© Wuhan University 2024

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 $N$ generalized coordinates $q_k\left(k=\mathrm{1,2},\cdots ,N\right)$, the action with exponential Lagrangians is:

$S_{\mathrm{E}}={\int }_{t_{\mathrm{1}}}^{t_{\mathrm{2}}}\mathrm{e}\mathrm{x}\mathrm{p}\left[L\left(t,q_k,{\dot{q}}_k\right)\right]\mathrm{d}t$(1)

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

$\delta S_{\mathrm{E}}=\mathrm{0}$(2)

And the dynamic equations are ^[6]

$\mathrm{e}\mathrm{x}\mathrm{p}\left(L\right)\left(\frac{\partial L}{\partial q_k}-\frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial L}{\partial {\dot{q}}_k}-\frac{\partial L}{\partial {\dot{q}}_k}\frac{\mathrm{d}L}{\mathrm{d}t}\right)=\mathrm{0}\left(k=\mathrm{1,2},\cdots ,N\right)$(3)

Define the generalized momentum corresponding to exponential Lagrangians as:

$p_k=\frac{\partial \mathrm{e}\mathrm{x}\mathrm{p}L}{\partial {\dot{q}}_k}=\mathrm{e}\mathrm{x}\mathrm{p}L\frac{\partial L}{\partial {\dot{q}}_k}$(4)

and the exponential Hamiltonian as:

$\mathrm{e}\mathrm{x}\mathrm{p}\left[H\left(t,q_k,p_k\right)\right]=p_k{\dot{q}}_k-\mathrm{e}\mathrm{x}\mathrm{p}\left[L\left(t,q_k,{\dot{q}}_k\right)\right]$(5)

After canonicalization, the equations (3) read:

${\dot{q}}_k=\mathrm{e}\mathrm{x}\mathrm{p}\left(H\right)\frac{\partial H}{\partial p_k},{\dot{p}}_k=-\mathrm{e}\mathrm{x}\mathrm{p}\left(H\right)\frac{\partial H}{\partial q_k}$(6)

We study the transformation from variables $q_k$ and $p_k$ to new variables $Q_k$ and $P_k$ , namely:

$Q_k=Q_k\left(t,q_s,p_s\right),P_k=P_k\left(t,q_s,p_s\right)$(7)

Let $\mathrm{\Delta }=\frac{\partial \left(Q_k,P_k\right)}{\partial \left(q_k,p_k\right)}\ne \mathrm{0}$, then, the transformation is invertible. If under transformation (7), the form of equations (6) remains unchanged, i.e.,

${\dot{Q}}_k=\mathrm{e}\mathrm{x}\mathrm{p}\left(K\right)\frac{\partial K}{\partial P_k},{\dot{P}}_k=-\mathrm{e}\mathrm{x}\mathrm{p}\left(K\right)\frac{\partial K}{\partial Q_k}$(8)

where $K=K\left(t,Q_k,P_k\right)$ 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:

$\delta {\int }_{t_{\mathrm{1}}}^{t_{\mathrm{2}}}\left[p_k{\dot{q}}_k-\mathrm{e}\mathrm{x}\mathrm{p}\left(H\right)\right]\mathrm{d}t=\mathrm{0}$(9)

$\delta {\int }_{t_{\mathrm{1}}}^{t_{\mathrm{2}}}\left[P_k{\dot{Q}}_k-\mathrm{e}\mathrm{x}\mathrm{p}\left(K\right)\right]\mathrm{d}t=\mathrm{0}$(10)

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

$\left[p_k{\dot{q}}_k-\mathrm{e}\mathrm{x}\mathrm{p}\left(H\right)\right]-\left[P_k{\dot{Q}}_k-\mathrm{e}\mathrm{x}\mathrm{p}\left(K\right)\right]=\frac{\mathrm{d}F}{\mathrm{d}t}$(11)

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

$p_k\mathrm{d}{q}_k-P_k\mathrm{d}{Q}_k+\left[\mathrm{e}\mathrm{x}\mathrm{p}\left(K\right)-\mathrm{e}\mathrm{x}\mathrm{p}\left(H\right)\right]\mathrm{d}t=\mathrm{d}F$(12)

Equation (12) is the criterion equation for the canonical transformation. Since $\mathrm{4}N$ canonical variables $q_k$, $p_k$ and $Q_k$, $P_k$ are related by $\mathrm{2}N$ transformation relations (7), only $\mathrm{2}N$ variables are independent. We can select the generating function $F$ as a function of $\mathrm{2}N$ variables and thus obtain the following four basic types of canonical transformations.

