is:

(109)

Then we have:

(110)

The fourth type of generating function is:

(111)

Then we have:

(112)

3.2 Poisson Theory

Let ,, then equation (97) can be transformed into a contravariant algebraic form:

(113)

where is a contravariant tensor.

Let be some function. According to equation (113), we define a product

(114)

The product (114) conforms to Lie algebra axioms and thus has Theorem 13.

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:

Theorem 14

The sufficient and necessary condition for to be a first integral of the system (113) is that:

(115)

Formula (115) is called the generalized Poisson condition with logarithmic Lagrangians.

Theorem 15

If the logarithmic Hamiltonian does not depend on , is a first integral of the system (113).

Theorem 16

If and are two first integrals of the system (113), which are not in involution, then is a first integral.

Theorem 17

If is a first integral of the system (113) which involves , but the logarithmic Hamiltonian does not depend on , then are all first integrals.

Theorem 18

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

Example 7

The action with logarithmic Lagrangians is [4]

(116)

In this problem, we have . The differential equation of motion reads

(117)

According to Eqs. (95) and (96), we get:

(118)

(119)

Select the generating function:

(120)

From Eq. (105), we get:

(121)

and

(122)

The canonical equation with as the new variables is:

(123)

From equation (123), we get:

(124)

Solving the differential equation (124) gives:

(125)

From Eqs. (121) and (125), we get:

(126)

It can be concluded that the damped harmonic vibration law of the critical damped system is:

(127)

where,

(128)

Example 8

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

(129)

Then, the transformation (108) gives

(130)

In this case, the generating function is:

(131)

Take

(132)

Then the transformation (112) gives:

(133)

Example 9

Try to study the first integral in Example 7.

Let

(134)

The Hamiltonian (119) can be expressed as

(135)

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

(136)

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.

thumbnail Fig. 5 Simulation of the trajectory of motion in Eq. (127)

thumbnail 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 non-standard 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 non-standard 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.

