© Wuhan University 2025

0 Introduction

Conservation laws play a key role in exploring the dynamics of constrained mechanical systems. There are also plenty of ways to find conserved quantities, such as analytical mechanics methods^[1], symmetry methods^[2], and symmetry methods which includes Lie symmetry^[3], Noether symmetry^[4] and so on. Djukić et al^[5] proposed to directly construct the conservation laws of nonconservative systems by multiplying the differential equations of motion by the integrating factors. Conserved quantities are calculated easily in this way. The integrating factors method has achieved a series of results in solving conserved quantities. Qiao et al^[6,7] applied the integrating factors method to solve the conserved quantities in holonomic nonconservative systems. Since then, the integrating factors method has been generalized and applied to nonholonomic systems^[8-10] and constrained Hamiltonian systems^[11]. Birkhoffian mechanics^[12,13] is a more general mechanics into which both holonomic and nonholonomic mechanics can be incorporated. Zhang^[14,15] proposed to construct conservation laws of the Birkhoffian system by using the integrating factors method. The results are further generalized to constrained Birkhoffian systems^[16] and fractional Birkhoffian systems^[17].

Hamilton's principle, which is generally applicable to conservative systems, has encountered difficulties in generalizing to nonconservative systems. Existing nonconservative generalizations of Hamilton's principle are no longer variational principles, although differential equations of motion can be derived from them. Herglotz's principle^[18], which defines its function through a differential equation, provides a variational principle for understanding systems that are by no means conservative. In 2002, Georgieva and Guenther^[19] investigated the invariance of the Lagrangian function, a statement of Noether symmetry, and proved the Noether theorem of Herglotz type. Noether conservation laws of the Herglotz type were given, which marks the beginning of the research on the Noether symmetry of the Herglotz type. Recently, Herglotz type Noether theorems have been extended to nonconservative Lagrange systems^[20-22], nonconservative Hamilton systems^[23,24], nonholonomic systems^[25-27], Birkhoffian systems^[28,29], time-delay dynamics^[30-33], fractional order dynamics ^[34-38], time-scale dynamics^[39,40], etc.

Combining the integrating factors method with Birkhoffian systems and the Herglotz principle, this paper studies how to construct the conservation laws of the Herglotz type Birkhoffian system by using the integrating factors method. In Section 1, we define the integral factors of Herglotz Birkhoff's equation. Section 2 proposes the conservation theorems. The generalized Killing equations are derived in Section 3. The inverse theorems are established in Section 4. Herglotz type Birkhoff's equations are a generalization of classical Birkhoff's equations, and the latter is the generalization of Hamilton's equations. Therefore, we apply the obtained results to classical Birkhoff's equations and Hamilton's equations and discuss the corresponding theorems. In Section 5, the damped harmonic oscillator is taken as an example to demonstrate the application of the results. The last section is the conclusion of the article.

1 Herglotz Type Birkhoff's Equations and Its Integrating Factors

The Herglotz type Birkhoff's equations^[28] are

$\begin{array}{l}\alpha \left(t\right)[{\tilde{\mathrm{\Omega }}}_{\mu v}{\dot{a}}^v-\frac{\partial B}{\partial a^{\mu }}-\frac{\partial R_{\mu }}{\partial t}+\frac{\partial R_{\mu }}{\partial z}B-R_{\mu }\frac{\partial B}{\partial z}]=\mathrm{0} \\ \left(\mu ,v=\mathrm{1,2},\cdots ,\mathrm{2}n\right)\end{array}$(1)

where $\alpha \left(t\right)=\mathrm{e}\mathrm{x}\mathrm{p}\left[-{\int }_{t_{\mathrm{0}}}^t\left(\frac{\partial R_{\mu }}{\partial z}{\dot{a}}^{\mu }-\frac{\partial B}{\partial z}\right)\mathrm{d}\theta \right]$, and $z$ is the Pfaff-Herglotz action which is defined by

$\dot{z}=R_{\mu }\left(t,a^v,z\right){\dot{a}}^{\mu }-B\left(t,a^v,z\right)$(2)

and $B\left(t,a^v,z\right)$ and $R_{\mu }\left(t,a^v,z\right)$ are the Birkhoffian and Birkhoff's functions in the sense of Herglotz, respectively. And $a^{\mu }$ are Birkhoff's variables. Eq. (1) can be reduced to

${\dot{a}}^{\mu }={\tilde{\mathrm{\Omega }}}^{\mu v}\left(\frac{\partial B}{\partial a^v}+\frac{\partial R_v}{\partial t}-\frac{\partial R_v}{\partial z}B+R_v\frac{\partial B}{\partial z}\right)$(3)

where ${\tilde{\mathrm{\Omega }}}_{\mu v}$ are called Herglotz type Birkhoff's tensor, there is

${\tilde{\mathrm{\Omega }}}_{\mu v}=\frac{\partial R_v}{\partial a^{\mu }}-\frac{\partial R_{\mu }}{\partial a^v}+R_{\mu }\frac{\partial R_v}{\partial z}-\frac{\partial R_{\mu }}{\partial z}{R}_v$(4)

and $\mathrm{d}\mathrm{e}\mathrm{t}\left({\tilde{\mathrm{\Omega }}}_{\mu v}\right)\ne \mathrm{0}$, ${\tilde{\mathrm{\Omega }}}^{\mu v}{\tilde{\mathrm{\Omega }}}_{v\rho }={\delta }_{\mu \rho }$.