The first type of generating function, denoted by $F_{\mathrm{1}}$, has the form:

$F=F_{\mathrm{1}}\left(t,q_k,Q_k\right)$(13)

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

$\begin{array}{r}\left(p_k-\frac{\partial F_{\mathrm{1}}}{\partial q_k}\right)\mathrm{d}{q}_k+\left(-P_k-\frac{\partial F_{\mathrm{1}}}{\partial Q_k}\right)\mathrm{d}{Q}_k\\ +\left(\mathrm{e}\mathrm{x}\mathrm{p}\left(K\right)-\mathrm{e}\mathrm{x}\mathrm{p}\left(H\right)-\frac{\partial F_{\mathrm{1}}}{\partial t}\right)\mathrm{d}t=\mathrm{0}\end{array}$(14)

Let the coefficients of $\mathrm{d}{q}_k$, $\mathrm{d}{Q}_k$, and $\mathrm{d}t$ be zero, respectively, and we get:

$p_k=\frac{\partial F_{\mathrm{1}}}{\partial q_k},P_k=-\frac{\partial F_{\mathrm{1}}}{\partial Q_k},\mathrm{e}\mathrm{x}\mathrm{p}\left(K\right)=\mathrm{e}\mathrm{x}\mathrm{p}\left(H\right)+\frac{\partial F_{\mathrm{1}}}{\partial t}$(15)

The second type of generating function $F_{\mathrm{2}}$ is:

$F_{\mathrm{2}}\left(t,q_k,P_k\right)=F_{\mathrm{1}}\left(t,q_k,Q_k\right)+Q_k{P}_k$(16)

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

$\begin{array}{r}\left(p_k-\frac{\partial F_{\mathrm{2}}}{\partial q_k}\right)\mathrm{d}{q}_k+\left(Q_k-\frac{\partial F_{\mathrm{2}}}{\partial P_k}\right)\mathrm{d}{P}_k\\ +\left(\mathrm{e}\mathrm{x}\mathrm{p}\left(K\right)-\mathrm{e}\mathrm{x}\mathrm{p}\left(H\right)-\frac{\partial F_{\mathrm{2}}}{\partial t}\right)\mathrm{d}t=\mathrm{0}\end{array}$(17)

So we have:

$p_k=\frac{\partial F_{\mathrm{2}}}{\partial q_k},Q_k=\frac{\partial F_{\mathrm{2}}}{\partial P_k},\mathrm{e}\mathrm{x}\mathrm{p}\left(K\right)=\mathrm{e}\mathrm{x}\mathrm{p}\left(H\right)+\frac{\partial F_{\mathrm{2}}}{\partial t}$(18)

The third type of generating function $F_{\mathrm{3}}$ is:

$F_{\mathrm{3}}\left(t,p_k,Q_k\right)=F_{\mathrm{1}}\left(t,q_k,Q_k\right)-q_k{p}_k$(19)

Then we get:

$q_k=-\frac{\partial F_{\mathrm{3}}}{\partial p_k},P_k=-\frac{\partial F_{\mathrm{3}}}{\partial Q_k},\mathrm{e}\mathrm{x}\mathrm{p}\left(K\right)=\mathrm{e}\mathrm{x}\mathrm{p}\left(H\right)+\frac{\partial F_{\mathrm{3}}}{\partial t}$(20)

The fourth type of generating function $F_{\mathrm{4}}$ is:

$F_{\mathrm{4}}\left(t,p_k,P_k\right)=F_{\mathrm{1}}\left(t,q_k,Q_k\right)-q_k{p}_k+Q_k{P}_k$(21)

Then we get:

$q_k=-\frac{\partial F_{\mathrm{4}}}{\partial p_k},Q_k=\frac{\partial F_{\mathrm{4}}}{\partial P_k},\mathrm{e}\mathrm{x}\mathrm{p}\left(K\right)=\mathrm{e}\mathrm{x}\mathrm{p}\left(H\right)+\frac{\partial F_{\mathrm{4}}}{\partial t}$(22)

1.2 Poisson Theory

Let $a^k=q_k$, $a^{n+k}=p_k$$\left(k=\mathrm{1,2},\cdots ,N\right)$, then equations (6) can be transformed into contravariant algebraic form:

