© Wuhan University 2025

0 Introduction

The system of differential equations $\mathrm{d}\mathit{x}/\mathrm{d}t=-\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{d}V\left(\mathit{x}\right)$ is called the gradient system, where the vector field $\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{d}V=\left(\partial V/\partial x_{\mathrm{1}},\partial V/\partial x_{\mathrm{2}},\cdots ,\partial V/\partial x_n\right)$ is the gradient of $V$. Hirsch et al ^[1] studied the gradient system and its properties and gave three important propositions about equilibrium point and stability. If a mechanical system can be transformed into a gradient system, then the characteristics of gradient system can be used to study the properties of mechanical system, especially the kinematic stability^[2]. Mei and Wu^[3-4] divided gradient systems into two categories: one is those whose equations do not explicitly contain time, including general gradient systems, skew-gradient systems, gradient systems with symmetric negative definite matrices, gradient systems with semi-negative definite matrices, and combined gradient systems; the other class is those whose equations are explicitly time-dependent, called generalized gradient systems. In their monographs^[3-4], Mei and Wu comprehensively discussed the gradient representation of constrained mechanical systems, transformed various constrained mechanical systems into various gradient systems, and then studied the integrals and the stability of their solutions by using the properties of gradient systems. Chen et al^[5] studied the generalized gradient representation of nonholonomic systems, gradient systems with negative definite asymmetric matrices and generalized Birkhoff systems^[6], triple combined gradient systems^[7], etc. Tarasov^[8-9], Mei et al^[10], Lou and Mei^[11], Wang and Qi^[12-13] studied the fractional-order gradient representation of mechanical systems. Wu and Mei^[14], Wang and Bao^[15] studied the gradient system in event space. In the past decade, many excellent results have been published in the discussion of gradient systems and constrained mechanical systems^[16-20]. Because the gradient system has a lot of good properties, it is very suitable to study with Lyapunov function, so the study of mechanical systems and gradient systems is still an open and important subject.

Herglotz's principle is a generalization of Hamilton's principle to non-conservative systems, which provides an alternative way to model non-conservative problems^[21-24]. Based on the Herglotz principle, Donchev^[25] and Lazo et al^[26-27] studied the variational description of some non-conservative equations, such as electron beam propagation equation, nonlinear Schrodinger equation, nonlinear dissipative oscillators, strings vibrating under viscous forces, etc. Recently, we have applied Herglotz's principle to establish Herglotz-type Lagrange equations^[28-29], Hamilton equations^[30-32], Birkhoff equations^[33-35] and so on for non-conservative mechanical systems. In this paper, we combine Herglotz-type equations with gradient systems to explore the gradientization of Herglotz-type equations, and then study the stability of solution of Herglotz equations.

1 Herglotz-Type Equations

Consider a system of particles described with generalized coordinates $q_s\left(s=\mathrm{1,2},\cdots ,n\right)$, the Lagrange equations of Herglotz-type are^[24]

$\frac{\partial \mathrm{ℒ}}{\partial q_s}-\frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial \mathrm{ℒ}}{\partial {\dot{q}}_s}+\frac{\partial \mathrm{ℒ}}{\partial z}\frac{\partial \mathrm{ℒ}}{\partial {\dot{q}}_s}=\mathrm{0} , s=\mathrm{1,2},\cdots ,n,$(1)

where $\mathrm{ℒ}=\mathrm{ℒ}\left(q_s,{\dot{q}}_s,z\right)$ is the Lagrangian function of Herglotz-type. $z$ is the Hamilton-Herglotz action determined by

$\frac{\mathrm{d}z}{\mathrm{d}t}=\mathrm{ℒ}\left(q_s,{\dot{q}}_s,z\right).$(2)

Define

$\mathrm{\mathscr{H}}\left(q_s,p_s,z\right)=p_s{\dot{q}}_s-\mathrm{ℒ}\left(q_s,{\dot{q}}_s,z\right),p_s=\frac{\partial \mathrm{ℒ}}{\partial {\dot{q}}_s}$(3)

as the Hamiltonian function of Herglotz-type, and the generalized momentum. By using the $\mathrm{\mathscr{H}}$ function, equation (1) becomes^[30]

${\dot{q}}_s=\frac{\partial \mathrm{\mathscr{H}}}{\partial p_s}, {\dot{p}}_s=-\frac{\partial \mathrm{\mathscr{H}}}{\partial q_s}-p_s\frac{\partial \mathrm{\mathscr{H}}}{\partial z}, s=\mathrm{1,2},\cdots ,n.$(4)

equation (4) is the Hamilton equation of Herglotz-type for non-conservative systems.