If $R_{\mu }$ does not explicitly contain $z$, then ${\tilde{\mathrm{\Omega }}}_{\mu v}$ is reduced to Birkhoff's tensor ${\mathrm{\Omega }}_{\mu v}$, i. e,

${\mathrm{\Omega }}_{\mu v}=\frac{\partial R_v}{\partial a^{\mu }}-\frac{\partial R_{\mu }}{\partial a^v}.$(5)

Definition 1   If there is a set of functions $G^{\mu }\left(t,a^v,z\right)$, such that this invariant

$\alpha \left(t\right)[{\tilde{\mathrm{\Omega }}}_{\mu v}{\dot{a}}^v-\frac{\partial B}{\partial a^{\mu }}-\frac{\partial R_{\mu }}{\partial t}+\frac{\partial R_{\mu }}{\partial z}B-R_{\mu }\frac{\partial B}{\partial z}]G^{\mu }$(6)

is reduced identically to

$\begin{array}{l}\alpha \left(t\right)[{\tilde{\mathrm{\Omega }}}_{\mu v}{\dot{a}}^v-\frac{\partial B}{\partial a^{\mu }}-\frac{\partial R_{\mu }}{\partial t}+\frac{\partial R_{\mu }}{\partial z}B-R_{\mu }\frac{\partial B}{\partial z}]G^{\mu }\\ \equiv {\lambda }^{\mu }\alpha \left(t\right)[{\tilde{\mathrm{\Omega }}}_{\mu v}{\dot{a}}^v-\frac{\partial B}{\partial a^{\mu }}-\frac{\partial R_{\mu }}{\partial t}+\frac{\partial R_{\mu }}{\partial z}B-R_{\mu }\frac{\partial B}{\partial z}]\end{array}$

$-\frac{\mathrm{d}}{\mathrm{d}t}\left[\alpha \left(t\right)\left(R_{\mu }{G}^{\mu }-B\tau -\mathrm{\Lambda }\right)\right]$(7)

where $\tau ,$$\mathrm{\Lambda }$ and ${\lambda }^{\mu }$ are functions of Birkhoff's variables $a_{\mu }$, time $t$ and functional $z$, then $G^{\mu }=G^{\mu }\left(t,a^v,z\right)$ are named the integrating factors of Herglotz type Birkhoff's equations (1).

2 Conservation Theorems

Using Eqs. (1), we rewrite the formula (7) as

$\frac{\mathrm{d}}{\mathrm{d}t}\left[\alpha \left(t\right)\left(R_{\mu }{G}^{\mu }-B\tau -\mathrm{\Lambda }\right)\right]={\lambda }^{\mu }\alpha \left(t\right)[{\tilde{\mathrm{\Omega }}}_{\mu v}{\dot{a}}^v-\frac{\partial B}{\partial a^{\mu }}-\frac{\partial R_{\mu }}{\partial t}+\frac{\partial R_{\mu }}{\partial z}B-R_{\mu }\frac{\partial B}{\partial z}]$(8)

So, we can easily arrive at the theorem below:

Theorem 1   If $G^{\mu }$ are integrating factors of Herglotz type Birkhoff's equations (1), then

$I_{\mathrm{H}\mathrm{e}\mathrm{r}\mathrm{g}\mathrm{l}\mathrm{o}\mathrm{t}\mathrm{z}}=\alpha \left(t\right)\left(R_{\mu }{G}^{\mu }-B\tau -\mathrm{\Lambda }\right)$(9)

is the first integral along the trajectory of the Herglotz type Birkhoffian system.

For the Herglotz type Birkhoffian system (1), by Definition 1, if $G^{\mu }$ are its integrating factors, the necessary condition (7) should be satisfied. Expanding (7), we get

$-\frac{\partial B}{\partial t}\tau +\left(\frac{\partial R_{\nu }}{\partial a^{\mu }}{G}^{\mu }-\frac{\partial B}{\partial a^{\nu }}\tau \right){\dot{a}}^{\nu }+\left(\frac{\partial R_{\nu }}{\partial z}B-\frac{\partial B}{\partial z}{R}_{\nu }\right){\dot{a}}^{\nu }\tau$

$+\left(\frac{\partial R_{\nu }}{\partial z}{\dot{a}}^{\nu }-\frac{\partial B}{\partial z}\right)\mathrm{\Lambda }+R_{\mu }{\dot{G}}^{\mu }-\frac{\partial B}{\partial a^{\mu }}{G}^{\mu }-B\dot{\tau }-\dot{\mathrm{\Lambda }}+\Phi =\mathrm{0}$(10)

where

$\Phi =-{\lambda }^{\mu }\left[{\tilde{\mathrm{\Omega }}}_{\mu v}{\dot{a}}^v-\frac{\partial B}{\partial a^{\mu }}-\frac{\partial R_{\mu }}{\partial t}+\frac{\partial R_{\mu }}{\partial z}B-R_{\mu }\frac{\partial B}{\partial z}\right].$(11)

Hence, we have

Theorem 2   For the Herglotz type Birkhoffian system (1), if there exists a non-singular set of functions $G^{\mu },\tau ,\mathrm{\Lambda }$ and ${\lambda }^{\mu }$ satisfying condition (10), then the system has a first integral such as (9).

Remark 1   In deriving formula (10), we have adopted the operator to calculate ${\dot{R}}_{\mu }$ and $\dot{B}$, namely:

$\frac{\mathrm{d}}{\mathrm{d}t}=\frac{\partial }{\partial t}+{\dot{a}}^v\frac{\partial }{\partial a^v}+\left(R_{\nu }{\dot{a}}^{\nu }-B\right)\frac{\partial }{\partial z}.$(12)

Remark 2   The functions $G^{\mu },\tau ,\mathrm{\Lambda }$ and ${\lambda }^{\mu }$ are called a singular set if they make the right side of the equation (9) a constant.