References

  1. Arnold V I. Mathematical Methods of Classical Mechanics [M]. New York: Springer-Verlag, 1978. [Google Scholar]
  2. El-Nabulsi R A. Nonlinear dynamics with non-standard Lagrangians [J]. Qualitative Theory of Dynamical Systems, 2013, 12(2): 273-291. [Google Scholar]
  3. Saha A, Talukdar B. Inverse variational problem for nonstandard Lagrangians [J]. Reports on Mathematical Physics, 2014, 73(3): 299-309. [Google Scholar]
  4. Saha A, Talukdar B. On the non-standard Lagrangian equations [EB/OL]. [2013-01-12]. https://arxiv.org/ftp/arxiv/papers/1301/1301.2667.pdf. [Google Scholar]
  5. Muslelak Z E. Standard and non-standard Lagrangians for dissipative dynamical systems with variable coefficients [J]. Journal of Physics A: Mathematical and Theoretical, 2008, 41(5): 055205. [Google Scholar]
  6. Zhang Y, Cai J X. Noether theorem of Herglotz type for nonconservative Hamilton systems in event space [J]. Wuhan University Journal of Natural Sciences, 2021, 26(5): 376-382. [Google Scholar]
  7. Chen J Y, Zhang Y. Lie symmetry theorem for non-shifted Birkhoffian systems on time scales [J]. Wuhan University Journal of Natural Sciences, 2022, 27(3): 211-217. [Google Scholar]
  8. Zhang Y. A study on time scale non-shifted Hamiltonian dynamics and Noether's theorems [J]. Wuhan University Journal of Natural Sciences, 2023, 28(2):106-116. [Google Scholar]
  9. Zhang Y, Zhou X S. Noether theorem and its inverse for nonlinear dynamical systems with nonstandard Lagrangians [J]. Nonlinear Dynamics, 2016, 84(4): 1867-1876. [Google Scholar]
  10. Song J, Zhang Y. Noether symmetry and conserved quantity for dynamical system with non-standard Lagrangians on time scales [J]. Chinese Physics B, 2017, 26(8): 084501. [Google Scholar]
  11. Song J, Zhang Y. Noether's theorems for dynamical systems of two kinds of non-standard Hamiltonians [J]. Acta Mechanica, 2018, 229(1): 285-297. [Google Scholar]
  12. Zhang L J, Zhang Y. Non-standard Birkhoffian dynamics and its Noether's theorems [J]. Communications in Nonlinear Science and Numerical Simulation, 2020, 91: 105435. [Google Scholar]
  13. Zhang Y, Wang X P. Lie symmetry perturbation and adiabatic invariants for dynamical system with non-standard Lagrangians [J]. International Journal of Non-Linear Mechanics, 2018, 105: 165-172. [Google Scholar]
  14. Jia Y D, Zhang Y. Fractional Birkhoffian dynamics based on quasi⁃fractional dynamics models and its Lie symmetry [J]. Transactions of Nanjing University of Aeronautics and Astronautics, 2021, 38(1): 84-95. [Google Scholar]
  15. Zhang Y, Wang X P. Mei symmetry and invariants of quasi-fractional dynamical systems with non-standard Lagrangians [J]. Symmetry, 2019, 11 (8):1061. [Google Scholar]
  16. Zhou X S, Zhang Y. Generalized energy integral and Whittaker method of reduction for dynamics systems with Non-standard Lagrangians [J]. Transactions of Nanjing University of Aeronautics and Astronautics, 2017, 49(2): 269-275(Ch). [Google Scholar]
  17. Zhou X S, Zhang Y. Routh method of reduction for dynamic systems with non-standard Lagrangians [J]. Chinese Quarterly of Mechanics, 2016, 37(1): 5-21(Ch). [Google Scholar]
  18. Cieśliński J I, Nikiciuk T A. Direct approach to the construction of standard and non-standard Lagrangians for dissipative-like dynamical systems with variable coefficients [J]. Journal of Physics A: Mathematical and Theoretical, 2010, 43(17): 175205. [Google Scholar]
  19. Bagchi B, Ghosh D, Modak S, et al. Nonstandard Lagrangians and branching: The case of some nonlinear Liénard systems [J]. Modern Physics Letters A, 2019, 34(14): 1950110. [Google Scholar]
  20. Liu S X, Guan F, Wang Y. The nonlinear dynamics based on the non-standard Hamiltonians [J]. Nonlinear Dynamics, 2017, 88(2):1229-1236. [Google Scholar]
  21. Chandrasekar V K, Senthilvelan M, Lakshmanan M. Unusual Liénard-type nonlinear oscillator [J]. Physics Review E, 2005, 72(6): 066203. [Google Scholar]
  22. Chandrasekar V K, Senthilvelan M, Lakshmanan M. On the Lagrangian and Hamiltonian description of the damped linear harmonic oscillator [J]. Journal of Mathematical Physics, 2007, 48(3): 032701. [Google Scholar]
  23. Muslelak Z E, Roy D, Swift L D. Method to drive Lagrangian and Hamiltonian for a nonlinear dynamical system with variable coefficients [J]. Chaos, Solitons & Fractals, 2008, 38(3): 894-902. [Google Scholar]
  24. EI-Nabulsi R A. Non-standard fractional Lagrangians [J]. Nonlinear Dynamics, 2013, 74(1): 381-394. [Google Scholar]
  25. EI-Nabulsi R A. Non-standard power-law Lagrangians in classical and quantum dynamics [J]. Applied Mathematics Letters, 2015, 43: 120-127. [Google Scholar]
  26. EI-Nabulsi R A. Gravitational field as a pressure force from logarithmic Lagrangians and non-standard Hamiltonians: The case of stellar halo of Milky Way [J]. Communications in Theoretical Physics, 2018, 69(3): 233-240. [Google Scholar]
  27. Chen B. Analytical Dynamics[M]. 2nd Ed. Beijing: Peking University Press, 2012(Ch). [Google Scholar]
  28. Mei F X, Wu H B, Li Y M. A Brief History of Analytical Mechanics [M]. Beijing: Science Press, 2019(Ch). [Google Scholar]
  29. Mei F X. Canonical transformation for weak nonholonomic systems [J]. Chinese Science Bulletin, 1993, 38(4): 281-285. [Google Scholar]
  30. Zhang Y. Theory of canonical transformation for a fractional mechanical system [J]. Acta Mathematicae Applicatae Sinica, 2016, 39(2): 249-260(Ch). [Google Scholar]
  31. Zhang Y. The generalized canonical transformations of Birkhoffian systems and their basic formulations [J]. Chinese Quarterly of Mechanics, 2019, 40(4): 656-665(Ch). [Google Scholar]
  32. Zhang Y. Theory of generalized canonical transformations for Birkhoff systems [J]. Advances in Mathematical Physics, 2020, 2020: 9482356. [Google Scholar]
  33. Zhang Y, Zhai X H. Generalized canonical transformation for second order Birkhoffian systems on time scales[J]. Theoretical and Applied Mechanics Letters, 2019, 9(6): 353-357. [Google Scholar]
  34. Mei F X. Applications of Lie Groups and Lie Algebras to Constrained Mechanical Systems [M]. Beijing: Science Press, 1999 (Ch). [Google Scholar]
  35. Zhang Y, Mei F X. Algebraic structure of the dynamical equations of holonomic mechanical system in relative motion [J]. Journal of Beijing Institute of Technology, 1998, 7(1): 12-18. [Google Scholar]
  36. Mei F X, Shi R C, Zhang Y F. Poisson's theory of the Chaplygin's equations [J]. Journal of Beijing Institute of Technology, 1995, 4(2): 123-129. [Google Scholar]
  37. Zhang Y, Shang M. Poisson theory and integration method for a dynamical system of relative motion [J]. Chinese Physics B, 2011, 20(2): 024501. [Google Scholar]
  38. Fu J L, Chen X W, Luo S K. Algebraic structures and Poisson integrals of relativistic dynamical equations for rotational systems [J]. Applied Mathematics and Mechanics, 1999, 10(11): 1266-1274. [Google Scholar]
  39. Mei F X. Poission's theory of Birkhoffian system [J]. Chinese Science Bulletin, 1996, 41(8): 641. [Google Scholar]
  40. Luo S K, Chen X W, Guo Y X. Algebraic structure and Poisson integrals of rotational relativistic Birkhoff system [J]. Chinese Physics, 2002, 11(6): 523-528. [Google Scholar]
  41. Zhang Y. Poisson theory and integration method of Birkhoffian systems in the event space [J]. Chinese Physics B, 2010, 19(8): 080301. [Google Scholar]

All Figures

thumbnail Fig. 1 Simulation of the trajectory of motion in Eq. (35)
In the text
thumbnail Fig. 2 The values of the Hamiltonian in Eq. (43) and the conserved quantities in Eqs. (44)-(45)
In the text
thumbnail Fig. 3 Simulation of the trajectory of motion in Eq. (79)
In the text
thumbnail Fig. 4 The values of the Hamiltonian in Eq. (86) and the conserved quantities in Eqs. (87)-(91)
In the text
thumbnail Fig. 5 Simulation of the trajectory of motion in Eq. (127)
In the text
thumbnail Fig. 6 The values of the Hamiltonian in Eq. (135) and the conserved quantity in Eq. (136)
In the text

Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.

Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.

Initial download of the metrics may take a while.