${\dot{a}}^{\mu }=\mathrm{e}\mathrm{x}\mathrm{p}\left(H\right){\omega }^{\mu \nu }\frac{\partial H}{\partial a^{\nu }}\left(\mu ,\nu =\mathrm{1,2},\cdots ,\mathrm{2}n\right)$(23)

where $\left({\omega }^{\mu \nu }\right)=\left[\begin{array}{cc}{\mathrm{0}}_{n\times n}& I_{n\times n}\\ -I_{n\times n}& {\mathrm{0}}_{n\times n}\end{array}\right]$ is a contravariant tensor.

Let $A\left(a\right)$ be some functions. According to equation (23), we define a product

$\dot{A}\left(a\right)=\frac{\partial A}{\partial a^{\mu }}{\omega }^{\mu \nu }\frac{\partial \mathrm{e}\mathrm{x}\mathrm{p}\left(H\right)}{\partial a^{\nu }}=A\circ \mathrm{e}\mathrm{x}\mathrm{p}\left(H\right)$(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:

Given that Eq. (23) possesses a Lie algebraic structure, we can establish Poisson theory as follows:

$\frac{\partial I}{\partial t}+I\circ \mathrm{e}\mathrm{x}\mathrm{p}\left(H\right)=\mathrm{0}$(25)

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

1.3 Examples

$S_{\mathrm{E}}={\int }_{t_{\mathrm{1}}}^{t_{\mathrm{2}}}\mathrm{e}\mathrm{x}\mathrm{p}\left(tq\dot{q}\right)\mathrm{d}t$(26)

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

From action (26), we get $L=tq\dot{q}$. Substituting $L=tq\dot{q}$ into equation (3), we get:

$tq\ddot{q}+q\dot{q}+t{\dot{q}}^{\mathrm{2}}+\frac{\mathrm{1}}{t}=\mathrm{0}$(27)

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

$p=tq\mathrm{e}\mathrm{x}\mathrm{p}\left(tq\dot{q}\right)$(28)

$H=\frac{p}{tq}\left(\mathrm{l}\mathrm{n}\frac{p}{q}-\mathrm{l}\mathrm{n}t-\mathrm{1}\right)$(29)

Select the generating function:

$F=F_{\mathrm{1}}\left(t,q,Q\right)=\frac{\mathrm{1}}{\mathrm{2}}{q}^{\mathrm{2}}{\mathrm{e}}^Q$(30)

From Eq. (15), we get:

$p=\frac{\partial F_{{}_{\mathrm{1}}}}{\partial q}=q{\mathrm{e}}^Q,P=-\frac{\partial F_{{}_{\mathrm{1}}}}{\partial Q}=-\frac{\mathrm{1}}{\mathrm{2}}{q}^{\mathrm{2}}{\mathrm{e}}^Q$(31)

And the new Hamiltonian $K$ is:

$K=\frac{\mathrm{1}}{t}{\mathrm{e}}^Q\left(Q-\mathrm{l}\mathrm{n}t-\mathrm{1}\right)$(32)

The canonical equation with $Q,P$ as the new variables is:

$\dot{Q}=\frac{\partial K}{\partial P}=\mathrm{0},\dot{P}=-\frac{\partial K}{\partial Q}=\frac{\mathrm{1}}{t}{\mathrm{e}}^Q\left(\mathrm{l}\mathrm{n}t-Q\right)$(33)

By integrating the above equation, we get:

$Q=c_{\mathrm{1}},P=\frac{\mathrm{1}}{\mathrm{2}}\mathrm{l}{\mathrm{n}}^{\mathrm{2}}t-c_{\mathrm{1}}\mathrm{l}\mathrm{n}t+c_{\mathrm{2}}$(34)

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

$q=\pm \sqrt[]{\mathrm{2}{c}_{\mathrm{1}}\mathrm{l}\mathrm{n}t-\mathrm{l}{\mathrm{n}}^{\mathrm{2}}t+\mathrm{2}{c}_{\mathrm{3}}}$(35)

Take the generating function as:

$F_{\mathrm{2}}\left(t,q,P\right)=F_{\mathrm{1}}+QP=-P+P\mathrm{l}\mathrm{n}\frac{-\mathrm{2}P}{q^{\mathrm{2}}}$(36)