If there is no action z in the Birkhoffian functions $B$ and $R_{\mu }$, Eq. (1) degenerate into the classical Birkhoff's equations

${\mathrm{\Omega }}_{\mu \nu }{\dot{a}}^{\nu }-\frac{\partial B}{\partial a^{\mu }}-\frac{\partial R_{\mu }}{\partial t}=\mathrm{0}$(13)

and Theorem 2 reduces to

Theorem 3   For the Birkhoffian system (13), if there exists a non-singular set of functions $G^{\mu },\tau ,\mathrm{\Lambda }$ and ${\lambda }^{\mu }$ satisfying condition below:

$-\frac{\partial B}{\partial t}\tau +\left(\frac{\partial R_{\nu }}{\partial a^{\mu }}{G}^{\mu }-\frac{\partial B}{\partial a^{\nu }}\tau \right){\dot{a}}^{\nu }+R_{\mu }{\dot{G}}^{\mu }-\frac{\partial B}{\partial a^{\mu }}{G}^{\mu }-B\dot{\tau }-\dot{\mathrm{\Lambda }}+{\Phi }_B=\mathrm{0}$(14)

where

${\Phi }_B=-{\lambda }^{\mu }\left({\mathrm{\Omega }}_{\mu \nu }{\dot{a}}^{\nu }-\frac{\partial B}{\partial a^{\mu }}-\frac{\partial R_{\mu }}{\partial t}\right)$(15)

then the system has a first integral such as

$I_{\mathrm{B}\mathrm{i}\mathrm{r}\mathrm{k}\mathrm{h}\mathrm{o}\mathrm{f}\mathrm{f}}=R_{\mu }{G}^{\mu }-B\tau -\mathrm{\Lambda }=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}.$(16)

Further, if^[13]

$a^{\mu }=\{\begin{array}{cc}{q}_{\mu },& \mu =\mathrm{1,2},\cdots ,n\\ p_{\mu -n},& \mu =n+\mathrm{1},n+\mathrm{2},\cdots ,\mathrm{2}n\end{array}$

$R_{\mu }=\{\begin{array}{cc}{p}_{\mu },& \mu =\mathrm{1,2},\cdots ,n\\ \mathrm{0},& \mu =n+\mathrm{1},n+\mathrm{2},\cdots ,\mathrm{2}n\end{array}$

$B=H\left(t,q_s,p_s\right)$(17)

then Eqs. (13) degenerate into the classical Hamilton's equations

${\dot{q}}_s=\frac{\partial H}{\partial p_s},\hspace{1em}{\dot{p}}_s=-\frac{\partial H}{\partial q_s},s=\mathrm{1,2},\cdots ,n$(18)

and Theorem 3 reduces to

Theorem 4   For the Hamiltonian system (18), if there exists a non-singular set of functions $G^s,\tau ,\mathrm{\Lambda }$ and ${\lambda }^s$ satisfying condition below:

$-\frac{\partial H}{\partial t}\tau +p_s{\dot{G}}^s-\frac{\partial H}{\partial q_s}{G}^s-H\dot{\tau }-\dot{\mathrm{\Lambda }}+\varphi =\mathrm{0}$(19)

where

$\varphi ={\lambda }^s\left({\dot{p}}_s+\frac{\partial H}{\partial q_s}\right)$(20)

then the system has a first integral such as

$I_{\mathrm{H}\mathrm{a}\mathrm{m}\mathrm{i}\mathrm{l}\mathrm{t}\mathrm{o}\mathrm{n}}=p_s{G}^s-H\tau -\mathrm{\Lambda }=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}.$(21)

Remark 3   The non-singular set of functions $G^{\mu },\tau ,\mathrm{\Lambda }$ and ${\lambda }^{\mu }$ can be obtained by solving the necessary condition (10) (or (14), or (19)), and once a set of solutions is found, we can use Theorem 2 (or Theorem 3, or Theorem 4) to obtain a first integral of Herglotz type Birkhoffian systems (or classical Birkhoffian systems, or Hamiltonian systems).

3 Generalized Killing Equations

The key to finding first integrals through Theorems 1 and 2 is to find non-singular sets of functions $G^{\mu },\tau ,\mathrm{\Lambda }$ and ${\lambda }^{\mu }$ by solving equation (10).

By substituting Eq. (3) into Eq. (10), we get

$-\frac{\partial B}{\partial t}\tau +K_{\mu }{\tilde{\mathrm{\Omega }}}^{\mu \nu }\left(\frac{\partial B}{\partial a^{\nu }}+\frac{\partial R_{\nu }}{\partial t}-\frac{\partial R_{\nu }}{\partial z}B+\frac{\partial B}{\partial z}{R}_{\nu }\right)$

$-\frac{\partial B}{\partial a^{\mu }}{G}^{\mu }-\frac{\partial B}{\partial z}\mathrm{\Lambda }+R_{\mu }{\dot{G}}^{\mu }-B\dot{\tau }-\dot{\mathrm{\Lambda }}+\Phi =\mathrm{0},$(22)

where

$K_{\mu }=\frac{\partial R_{\mu }}{\partial a^{\nu }}{G}^{\nu }-\frac{\partial B}{\partial a^{\mu }}\tau +\frac{\partial R_{\mu }}{\partial z}B\tau -\frac{\partial B}{\partial z}{R}_{\mu }\tau +\frac{\partial R_{\mu }}{\partial z}\mathrm{\Lambda }.$(23)

We expand Eq. (22) and decompose it into first-order partial differential equations for $G^{\mu },\tau$ and $\mathrm{\Lambda },$ which are called generalized Killing equations. Solving the generalized Killing equations, it is possible to find these functions. In practice, we have the following two ways.