Let $a^s=q_s$, $a^{n+s}=p_s$, then equation (4) can be expressed as

${\dot{a}}^{\mu }={\mathrm{\Omega }}^{\mu \nu }\frac{\partial \mathrm{\mathscr{H}}}{\partial a^{\nu }}+{\mathrm{\Pi }}_{\mu } , \mu ,\nu =\mathrm{1,2},\cdots ,\mathrm{2}n,$(5)

where

$\left({\mathrm{\Omega }}^{\mu \nu }\right)=\left(\begin{array}{cc}{\mathrm{0}}_{n\times n}& {\mathrm{1}}_{n\times n}\\ -{\mathrm{1}}_{n\times n}& {\mathrm{0}}_{n\times n}\end{array}\right),$(6)

${\mathrm{\Pi }}_s=\mathrm{0},{\mathrm{\Pi }}_{n+s}=-a^{n+s}\frac{\partial \mathrm{\mathscr{H}}}{\partial z}.$(7)

2 Basic Gradient Systems

Let there be a function $V\left(\mathit{x}\right)$, so as to

${\dot{x}}_s=-\frac{\partial V\left(\mathit{x}\right)}{\partial x_s}, s=\mathrm{1,2},\cdots ,n.$(8)

Eq.(8) are then called the general gradient system^[1], where $V$ is called the potential function, $\mathit{x}=\left(x_{\mathrm{1}},x_{\mathrm{2}},\cdots ,x_n\right)$.

Let there be a function $V\left(\mathit{x}\right)$, so as to

${\dot{x}}_s={\beta }_{sk}\left(\mathit{x}\right)\frac{\partial V\left(\mathit{x}\right)}{\partial x_k}, s,k=\mathrm{1,2},\cdots ,n,$(9)

where $\left({\beta }_{sk}\right)=\left({\beta }_{sk}\left(\mathit{x}\right)\right)$ is an antisymmetric matrix, i.e.,

${\beta }_{sk}\left(\mathit{x}\right)=-{\beta }_{ks}\left(\mathit{x}\right).$(10)

Eq.(9) are then called the skew-gradient system^[3].

Let there be a function $V\left(\mathit{x}\right)$, so as to

${\dot{x}}_s={\gamma }_{sk}\left(\mathit{x}\right)\frac{\partial V\left(\mathit{x}\right)}{\partial x_k} , s,k=\mathrm{1,2},\cdots ,n,$(11)

where the matrix $\left({\gamma }_{sk}\right)=\left({\gamma }_{sk}\left(\mathit{x}\right)\right)$ is symmetric negative definite, then Eq.(11) are called the gradient system with a symmetric negative definite matrix^[3].

Let there be a function $V\left(\mathit{x}\right)$, so as to

${\dot{x}}_s={\vartheta }_{sk}\left(\mathit{x}\right)\frac{\partial V\left(\mathit{x}\right)}{\partial x_k} , s,k=\mathrm{1,2},\cdots ,n,$(12)

where the matrix $\left({\vartheta }_{sk}\right)=\left({\vartheta }_{sk}\left(\mathit{x}\right)\right)$ is semi-negative definite, then Eq.(12) are called the gradient system with a semi-negative definite matrix^[3].

For the sake of discussion, the above four kinds of gradient systems may be called basic gradient systems. If the equations of constrained mechanical systems can be expressed in the form of Eqs.(8), (9), (11) or (12), then we can analyze the stability of solutions of the systems and study their integrals according to the properties of basic gradient systems.

3 Gradientization of Herglotz-Type Equations

Herglotz-type equations may not necessarily be reduced to the four basic gradient systems mentioned above. The conditions for gradientization of Herglotz-type equations are given below.