Then the transformation (18) gives:

$p=\frac{\partial F_{\mathrm{2}}}{\partial q}=-\frac{\mathrm{2}P}{q},Q=\frac{\partial F_{\mathrm{2}}}{\partial P}=\mathrm{l}\mathrm{n}\frac{-\mathrm{2}P}{q^{\mathrm{2}}}$(37)

Take

$F_{\mathrm{3}}\left(t,Q,p\right)=F_{\mathrm{1}}-qp=-\frac{p^{\mathrm{2}}}{\mathrm{2}{\mathrm{e}}^Q}$(38)

Then the transformation (20) gives:

$q=-\frac{\partial F_{\mathrm{3}}}{\partial p}=\frac{p}{{\mathrm{e}}^Q},P=-\frac{\partial F_{\mathrm{3}}}{\partial Q}=-\frac{p^{\mathrm{2}}}{\mathrm{2}{\mathrm{e}}^Q}$(39)

Take

$F_{\mathrm{4}}\left(t,P,p\right)=F_{\mathrm{1}}-qp+QP=P+P\mathrm{l}\mathrm{n}\frac{p^{\mathrm{2}}}{-\mathrm{2}P}$(40)

Then the transformation (22) gives:

$q=-\frac{\partial F_{\mathrm{4}}}{\partial p}=\frac{p}{{\mathrm{e}}^Q},Q=\frac{\partial F_{\mathrm{4}}}{\partial P}=\mathrm{l}\mathrm{n}\frac{p^{\mathrm{2}}}{-\mathrm{2}P}$(41)

Let

$q=a^{\mathrm{1}},p=a^{\mathrm{2}}$(42)

The Hamiltonian (29) can be written as

$H=\frac{a^{\mathrm{2}}}{t{a}^{\mathrm{1}}}\left(\mathrm{l}\mathrm{n}\frac{a^{\mathrm{2}}}{a^{\mathrm{1}}}-\mathrm{l}\mathrm{n}t-\mathrm{1}\right)$(43)

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

$I_{\mathrm{1}}=\frac{a^{\mathrm{2}}}{a^{\mathrm{1}}}+\frac{a^{\mathrm{2}}}{a^{\mathrm{1}}}\mathrm{l}\mathrm{n}\frac{a^{\mathrm{2}}}{a^{\mathrm{1}}}=c_{\mathrm{1}}$(44)

$I_{\mathrm{2}}={\mathrm{e}}^{\frac{a^{\mathrm{2}}}{a^{\mathrm{1}}}}=c_{\mathrm{2}}$(45)

are the first integrals of the system.

Assuming the initial conditions $q=\mathrm{1}$ and $\dot{q}=\mathrm{1}$, the trajectory of motion $q$, the Hamiltonian $H$, and the conserved quantities $I_{\mathrm{1}}$ and $I_{\mathrm{2}}$ can be easily calculated, as shown in Fig. 1 and Fig. 2.

Fig. 1 Simulation of the trajectory of motion $\mathit{q}\left(\mathit{t}\right)$ 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 Power-Law Lagrangians

2.1 Canonical Transformations

Let the configuration of the mechanical system be determined by $N$ generalized coordinates $q_s\left(s=\mathrm{1,2},\cdots ,N\right)$, the action with power-law Lagrangians is:

$S_{\mathrm{P}}={\int }_{t_{\mathrm{1}}}^{t_{\mathrm{2}}}{L}^{\mathrm{1}+\gamma }\left(t,q_s,{\dot{q}}_s\right)\mathrm{d}t$(46)

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

$\delta S_{\mathrm{P}}=\mathrm{0}$(47)

And the dynamic equations are ^[6]

$\left(\mathrm{1}+\gamma \right)L^{\gamma }\left[\frac{\partial L}{\partial q_s}-\frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial L}{\partial {\dot{q}}_s}-\frac{\gamma }{L}\frac{\partial L}{\partial {\dot{q}}_s}\frac{\mathrm{d}L}{\mathrm{d}t}\right]=\mathrm{0}\left(s=\mathrm{1,2},\cdots ,N\right)$(48)

where $\gamma \ne -\mathrm{1}$.