In the first way, we consider equation (22) to be true for arbitrary $t,a^{\mu }$ and ${\dot{a}}^{\mu }$. Expand the Eq. (22) and let the coefficients of the terms ${\dot{a}}^{\mu }$ and the terms without ${\dot{a}}^{\mu }$ equal to zero respectively, we have

$\begin{array}{l}{R}_{\nu }\left(\frac{\partial G^{\nu }}{\partial a^{\mu }}+\frac{\partial G^{\nu }}{\partial z}{R}_{\mu }\right)-B\left(\frac{\partial \tau }{\partial a^{\mu }}+\frac{\partial \tau }{\partial z}{R}_{\mu }\right)-\frac{\partial \mathrm{\Lambda }}{\partial z}{R}_{\mu }-{\lambda }^{\nu }{\tilde{\mathrm{\Omega }}}_{\nu \mu }\\ =\mathrm{0},\mu =\mathrm{1,2},\cdots ,\mathrm{2}n\end{array}$(24)

and

$\begin{array}{l}-\frac{\partial B}{\partial t}\tau +K_{\mu }{\tilde{\mathrm{\Omega }}}^{\mu \nu }\left(\frac{\partial B}{\partial a^{\nu }}+\frac{\partial R_{\nu }}{\partial t}-\frac{\partial R_{\nu }}{\partial z}B+\frac{\partial B}{\partial z}{R}_{\nu }\right)-\frac{\partial B}{\partial a^{\mu }}{G}^{\mu }\\ -\frac{\partial B}{\partial z}\mathrm{\Lambda }+R_{\mu }\left(\frac{\partial G^{\mu }}{\partial t}-\frac{\partial G^{\mu }}{\partial z}B\right)\end{array}$

$-B\left(\frac{\partial \tau }{\partial t}-\frac{\partial \tau }{\partial z}B\right)-\frac{\partial \mathrm{\Lambda }}{\partial t}+\frac{\partial \mathrm{\Lambda }}{\partial z}B+{\lambda }^{\mu }\left(\frac{\partial B}{\partial a^{\mu }}+\frac{\partial R_{\mu }}{\partial t}-\frac{\partial R_{\mu }}{\partial z}B+\frac{\partial B}{\partial z}{R}_{\mu }\right)=\mathrm{0}$(25)

Eqs. (24) and (25) are named generalized Killing equations of Herglotz type Birkhoffian systems. So, we have

Theorem 5   For the Herglotz type Birkhoffian system (1), if there exists a non-singular set of functions $G^{\mu },\tau ,\mathrm{\Lambda }$ and ${\lambda }^{\mu }$ satisfying generalized Killing equations (24) and (25), then the system has a first integral such as (9).

If $B$ and $R_{\mu }$ do not contain $z$, Eqs. (24) and (25) degenerate into

$R_{\nu }\frac{\partial G^{\nu }}{\partial a^{\mu }}-B\frac{\partial \tau }{\partial a^{\mu }}-\frac{\partial \mathrm{\Lambda }}{\partial a^{\mu }}-{\lambda }^{\nu }{\mathrm{\Omega }}_{\nu \mu }=\mathrm{0},\mu =\mathrm{1,2},\cdots ,\mathrm{2}n,$(26)

and

$-\frac{\partial B}{\partial t}\tau +\left(\frac{\partial R_{\mu }}{\partial a^{\rho }}{G}^{\rho }-\frac{\partial B}{\partial a^{\mu }}\tau \right){\mathrm{\Omega }}^{\mu \nu }\left(\frac{\partial B}{\partial a^{\nu }}+\frac{\partial R_{\nu }}{\partial t}\right)-\frac{\partial B}{\partial a^{\mu }}{G}^{\mu }+R_{\mu }\frac{\partial G^{\mu }}{\partial t}-B\frac{\partial \tau }{\partial t}-\frac{\partial \mathrm{\Lambda }}{\partial t}+{\lambda }^{\mu }\left(\frac{\partial B}{\partial a^{\mu }}+\frac{\partial R_{\mu }}{\partial t}\right)=\mathrm{0}$(27)

then Theorem 5 reduces to

Theorem 6   For the Birkhoffian system (13), if there exists a non-singular set of functions $G^{\mu },\tau ,\mathrm{\Lambda }$ and ${\lambda }^{\mu }$ satisfying generalized Killing equations (26) and (27), then the system has a first integral such as (16).

For classical Hamiltonian system (18), Eqs. (26) and (27) degenerate into

$p_k\frac{\partial G^k}{\partial p_s}-H\frac{\partial \tau }{\partial p_s}-\frac{\partial \mathrm{\Lambda }}{\partial p_s}+{\lambda }^s=\mathrm{0},s=\mathrm{1,2},\cdots ,n$(28)

and

$-\frac{\partial H}{\partial t}\tau -\frac{\partial H}{\partial q_k}{G}^k+p_k\left(\frac{\partial G^k}{\partial q_s}\frac{\partial H}{\partial p_s}+\frac{\partial G^k}{\partial t}\right)-\frac{\partial \mathrm{\Lambda }}{\partial t}$

$-H\left(\frac{\partial \tau }{\partial q_s}\frac{\partial H}{\partial p_s}+\frac{\partial \tau }{\partial t}\right)-\frac{\partial \mathrm{\Lambda }}{\partial q_s}\frac{\partial H}{\partial p_s}+{\lambda }^s\frac{\partial H}{\partial q_s}=\mathrm{0}$(29)

then Theorem 6 reduces to

Theorem 7   For the Hamiltonian system (20), if there exists a non-singular set of functions $G^s,\tau ,\mathrm{\Lambda }$ and ${\lambda }^s$ satisfying generalized Killing equations (28) and (29), then the system has a first integral such as (21).