Comparing Eqs. (5) and (8), if the condition

$\frac{\partial }{\partial a^{\rho }}\left({\mathrm{\Omega }}^{\mu \nu }\frac{\partial \mathrm{\mathscr{H}}}{\partial a^{\nu }}+{\mathrm{\Pi }}_{\mu }\right)-\frac{\partial }{\partial a^{\mu }}\left({\mathrm{\Omega }}^{\rho \nu }\frac{\partial \mathrm{\mathscr{H}}}{\partial a^{\nu }}+{\mathrm{\Pi }}_{\rho }\right)=\mathrm{0}$(13)

is satisfied, then there is a potential function $V\left(\mathit{a}\right)$ that makes the equations

$-\frac{\partial V}{\partial a^{\mu }}={\mathrm{\Omega }}^{\mu \nu }\frac{\partial \mathrm{\mathscr{H}}}{\partial a^{\nu }}+{\mathrm{\Pi }}_{\mu } , \mu ,\nu =\mathrm{1,2},\cdots ,\mathrm{2}n$(14)

valid, then Eq.(5) can be reduced to the first kind of basic gradient system (8), where $\mathit{a}=\left(a^{\mathrm{1}},a^{\mathrm{2}},\cdots ,a^{\mathrm{2}n}\right)$.

Comparing Eqs.(5) and (9), if there exist $\left({\beta }_{\mu \rho }\left(\mathit{a}\right)\right)$ and $V\left(\mathit{a}\right)$, such that the equations

${\beta }_{\mu \rho }\frac{\partial V}{\partial a^{\rho }}={\mathrm{\Omega }}^{\mu \nu }\frac{\partial \mathrm{\mathscr{H}}}{\partial a^{\nu }}+{\mathrm{\Pi }}_{\mu } , \mu ,\nu ,\rho =\mathrm{1,2},\cdots ,\mathrm{2}n$(15)

are satisfied, then Eq.(5) can be reduced to the second kind of basic gradient system (9).

Comparing Eqs.(5) and (11), if there exist $\left({\gamma }_{\mu \rho }\left(\mathit{a}\right)\right)$ and $V\left(\mathit{a}\right)$, such that the equations

${\gamma }_{\mu \rho }\frac{\partial V}{\partial a^{\rho }}={\mathrm{\Omega }}^{\mu \nu }\frac{\partial \mathrm{\mathscr{H}}}{\partial a^{\nu }}+{\mathrm{\Pi }}_{\mu } , \mu ,\nu ,\rho =\mathrm{1,2},\cdots ,\mathrm{2}n$(16)

are satisfied, then Eq.(5) can be reduced to the third kind of basic gradient system (11).

Comparing Eqs.(5) and (12), if there exist $\left({\vartheta }_{\mu \rho }\left(\mathit{a}\right)\right)$ and $V\left(\mathit{a}\right)$, such that the equations

${\vartheta }_{\mu \rho }\frac{\partial V}{\partial a^{\rho }}={\mathrm{\Omega }}^{\mu \nu }\frac{\partial \mathrm{\mathscr{H}}}{\partial a^{\nu }}+{\mathrm{\Pi }}_{\mu } , \mu ,\nu ,\rho =\mathrm{1,2},\cdots ,\mathrm{2}n$(17)

are satisfied, then Eq.(5) can be reduced to the fourth basic gradient system (12).

It should be noted that conditions (13), (15), (16) and (17) are sufficient. If it is not satisfied, it cannot be determined that it is not a gradient system. In fact, this is related to the form of first-order equations resulting from Herglotz-type equations.

4 Stability of the Solution of Herglotz-Type Equations

Using the properties of gradient systems, we can explore the integrals of Herglotz-type equations and the stability of their solutions, with the following results:

If the Herglotz-type equations (5) can be reduced to the first kind of basic gradient system (8), then the function $V$ is a first integral of equations (5), which can be taken as the Lyapunov function. Let

$\frac{\partial V}{\partial a^{\mu }}=\mathrm{0} , \mu =\mathrm{1,2},\cdots ,\mathrm{2}n.$(18)

Solving the equations (18), we get the solution as

$a^{\mu }=a_{\mathrm{0}}^{\mu } , \mu =\mathrm{1,2},\cdots ,\mathrm{2}n.$(19)

If $V$ is positive definite, then the solution (19) is asymptotically stable. If the function $V$ cannot be taken as the Lyapunov function, we can linearize the equations (5) to find their eigenroots, if there are all negative real roots, then the solution (19) is asymptotically stable; If there are positive real roots, then the solution (19) is unstable.