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

$p_s=\frac{\partial L^{\mathrm{1}+\gamma }}{\partial {\dot{q}}_s}=\left(\mathrm{1}+\gamma \right)L^{\gamma }\frac{\partial L}{\partial {\dot{q}}_s}$(49)

And the power-law Hamiltonian as:

$H^{\mathrm{1}+\gamma }\left(t,q_s,p_s\right)=p_s{\dot{q}}_s-L^{{}^{\mathrm{1}+\gamma }}\left(t,q_s,{\dot{q}}_s\right)$(50)

After canonicalization, the equations (48) read:

${\dot{q}}_s=\left(\mathrm{1}+\gamma \right)H^{\gamma }\frac{\partial H}{\partial p_s},{\dot{p}}_s=-\left(\mathrm{1}+\gamma \right)H^{\gamma }\frac{\partial H}{\partial q_s}$(51)

We study the transformation from variables $q_k$ and $p_k$ to new variables $Q_k$ and $P_k$, namely:

$Q_s=Q_s\left(t,q_k,{\dot{q}}_k\right), P_s=P_s\left(t,q_k,{\dot{q}}_k\right)$(52)

Let $\mathrm{\Delta }=\frac{\partial \left(Q_s,P_s\right)}{\partial \left(q_s,p_s\right)}\ne \mathrm{0}$, then, the transformation is invertible. If under transformation (52), the form of equation (51) remains unchanged, i.e.,

${\dot{Q}}_s=\left(\mathrm{1}+\gamma \right)K^{\gamma }\frac{\partial K}{\partial P_s}, {\dot{P}}_s=-\left(\mathrm{1}+\gamma \right)K^{\gamma }\frac{\partial K}{\partial Q_s}$(53)

where $K=K\left(t,Q_s,P_s\right)$ 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:

$\delta {\int }_{t_{\mathrm{1}}}^{t_{\mathrm{2}}}\left(p_s{\dot{q}}_s-H^{\mathrm{1}+\gamma }\right)\mathrm{d}t=\mathrm{0}$(54)

$\delta {\int }_{t_{\mathrm{1}}}^{t_{\mathrm{2}}}\left(P_s{\dot{Q}}_s-K^{\mathrm{1}+\gamma }\right)\mathrm{d}t=\mathrm{0}$(55)

Therefore, the criterion equation for the canonical transformation is:

$p_s\mathrm{d}{q}_s-P_s\mathrm{d}{Q}_s+\left(K^{\mathrm{1}+\gamma }-H^{\mathrm{1}+\gamma }\right)\mathrm{d}t=\mathrm{d}F$(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 $F_{\mathrm{1}}$, has the form

$F=F_{\mathrm{1}}\left(t,q_s,Q_s\right)$(57)

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

$\begin{array}{l}\left(p_s-\frac{\partial F_{\mathrm{1}}}{\partial q_s}\right)\mathrm{d}{q}_s+\left(-P_s-\frac{\partial F_{\mathrm{1}}}{\partial Q_s}\right)\mathrm{d}{Q}_s\\ +\left(K^{\mathrm{1}+\gamma }-H^{\mathrm{1}+\gamma }-\frac{\partial F_{\mathrm{1}}}{\partial t}\right)\mathrm{d}t=\mathrm{0}\end{array}$(58)

Let the coefficients of $\mathrm{d}{q}_k$, $\mathrm{d}{Q}_k$, and $\mathrm{d}t$ be zero, respectively, we have:

$p_s=\frac{\partial F_{\mathrm{1}}}{\partial q_s},P_s=-\frac{\partial F_{\mathrm{1}}}{\partial Q_s},K^{\mathrm{1}+\gamma }=H^{\mathrm{1}+\gamma }+\frac{\partial F_{\mathrm{1}}}{\partial t}\left(s=\mathrm{1,2},\cdots ,N\right)$(59)

The second type of generating function $F_{\mathrm{2}}$ is:

$F_{\mathrm{2}}\left(t,q_s,P_s\right)=F_{\mathrm{1}}\left(t,q_s,Q_s\right)+Q_s{P}_s$(60)

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

$\begin{array}{l}\left(p_s-\frac{\partial F_{\mathrm{2}}}{\partial q_s}\right)\mathrm{d}{q}_s+\left(Q_s-\frac{\partial F_{\mathrm{2}}}{\partial P_s}\right)\mathrm{d}{P}_s\\ +\left(K^{\mathrm{1}+\gamma }-H^{\mathrm{1}+\gamma }-\frac{\partial F_{\mathrm{2}}}{\partial t}\right)\mathrm{d}t=\mathrm{0}\end{array}$(61)