In the second way, we consider that Eq. (10) is true along the trajectory of Birkhoffian system (1), which means that ${\dot{a}}^{\mu }$ in Eq. (10) is not arbitrary, but is a function of $a^{\mu }$ and $t$. Using Eqs. (3) and (11), Eq. (10) becomes

$\begin{array}{l}-\frac{\partial B}{\partial t}\tau -\frac{\partial B}{\partial a^{\mu }}{G}^{\mu }-\frac{\partial B}{\partial z}\mathrm{\Lambda }+R_{\mu }\left(\frac{\partial G^{\mu }}{\partial t}-\frac{\partial G^{\mu }}{\partial z}B\right)\\ +[R_{\rho }\left(\frac{\partial G^{\rho }}{\partial a^{\mu }}+\frac{\partial G^{\rho }}{\partial z}{R}_{\mu }\right)-B\left(\frac{\partial \tau }{\partial a^{\mu }}+\frac{\partial \tau }{\partial z}{R}_{\mu }\right)-\frac{\partial \mathrm{\Lambda }}{\partial a^{\mu }}\\ -\frac{\partial \mathrm{\Lambda }}{\partial z}{R}_{\mu }+K_{\mu }]{\tilde{\mathrm{\Omega }}}^{\mu \nu }\left(\frac{\partial B}{\partial a^{\nu }}+\frac{\partial R_{\nu }}{\partial t}-\frac{\partial R_{\nu }}{\partial z}B+\frac{\partial B}{\partial z}{R}_{\nu }\right)\end{array}$

$-B\left(\frac{\partial \tau }{\partial t}-\frac{\partial \tau }{\partial z}B\right)-\frac{\partial \mathrm{\Lambda }}{\partial t}+\frac{\partial \mathrm{\Lambda }}{\partial z}B=\mathrm{0}.$(30)

Eq. (30) is named generalized Killing equation of Herglotz type Birkhoffian systems along the dynamical trajectory. Then, we have

Theorem 8   For the Herglotz type Birkhoffian system (1), if there exists a non-singular set of functions $G^{\mu },\tau$ and $\mathrm{\Lambda }$ satisfying generalized Killing Eq. (30), then the system has a first integral such as (9).

If $B$ and $R_{\mu }$ do not contain $z$, Eq. (28) degenerates into

$\left(R_{\rho }\frac{\partial G^{\rho }}{\partial a^{\mu }}+\frac{\partial R_{\mu }}{\partial a^{\rho }}{G}^{\rho }-B\frac{\partial \tau }{\partial a^{\mu }}-\frac{\partial B}{\partial a^{\mu }}\tau -\frac{\partial \mathrm{\Lambda }}{\partial a^{\mu }}\right)\cdot {\mathrm{\Omega }}^{\mu \nu }\left(\frac{\partial B}{\partial a^{\nu }}+\frac{\partial R_{\nu }}{\partial t}\right)-\frac{\partial B}{\partial t}\tau -\frac{\partial B}{\partial a^{\mu }}{G}^{\mu }+R_{\mu }\frac{\partial G^{\mu }}{\partial t}-B\frac{\partial \tau }{\partial t}-\frac{\partial \mathrm{\Lambda }}{\partial t}=\mathrm{0}$(31)

and Theorem 8 reduces to

Theorem 9   For the Birkhoffian system (13), if there exists a non-singular set of functions $G^{\mu },\tau$ and $\mathrm{\Lambda }$ satisfying generalized Killing equations (31), then the system has a first integral such as (16).

For classical Hamiltonian system (18), Eq. (31) degenerates into

$-\frac{\partial H}{\partial t}\tau -\frac{\partial H}{\partial q_k}{G}^k+p_k\left(\frac{\partial G^k}{\partial q_s}\frac{\partial H}{\partial p_s}+\frac{\partial G^k}{\partial t}\right)-\frac{\partial \mathrm{\Lambda }}{\partial t}-H\left(\frac{\partial \tau }{\partial q_s}\frac{\partial H}{\partial p_s}+\frac{\partial \tau }{\partial t}\right)-\frac{\partial \mathrm{\Lambda }}{\partial q_s}\frac{\partial H}{\partial p_s}-\left(p_k\frac{\partial G^k}{\partial p_s}-H\frac{\partial \tau }{\partial p_s}-\frac{\partial \mathrm{\Lambda }}{\partial p_s}\right)\frac{\partial H}{\partial q_s}=\mathrm{0}$(32)

and Theorem 9 reduces to

Theorem 10   For the Hamiltonian system (18), if there exists a non-singular set of functions $G^s,\tau$ and $\mathrm{\Lambda }$ satisfying generalized Killing equation (32), then the system has a first integral such as (21).