If the Herglotz-type equations (5) can be reduced to the second kind of basic gradient system (9), then the potential function $V$ is the first integral. Let

${\beta }_{\mu \rho }\frac{\partial V}{\partial a^{\rho }}=\mathrm{0} , \mu ,\rho =\mathrm{1,2},\cdots ,\mathrm{2}n.$(20)

Solving the equations (20), we get the solution as

$a^{\mu }=a_{\mathrm{0}}^{\mu } , \mu =\mathrm{1,2},\cdots ,\mathrm{2}n.$(21)

Take $V$ as the Lyapunov function, and if $V$ is positive definite, the solution (21) is stable. If $V$ cannot be taken as a Lyapunov function, then the stability of solution (21) can be analyzed by using Lyapunov's first-order approximation theory, similar to the case mentioned above.

If the Herglotz-type equations (5) can be reduced to the third basic gradient system (11), let

${\gamma }_{\mu \rho }\frac{\partial V}{\partial a^{\rho }}=\mathrm{0} , \mu ,\rho =\mathrm{1,2},\cdots ,\mathrm{2}n.$(22)

Solving the equations (22), we get the solution as

$a^{\mu }=a_{\mathrm{0}}^{\mu } , \mu =\mathrm{1,2},\cdots ,\mathrm{2}n.$(23)

In the neighborhood of solution (23), if the potential function $V$ is positive definite, then the solution is asymptotically stable.

If the Herglotz-type equations (5) can be reduced to the fourth basic gradient system (12), let

${\vartheta }_{\mu \rho }\frac{\partial V}{\partial a^{\rho }}=\mathrm{0} , \mu ,\rho =\mathrm{1,2},\cdots ,\mathrm{2}n.$(24)

Solving the equations (24), we get the solution as

$a^{\mu }=a_{\mathrm{0}}^{\mu } , \mu =\mathrm{1,2},\cdots ,\mathrm{2}n.$(25)

In the neighborhood of the solution (25), if the potential function $V$ is positive definite, then the solution is stable.

5 Application of Results

Example 1 Suppose that the Lagrangian function of Herglotz-type is

$\mathrm{ℒ}=\frac{\mathrm{1}}{\mathrm{2}}{\dot{q}}^{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{2}}{q}^{\mathrm{2}}-cz,$(26)

where $\frac{\mathrm{d}z}{\mathrm{d}t}=\mathrm{ℒ}$, and the constant $c>\mathrm{0}$, we try to transform it into the first kind of basic gradient system and analyze its stability.

Using formulas (3), the Hamiltonian function and generalized momentum of Herglotz-type are

$\mathrm{\mathscr{H}}=\frac{\mathrm{1}}{\mathrm{2}}{p}^{\mathrm{2}}-\frac{\mathrm{1}}{\mathrm{2}}{q}^{\mathrm{2}}+cz,p=\frac{\partial \mathrm{ℒ}}{\partial \dot{q}}=\dot{q}.$(27)

From Eq.(4), the Herglotz-type equations of the system are

$\dot{q}=p,\dot{p}=q-cp.$(28)

Let $a^{\mathrm{1}}=q$, $a^{\mathrm{2}}=p$, then equations (28) can be expressed as

${\dot{a}}^{\mathrm{1}}=a^{\mathrm{2}}, {\dot{a}}^{\mathrm{2}}=a^{\mathrm{1}}-c{a}^{\mathrm{2}}.$(29)

It is easy to verify that condition (13) is satisfied, so equations (29) can be reduced to the first kind of gradient system. Let

$-\frac{\partial V}{\partial a^{\mathrm{1}}}=a^{\mathrm{2}}, -\frac{\partial V}{\partial a^{\mathrm{2}}}=a^{\mathrm{1}}-c{a}^{\mathrm{2}}.$(30)

Solveing this equation, we can get

$V=-a^{\mathrm{1}}{a}^{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{2}}c{\left(a^{\mathrm{2}}\right)}^{\mathrm{2}}.$(31)

This is the potential function of the system. Let

$\frac{\partial V}{\partial a^{\mathrm{1}}}=\mathrm{0}, \frac{\partial V}{\partial a^{\mathrm{2}}}=\mathrm{0}.$(32)