So we have:

$p_s=\frac{\partial F_{\mathrm{2}}}{\partial q_s},Q_s=\frac{\partial F_{\mathrm{2}}}{\partial P_s},K^{\mathrm{1}+\gamma }=H^{\mathrm{1}+\gamma }+\frac{\partial F_{\mathrm{2}}}{\partial t}\left(s=\mathrm{1,2},\cdots ,N\right)$(62)

The third type of generating function is:

$F_{\mathrm{3}}\left(t,p_s,Q_s\right)=F_{\mathrm{1}}\left(t,q_s,Q_s\right)-q_s{p}_s$(63)

Then we have:

$q_s=-\frac{\partial F_{\mathrm{3}}}{\partial p_s},P_s=-\frac{\partial F_{\mathrm{3}}}{\partial Q_s},K^{\mathrm{1}+\gamma }=H^{\mathrm{1}+\gamma }+\frac{\partial F_{\mathrm{3}}}{\partial t}\left(s=\mathrm{1,2},\cdots ,N\right)$(64)

The fourth type of generating function is:

$F_{\mathrm{4}}\left(t,p_s,P_s\right)=F_{\mathrm{1}}\left(t,q_s,Q_s\right)-q_s{p}_s+Q_s{P}_s$(65)

Then we have:

$q_s=-\frac{\partial F_{\mathrm{4}}}{\partial p_s},Q_s=\frac{\partial F_{\mathrm{4}}}{\partial P_s},K^{\mathrm{1}+\gamma }=H^{\mathrm{1}+\gamma }+\frac{\partial F_{\mathrm{4}}}{\partial t}\left(s=\mathrm{1,2},\cdots ,N\right)$(66)

2.2 Poisson Theory

Let $a^s=q_s$,$a^{n+s}=p_s$$\left(s=\mathrm{1,2},\cdots ,N\right)$, then equation (51) can be transformed into a contravariant algebraic form

${\dot{a}}^{\mu }={\omega }^{\mu \nu }\frac{\partial H^{\mathrm{1}+\gamma }}{\partial a^{\nu }}\left(\mu ,\nu =\mathrm{1,2},\cdots ,\mathrm{2}n\right)$(67)

where $\left({\omega }^{\mu \nu }\right)=\left[\begin{array}{cc}{\mathrm{0}}_{n\times n}& I_{n\times n}\\ -I_{n\times n}& {\mathrm{0}}_{n\times n}\end{array}\right]$ is a contravariant tensor.

Let $A\left(a\right)$ be some function. According to equation (67), we define a product

$\dot{A}\left(a\right)=\frac{\partial A}{\partial a^{\mu }}{\omega }^{\mu \nu }\frac{\partial H^{\mathrm{1}+\gamma }}{\partial a^{\nu }}=A\circ H^{\mathrm{1}+\gamma }$(68)

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

Since Eq. (67) has a Lie algebraic structure, we can establish Poisson theory as follows:

$\frac{\partial I}{\partial t}+I\circ H^{\mathrm{1}+\gamma }=\mathrm{0}$(69)

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

2.3 Examples

$S_{\mathrm{P}}={{\int }_{t_{\mathrm{1}}}^{t_{\mathrm{2}}}\left(\dot{q}+k{q}^{\mathrm{2}}\right)}^{-\mathrm{1}}\mathrm{d}t$(70)

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

In this problem, $L=\dot{q}+k{q}^{\mathrm{2}}$,$\gamma =-\mathrm{2}$. Let $k=\mathrm{1}$, then the dynamical equation reads:

$\ddot{q}+\mathrm{3}q\dot{q}+q^{\mathrm{3}}=\mathrm{0}$(71)

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

$p=-\frac{\mathrm{1}}{{\left(\dot{q}+q^{\mathrm{2}}\right)}^{\mathrm{2}}}$(72)

$H=-\mathrm{2}\sqrt[]{-p}-p{q}^{\mathrm{2}}$(73)

Select the generating function:

$F=F_{\mathrm{1}}\left(q,Q,t\right)=-q{Q}^{\mathrm{2}}$(74)

From Eq. (59), we get:

$p=\frac{\partial F_{{}_{\mathrm{1}}}}{\partial q}=-Q^{\mathrm{2}},P=-\frac{\partial F_{{}_{\mathrm{1}}}}{\partial Q}=\mathrm{2}qQ$(75)

and

$K=-\mathrm{2}Q+\frac{P^{\mathrm{2}}}{\mathrm{4}}$(76)