4 Inverse Theorems

Assume that the Herglotz type Birkhoffian system (1) has a first integral

$I=I\left(t,a^{\mu },z\right)=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}.$(33)

By differentiating the Eq. (33), we get

$\frac{\mathrm{d}I}{\mathrm{d}t}=\frac{\partial I}{\partial t}+{\dot{a}}^v\frac{\partial I}{\partial a^v}+\left(R_{\nu }{\dot{a}}^{\nu }-B\right)\frac{\partial I}{\partial z}=\mathrm{0}.$(34)

Multiply Eq. (1) by $\left(G^{\mu }-{\dot{a}}^{\mu }\tau \right)$, then add it to Eq. (34), and let the coefficient of ${\dot{a}}^{\mu }$ in the resulting equation equal to zero, we get

$G^{\mu }=-\tau \left(\frac{\partial B}{\partial a^{\nu }}+\frac{\partial R_{\nu }}{\partial t}-\frac{\partial R_{\nu }}{\partial z}B+R_{\nu }\frac{\partial B}{\partial z}\right){\tilde{\mathrm{\Omega }}}^{\nu \mu }$

$-\frac{\mathrm{1}}{\alpha \left(t\right)}\left(\frac{\partial I}{\partial a^{\nu }}+\frac{\partial I}{\partial z}{R}_{\nu }\right){\tilde{\mathrm{\Omega }}}^{\nu \mu },\mu =\mathrm{1,2},\cdots ,\mathrm{2}n.$(35)

Taking the partial derivative of Eq. (9), we get

$\frac{\partial \mathrm{\Lambda }}{\partial a^{\mu }}=\frac{\partial R_{\nu }}{\partial a^{\mu }}{G}^{\nu }+R_{\nu }\frac{\partial G^{\nu }}{\partial a^{\mu }}-\frac{\partial B}{\partial a^{\mu }}\tau -B\frac{\partial \tau }{\partial a^{\mu }}-\frac{\mathrm{1}}{\alpha \left(t\right)}\frac{\partial I}{\partial a^{\mu }},$(36)

and

$\frac{\partial \mathrm{\Lambda }}{\partial z}=\frac{\partial R_{\nu }}{\partial z}{G}^{\nu }+R_{\nu }\frac{\partial G^{\nu }}{\partial z}-\frac{\partial B}{\partial z}\tau -B\frac{\partial \tau }{\partial z}-\frac{\mathrm{1}}{\alpha \left(t\right)}\frac{\partial I}{\partial z}.$(37)

Substituting Eqs. (36) and (37) into (24), we get

$\frac{\partial R_{\nu }}{\partial a^{\mu }}{G}^{\nu }+\frac{\partial R_{\nu }}{\partial z}{G}^{\nu }{R}_{\mu }-\frac{\partial B}{\partial a^{\mu }}\tau -\frac{\partial B}{\partial z}\tau R_{\mu }+{\lambda }^{\nu }{\tilde{\mathrm{\Omega }}}_{\nu \mu }$

$-\frac{\mathrm{1}}{\alpha \left(t\right)}\left(\frac{\partial I}{\partial a^{\mu }}+\frac{\partial I}{\partial z}{R}_{\mu }\right)=\mathrm{0},\mu =\mathrm{1,2},\cdots ,\mathrm{2}n$(38)

Let Eq. (33) equal the first integral (9), we have

$\alpha \left(t\right)\left(R_{\mu }{G}^{\mu }-B\tau -\mathrm{\Lambda }\right)=I.$(39)

Hence, we have the theorem below.

Theorem 11   If Herglotz type Birkhoffian system (1) has a first integral (33), then its integrating factors, $G^{\mu }$ and functions $\tau ,\mathrm{\Lambda }$ and ${\lambda }^{\mu }$ are determined by Eqs. (35), (38) and (39).

Remark 4   Formulas (35), (38) and (39) are $\left(\mathrm{4}n+\mathrm{1}\right)$ equations with $\left(\mathrm{4}n+\mathrm{2}\right)$ variables $G^{\mu },\tau ,\mathrm{\Lambda }$ and ${\lambda }^{\mu }$, and obviously their solutions are not unique. In practical application, we can choose one variable appropriately to obtain the integrating factors $G^{\mu }$.

If $B$ and $R_{\mu }$ do not contain $z$, Eqs. (35), (38) and (39) degenerate into

$G^{\mu }=-\tau \left(\frac{\partial B}{\partial a^{\nu }}+\frac{\partial R_{\nu }}{\partial t}\right){\mathrm{\Omega }}^{\nu \mu }-\frac{\partial I}{\partial a^{\nu }}{\mathrm{\Omega }}^{\nu \mu },$(40)

$\frac{\partial R_{\nu }}{\partial a^{\mu }}{G}^{\nu }-\frac{\partial B}{\partial a^{\mu }}\tau +{\lambda }^{\nu }{\mathrm{\Omega }}_{\nu \mu }-\frac{\partial I}{\partial a^{\mu }}=\mathrm{0},$(41)

$R_{\mu }{G}^{\mu }-B\tau -\mathrm{\Lambda }=I,$(42)

and Theorem 11 reduces to

Theorem 12   If the Birkhoffian system (13) has a first integral

$I=I\left(t,a^{\mu }\right)=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}.$(43)

then its integrating factors $G^{\mu }$ and functions $\tau ,\mathrm{\Lambda }$ and ${\lambda }^{\mu }$ are determined by Eqs. (40), (41) and (42).

For classical Hamiltonian system (18), Eqs. (40)-(42) degenerate into

$G^s=\frac{\partial H}{\partial p_s}\tau +{\lambda }^s+\frac{\partial I}{\partial p_s},$(44)

$p_s{G}^s-H\tau -\mathrm{\Lambda }=I,$(45)

and Theorem 12 reduces to

Theorem 13   If the Hamiltonian system (18) has a first integral

$I=I\left(t,q_s,p_s\right)=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}$(46)

then its integrating factors $G^s$ and functions $\tau ,\mathrm{\Lambda }$ and ${\lambda }^s$ are determined by Eqs. (44) and (45).