The equilibrium position is solved as

$a_{\mathrm{0}}^{\mathrm{1}}=\mathrm{0}, a_{\mathrm{0}}^{\mathrm{2}}=\mathrm{0}.$(33)

Obviously, the potential function (31) cannot be taken as the Lyapunov function, and the characteristic equation of equations (29) is

$\left|\begin{array}{cc}\lambda & -\mathrm{1}\\ -\mathrm{1}& \lambda +c\end{array}\right|=\mathrm{0},$(34)

i.e.,

${\lambda }^{\mathrm{2}}+c\lambda -\mathrm{1}=\mathrm{0}.$(35)

Equation (35) has the solutions

$\lambda =\frac{-c\pm \sqrt[]{c^{\mathrm{2}}+\mathrm{4}}}{\mathrm{2}},$(36)

Since there are positive real roots, the solution (33) is unstable.

Example 2 Let the Lagrangian function of Herglotz-type be

$\mathrm{ℒ}=\frac{\mathrm{1}}{\mathrm{2}}{\dot{q}}^{\mathrm{2}}-\frac{\mathrm{1}}{\mathrm{2}}{q}^{\mathrm{2}}{\left(\mathrm{1}+q^{\mathrm{2}}\right)}^{\mathrm{2}}+\frac{\mathrm{4}q\dot{q}}{\mathrm{1}+q^{\mathrm{2}}}z.$(37)

The Herglotz-Hamilton action $z$ is defined as

$\frac{\mathrm{d}z}{\mathrm{d}t}=\frac{\mathrm{1}}{\mathrm{2}}{\dot{q}}^{\mathrm{2}}-\frac{\mathrm{1}}{\mathrm{2}}{q}^{\mathrm{2}}{\left(\mathrm{1}+q^{\mathrm{2}}\right)}^{\mathrm{2}}+\frac{\mathrm{4}q\dot{q}}{\mathrm{1}+q^{\mathrm{2}}}z.$(38)

Try to transform it into the second kind of basic gradient system and study its stability.

From formulas (3), we obtain

$\mathrm{\mathscr{H}}=\frac{\mathrm{1}}{\mathrm{2}}{\left(p-\frac{\mathrm{4}qz}{\mathrm{1}+q^{\mathrm{2}}}\right)}^{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{2}}{q}^{\mathrm{2}}{\left(\mathrm{1}+q^{\mathrm{2}}\right)}^{\mathrm{2}},p=\frac{\partial \mathrm{ℒ}}{\partial \dot{q}}=\dot{q}+\frac{\mathrm{4}qz}{\mathrm{1}+q^{\mathrm{2}}}.$(39)

Thus, the Hamilton equations of Herglotz-type are

$\dot{q}=p-\frac{\mathrm{4}qz}{\mathrm{1}+q^{\mathrm{2}}},\dot{p}=\frac{\mathrm{4}}{\mathrm{1}+q^{\mathrm{2}}}\left(p-\frac{\mathrm{4}qz}{\mathrm{1}+q^{\mathrm{2}}}\right)\left[z\left(\mathrm{1}-q^{\mathrm{2}}\right)+qp\right]-\left(\mathrm{1}+q^{\mathrm{2}}\right)\left(\mathrm{1}+\mathrm{3}{q}^{\mathrm{2}}\right)q.$(40)

Let

$a^{\mathrm{1}}=q,a^{\mathrm{2}}=\frac{\mathrm{1}}{\mathrm{1}+q^{\mathrm{2}}}\left(p-\frac{\mathrm{4}qz}{\mathrm{1}+q^{\mathrm{2}}}\right).$(41)

Then Eq.(40) can be rewritten as

${\dot{a}}^{\mathrm{1}}=a^{\mathrm{2}}\left[\mathrm{1}+{\left(a^{\mathrm{1}}\right)}^{\mathrm{2}}\right],{\dot{a}}^{\mathrm{2}}=-a^{\mathrm{1}}\left[\mathrm{1}+{\left(a^{\mathrm{1}}\right)}^{\mathrm{2}}\right].$(42)