The canonical equation with $Q,P$ as the new variables is:

$\dot{Q}=\frac{\partial K}{\partial P}=\frac{\mathrm{1}}{\mathrm{2}}P,\dot{P}=-\frac{\partial K}{\partial Q}=\mathrm{2}$(77)

By integrating the above equation, we get:

$Q=\mathrm{4}{t}^{\mathrm{2}}+c_{\mathrm{1}},P=\mathrm{2}t+c_{\mathrm{2}}$(78)

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

$q=\frac{\mathrm{2}t+c_{\mathrm{2}}}{\mathrm{2}\left(\mathrm{4}{t}^{\mathrm{2}}+c_{\mathrm{1}}\right)}$(79)

Take the generating function as:

$F_{\mathrm{2}}\left(t,q,P\right)=F_{\mathrm{1}}+QP=\frac{P^{\mathrm{2}}}{\mathrm{4}q}$(80)

Then the transformation (62) gives:

$p=\frac{\partial F_{\mathrm{2}}}{\partial q}=-\frac{p^{\mathrm{2}}}{\mathrm{4}{q}^{\mathrm{2}}},Q=\frac{\partial F_{\mathrm{2}}}{\partial P}=\frac{P}{\mathrm{2}q}$(81)

In this case, the generating function $F_{\mathrm{3}}$ is:

$F_{\mathrm{3}}\left(t,p,Q\right)=F_{\mathrm{1}}-qp=\mathrm{0}$(82)

Take

$F_{\mathrm{4}}\left(t,P,p\right)=F_{\mathrm{1}}-qp+QP=P\sqrt[]{-p}$(83)

Then the transformation (66) gives:

$q=-\frac{\partial F_{\mathrm{4}}}{\partial p}=\frac{P}{\mathrm{2}\sqrt[]{-p}},Q=\frac{\partial F_{\mathrm{4}}}{\partial P}=\sqrt[]{-p}$(84)

Let

$q=a^{\mathrm{1}},p=a^{\mathrm{2}}$(85)

The Hamiltonian (73) can be written as:

$H=-\mathrm{2}\sqrt[]{-a^{\mathrm{2}}}-a^{\mathrm{2}}{\left(a^{\mathrm{1}}\right)}^{\mathrm{2}}$(86)

The equation has a Lie algebraic structure.

Since $H$ does not explicitly contain $t$, the system has an integral $H=h$ by Theorem 9.

By using the generalized Poisson condition (69), we can easily verify that:

$I_{\mathrm{1}}=\sqrt[]{-a^{\mathrm{2}}}+\frac{a^{\mathrm{2}}{\left(a^{\mathrm{1}}\right)}^{\mathrm{2}}}{\mathrm{2}}=c_{\mathrm{1}}$(87)

$I_{\mathrm{2}}=\frac{\mathrm{1}}{\mathrm{2}}{t}^{\mathrm{2}}+\left(\mathrm{1}-t{a}^{\mathrm{1}}\right)\sqrt[]{-a^{\mathrm{2}}}=c_{\mathrm{2}}$(88)

$I_{\mathrm{3}}=-{\left({\left(-a^{\mathrm{2}}\right)}^{\frac{\mathrm{1}}{\mathrm{2}}}+\frac{\mathrm{1}}{\mathrm{2}}{\left(a^{\mathrm{1}}\right)}^{\mathrm{2}}\left(-a^{\mathrm{2}}\right)\right)}^{-\frac{\mathrm{1}}{\mathrm{2}}}=c_{\mathrm{3}}$(89)

are the first integrals of the system.

Calculate the Poisson parentheses:

$I_{\mathrm{4}}=I_{\mathrm{1}}\circ I_{\mathrm{2}}=\frac{\mathrm{1}}{\mathrm{2}}{a}^{\mathrm{1}}{\left(-a^{\mathrm{2}}\right)}^{\frac{\mathrm{1}}{\mathrm{2}}}-\frac{\mathrm{1}}{\mathrm{2}}t=c_{\mathrm{4}}$(90)

According to Theorem 10, $I_{\mathrm{4}}$ is also a first integral.