5 Example

Consider a nonconservative system

$\ddot{x}+\gamma \dot{x}+x=\mathrm{0},\gamma =\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}.$(47)

First, we establish Herglotz type Birkhoff's equations. Let $a^{\mathrm{1}}=x$ and $a^{\mathrm{2}}=\dot{x}$, then Eq. (47) is reduced to

${\dot{a}}^{\mathrm{1}}=a^{\mathrm{2}},{\dot{a}}^{\mathrm{2}}=-a^{\mathrm{1}}-\gamma a^{\mathrm{2}}.$(48)

Eq. (48) can be expressed as a Herglotz type Birkhoffian system^[28], where

$B=\frac{\mathrm{1}}{\mathrm{2}}\left[{\left(a^{\mathrm{1}}\right)}^{\mathrm{2}}+{\left(a^{\mathrm{2}}\right)}^{\mathrm{2}}\right]+\gamma z,R_{\mathrm{1}}=a^{\mathrm{2}},R_{\mathrm{2}}=\mathrm{0}$(49)

Using Eqs. (4) and (5), we get

$\left({\tilde{\mathrm{\Omega }}}_{\mu \nu }\right)=\left(\begin{array}{cc}\mathrm{0}& -\mathrm{1}\\ \mathrm{1}& \mathrm{0}\end{array}\right)=\left({\mathrm{\Omega }}_{\mu \nu }\right)$(50)

Substituting formulas (49) and (50) into Eq. (1), we get

${\mathrm{e}}^{\gamma \left(t-t_{\mathrm{0}}\right)}\left(-{\dot{a}}^{\mathrm{2}}-a^{\mathrm{1}}-\gamma a^{\mathrm{2}}\right)=\mathrm{0},{\mathrm{e}}^{\gamma \left(t-t_{\mathrm{0}}\right)}\left({\dot{a}}^{\mathrm{1}}-a^{\mathrm{2}}\right)=\mathrm{0}$(51)

This is Herglotz type Birkhoff's equations corresponding to Eqs. (1).

Second, we study the positive problem of finding conserved quantities. From (23), we have

$K_{\mathrm{1}}=G^{\mathrm{2}}-a^{\mathrm{1}}\tau -\gamma a^{\mathrm{2}}\tau ,K_{\mathrm{2}}=-a^{\mathrm{2}}\tau$(52)

The generalized Killing Eqs. (24) and (25) read

$a^{\mathrm{2}}\left(\frac{\partial G^{\mathrm{1}}}{\partial a^{\mathrm{1}}}+\frac{\partial G^{\mathrm{1}}}{\partial z}{a}^{\mathrm{2}}\right)-B\left(\frac{\partial \tau }{\partial a^{\mathrm{1}}}+\frac{\partial \tau }{\partial z}{a}^{\mathrm{2}}\right)-\frac{\partial \mathrm{\Lambda }}{\partial a^{\mathrm{1}}}-\frac{\partial \mathrm{\Lambda }}{\partial z}{a}^{\mathrm{2}}-{\lambda }^{\mathrm{2}}=\mathrm{0},$(53)

$a^{\mathrm{2}}\frac{\partial G^{\mathrm{1}}}{\partial a^{\mathrm{2}}}-B\frac{\partial \tau }{\partial a^{\mathrm{2}}}-\frac{\partial \mathrm{\Lambda }}{\partial a^{\mathrm{2}}}+{\lambda }^{\mathrm{1}}=\mathrm{0}$(54)

and

$a^{\mathrm{2}}\tau \left(a^{\mathrm{1}}+\gamma a^{\mathrm{2}}\right)+a^{\mathrm{2}}\left(G^{\mathrm{2}}-a^{\mathrm{1}}\tau -\gamma a^{\mathrm{2}}\tau \right)-a^{\mathrm{1}}{G}^{\mathrm{1}}-a^{\mathrm{2}}{G}^{\mathrm{2}}$

$-\gamma \mathrm{\Lambda }+a^{\mathrm{2}}\left(\frac{\partial G^{\mathrm{1}}}{\partial t}-\frac{\partial G^{\mathrm{1}}}{\partial z}B\right)-B\left(\frac{\partial \tau }{\partial t}-\frac{\partial \tau }{\partial z}B\right)-\frac{\partial \mathrm{\Lambda }}{\partial t}$

$+\frac{\partial \mathrm{\Lambda }}{\partial z}B+{\lambda }^{\mathrm{1}}\left(a^{\mathrm{1}}+\gamma a^{\mathrm{2}}\right)+{\lambda }^{\mathrm{2}}{a}^{\mathrm{2}}=\mathrm{0}$(55)

Eqs. (53)-(55) have the following solutions

$G^{\mathrm{1}}=\frac{\mathrm{1}}{\mathrm{2}}\gamma a^{\mathrm{1}},G^{\mathrm{2}}=\frac{\mathrm{1}}{\mathrm{2}}\gamma a^{\mathrm{2}},\tau =\mathrm{0},\mathrm{\Lambda }=\gamma z,{\lambda }^{\mathrm{1}}=\mathrm{0},{\lambda }^{\mathrm{2}}=-\frac{\mathrm{1}}{\mathrm{2}}\gamma a^{\mathrm{2}},$(56)

$G^{\mathrm{1}}=-\frac{\mathrm{1}}{\mathrm{2}}{a}^{\mathrm{2}},G^{\mathrm{2}}=\mathrm{0},\tau =\mathrm{0},\mathrm{\Lambda }=\frac{\mathrm{1}}{\mathrm{2}}{\left(a^{\mathrm{1}}\right)}^{\mathrm{2}}+\gamma z,{\lambda }^{\mathrm{1}}=\frac{\mathrm{1}}{\mathrm{2}}{a}^{\mathrm{2}},{\lambda }^{\mathrm{2}}=-a^{\mathrm{1}}-\gamma a^{\mathrm{2}}$(57)

$G^{\mathrm{1}}=\mathrm{0},G^{\mathrm{2}}=\mathrm{1},\tau =\mathrm{0},\mathrm{\Lambda }=\frac{\mathrm{1}}{\mathrm{2}}{\left(a^{\mathrm{1}}\right)}^{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{2}}{\left(a^{\mathrm{2}}\right)}^{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{2}}{a}^{\mathrm{1}}{a}^{\mathrm{2}}\gamma ,{\lambda }^{\mathrm{1}}=a^{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{2}}\gamma a^{\mathrm{1}},{\lambda }^{\mathrm{2}}=-a^{\mathrm{1}}-\frac{\mathrm{1}}{\mathrm{2}}\gamma a^{\mathrm{2}}$(58)

$G^{\mathrm{1}}=\frac{\gamma z}{a^{\mathrm{2}}}-a^{\mathrm{1}}\gamma ,G^{\mathrm{2}}=\mathrm{0},\tau =\mathrm{0},\mathrm{\Lambda }=\frac{\mathrm{1}}{\mathrm{2}}{\left(a^{\mathrm{1}}\right)}^{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{2}}{\left(a^{\mathrm{2}}\right)}^{\mathrm{2}},$

${\lambda }^{\mathrm{1}}=a^{\mathrm{2}}+\gamma \frac{z}{a^{\mathrm{2}}},{\lambda }^{\mathrm{2}}=-a^{\mathrm{1}}$(59)

According to Theorem 2, the system has first integrals as follows

$I_{\mathrm{1}}=\frac{{\mathrm{e}}^{\gamma t}}{{\mathrm{e}}^{\gamma t_{\mathrm{0}}}}\left(\frac{\mathrm{1}}{\mathrm{2}}{a}^{\mathrm{1}}{a}^{\mathrm{2}}\gamma -\gamma z\right)$(60)

$I_{\mathrm{2}}=-\frac{{\mathrm{e}}^{\gamma t}}{{\mathrm{e}}^{\gamma t_{\mathrm{0}}}}\left[\frac{\mathrm{1}}{\mathrm{2}}{\left(a^{\mathrm{1}}\right)}^{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{2}}{\left(a^{\mathrm{2}}\right)}^{\mathrm{2}}+\gamma z\right]$(61)

$I_{\mathrm{3}}=-\frac{{\mathrm{e}}^{\gamma t}}{\mathrm{2}{\mathrm{e}}^{\gamma t_{\mathrm{0}}}}\left[{\left(a^{\mathrm{1}}\right)}^{\mathrm{2}}+{\left(a^{\mathrm{2}}\right)}^{\mathrm{2}}+a^{\mathrm{1}}{a}^{\mathrm{2}}\gamma \right]$(62)

$I_{\mathrm{4}}=-\frac{{\mathrm{e}}^{\gamma t}}{{\mathrm{e}}^{\gamma t_{\mathrm{0}}}}\left[\frac{\mathrm{1}}{\mathrm{2}}{\left(a^{\mathrm{1}}\right)}^{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{2}}{\left(a^{\mathrm{2}}\right)}^{\mathrm{2}}+a^{\mathrm{1}}{a}^{\mathrm{2}}\gamma -\gamma z\right]$(63)

Finally, we study the inverse problem. Suppose that the system has a first integral

$I=\frac{{\mathrm{e}}^{\gamma t}}{{\mathrm{e}}^{\gamma t_{\mathrm{0}}}}\left(\frac{\mathrm{1}}{\mathrm{2}}{a}^{\mathrm{1}}{a}^{\mathrm{2}}\gamma -\gamma z\right)=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}.$(64)

By using equation (35), we get

$G^{\mathrm{1}}=\tau a^{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{2}}\gamma a^{\mathrm{1}},G^{\mathrm{2}}=-\tau a^{\mathrm{1}}-\tau a^{\mathrm{2}}\gamma +\frac{\mathrm{1}}{\mathrm{2}}\gamma a^{\mathrm{2}}$(65)

and by equations (38) and (39), we get

$-\tau a^{\mathrm{1}}-\tau a^{\mathrm{2}}\gamma +{\lambda }^{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{2}}\gamma a^{\mathrm{2}}=\mathrm{0}$(66)

$G^{\mathrm{1}}-\tau a^{\mathrm{2}}-{\lambda }^{\mathrm{1}}-\frac{\mathrm{1}}{\mathrm{2}}\gamma a^{\mathrm{1}}=\mathrm{0}$(67)

${\mathrm{e}}^{\gamma \left(t-t_{\mathrm{0}}\right)}\left(a^{\mathrm{2}}{G}^{\mathrm{1}}-B\tau -\mathrm{\Lambda }\right)={\mathrm{e}}^{\gamma \left(t-t_{\mathrm{0}}\right)}\left(\frac{\mathrm{1}}{\mathrm{2}}\gamma a^{\mathrm{1}}{a}^{\mathrm{2}}-\gamma z\right)$(68)