Since $I_{\mathrm{2}}$ involves time $t$, while the power-law Hamiltonian $H^{\mathrm{1}+\gamma }$ does not explicitly involve time $t$, according to Theorem 11, we get

$I_{\mathrm{5}}=\frac{\partial I_{\mathrm{2}}}{\partial t}=t-a^{\mathrm{1}}\sqrt[]{-a^{\mathrm{2}}}=c_{\mathrm{5}}$(91)

Formula (91) is the first integral.

Assuming the initial conditions $q=\mathrm{1}$ and $\dot{q}=\mathrm{1}$, the trajectory of motion $q$, the Hamiltonian $H$, and the conserved quantities $I_{\mathrm{1}},I_{\mathrm{2}},I_{\mathrm{3}},I_{\mathrm{4}}$ and $I_{\mathrm{5}}$ can be easily calculated, as shown in Fig. 3 and Fig. 4.

Fig. 3 Simulation of the trajectory of motion $\mathit{q}\left(\mathit{t}\right)$ 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 $N$ generalized coordinates $q_k\left(k=\mathrm{1,2},\cdots ,N\right)$, the action with logarithmic Lagrangians is:

$S_L={\int }_{t_{\mathrm{1}}}^{t_{\mathrm{2}}}\mathrm{l}\mathrm{o}{\mathrm{g}}_{b}L\left(t,q_k,{\dot{q}}_k\right)\mathrm{d}t,b>\mathrm{0}$(92)

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

$\delta S_L=\mathrm{0}$(93)

And the dynamic equations are:

$\frac{\mathrm{1}}{L\mathrm{l}\mathrm{n}b}\left(\frac{\partial L}{\partial q_k}+\frac{\mathrm{1}}{L}\frac{\partial L}{\partial {\dot{q}}_k}-\frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial L}{\partial q_k}\right)=\mathrm{0},k=\mathrm{1,2},\cdots ,N$(94)

Define the generalized momentum corresponding to logarithmic Lagrangians as:

$p_k=\frac{\partial \mathrm{l}\mathrm{o}{\mathrm{g}}_{b}L}{\partial {\dot{q}}_k}=\frac{\mathrm{1}}{L\mathrm{l}\mathrm{n}b}\frac{\partial L}{\partial {\dot{q}}_k}$(95)

And the logarithmic Hamiltonian as:

$\mathrm{l}\mathrm{o}{\mathrm{g}}_{b}H\left(t,q_k,p_k\right)=p_k{\dot{q}}_k-\mathrm{l}\mathrm{o}{\mathrm{g}}_{b}L\left(t,q_k,{\dot{q}}_k\right)$(96)

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

${\dot{q}}_k=\frac{\mathrm{1}}{H\mathrm{l}\mathrm{n}b}\frac{\partial H}{\partial p_k},{\dot{q}}_k=-\frac{\mathrm{1}}{H\mathrm{l}\mathrm{n}b}\frac{\partial H}{\partial q_k}$(97)

We study the transformation from variables $q_k$ and $p_k$ to new variables $Q_k$ and $P_k$, namely:

$Q_k=Q_k\left(t,q_s,p_s\right),P_k=P_k\left(t,q_s,p_s\right)$(98)

Let $\mathrm{\Delta }=\frac{\partial \left(Q_k,P_k\right)}{\partial \left(q_k,p_k\right)}\ne \mathrm{0}$, then, the transformation is invertible. If under transformation (98), the form of equations (97) remains unchanged, i.e.,

${\dot{Q}}_k=\frac{\mathrm{1}}{K\mathrm{l}\mathrm{n}b}\frac{\partial K}{\partial P_k},{\dot{P}}_k=-\frac{\mathrm{1}}{K\mathrm{l}\mathrm{n}b}\frac{\partial K}{\partial Q_k}$(99)

where $K=K\left(t,Q_k,P_k\right)$ 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:

$\delta {\int }_{t_{\mathrm{1}}}^{t_{\mathrm{2}}}\left(p_k{\dot{q}}_k-\mathrm{l}\mathrm{o}{\mathrm{g}}_{b}H\right)\mathrm{d}t=\mathrm{0}$(100)

$\delta {\int }_{t_{\mathrm{1}}}^{t_{\mathrm{2}}}\left(P_k{\dot{Q}}_k-\mathrm{l}\mathrm{o}{\mathrm{g}}_{b}K\right)\mathrm{d}t=\mathrm{0}$(101)