According to condition (15), we have

$\left(\begin{array}{cc}\mathrm{0}& {\beta }_{\mathrm{12}}\\ {\beta }_{\mathrm{21}}& \mathrm{0}\end{array}\right)\left(\begin{array}{c}\frac{\partial V}{\partial a^{\mathrm{1}}}\\ \frac{\partial V}{\partial a^{\mathrm{2}}}\end{array}\right)=\left(\begin{array}{c}{a}^{\mathrm{2}}\left[\mathrm{1}+{\left(a^{\mathrm{1}}\right)}^{\mathrm{2}}\right]\\ -a^{\mathrm{1}}\left[\mathrm{1}+{\left(a^{\mathrm{1}}\right)}^{\mathrm{2}}\right]\end{array}\right),$(43)

where ${\beta }_{\mathrm{21}}=-{\beta }_{\mathrm{12}}$. Eq.(43) has a solution

${\beta }_{\mathrm{12}}=\mathrm{1}+{\left(a^{\mathrm{1}}\right)}^{\mathrm{2}},V=\frac{\mathrm{1}}{\mathrm{2}}{\left(a^{\mathrm{1}}\right)}^{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{2}}{\left(a^{\mathrm{2}}\right)}^{\mathrm{2}}.$(44)

Therefore, this belongs to the second kind of basic gradient system (9), namely, skew-gradient system. It is easy to verify that $V$ is the first integral. From

${\beta }_{\mathrm{12}}\frac{\partial V}{\partial a^{\mathrm{2}}}=\mathrm{0}, {\beta }_{\mathrm{21}}\frac{\partial V}{\partial a^{\mathrm{1}}}=\mathrm{0},$(45)

the equilibrium position can be solved as

$a_{\mathrm{0}}^{\mathrm{1}}=\mathrm{0}, a_{\mathrm{0}}^{\mathrm{2}}=\mathrm{0}.$(46)

Since $V$ is positive definite, the solution (46) is stable.

Example 3 The Lagrangian function of Herglotz- type for non-conservative two-degree-of-freedom systems is

$\mathrm{ℒ}=\frac{\mathrm{1}}{\mathrm{2}}\left({\dot{q}}_{\mathrm{1}}^{\mathrm{2}}+{\dot{q}}_{\mathrm{2}}^{\mathrm{2}}\right)-\frac{\mathrm{1}}{\mathrm{2}}\left(q_{\mathrm{1}}^{\mathrm{2}}+q_{\mathrm{2}}^{\mathrm{2}}\right)-\mathrm{3}z,$(47)

where $z$ is determined by

$\frac{\mathrm{d}z}{\mathrm{d}t}=\frac{\mathrm{1}}{\mathrm{2}}\left({\dot{q}}_{\mathrm{1}}^{\mathrm{2}}+{\dot{q}}_{\mathrm{2}}^{\mathrm{2}}\right)-\frac{\mathrm{1}}{\mathrm{2}}\left(q_{\mathrm{1}}^{\mathrm{2}}+q_{\mathrm{2}}^{\mathrm{2}}\right)-\mathrm{3}z.$(48)

Try to transform it into the third kind of basic gradient system and study the stability of the solution.

The Hamiltonian function of Herglotz-type and its generalized momentum are

$\mathrm{\mathscr{H}}=\frac{\mathrm{1}}{\mathrm{2}}\left(p_{\mathrm{1}}^{\mathrm{2}}+p_{\mathrm{2}}^{\mathrm{2}}\right)+\frac{\mathrm{1}}{\mathrm{2}}\left(q_{\mathrm{1}}^{\mathrm{2}}+q_{\mathrm{2}}^{\mathrm{2}}\right)+\mathrm{3}z,p_{\mathrm{1}}=\frac{\partial \mathrm{ℒ}}{\partial {\dot{q}}_{\mathrm{1}}}={\dot{q}}_{\mathrm{1}}, p_{\mathrm{2}}=\frac{\partial \mathrm{ℒ}}{\partial {\dot{q}}_{\mathrm{2}}}={\dot{q}}_{\mathrm{2}}.$(49)

The Herglotz-type equations of the system are

${\dot{q}}_{\mathrm{1}}=p_{\mathrm{1}},{\dot{q}}_{\mathrm{2}}=p_{\mathrm{2}},{\dot{p}}_{\mathrm{1}}=-q_{\mathrm{1}}-\mathrm{3}{p}_{\mathrm{1}},{\dot{p}}_{\mathrm{2}}=-q_{\mathrm{2}}-\mathrm{3}{p}_{\mathrm{2}}.$(50)

Let

$a^{\mathrm{1}}=q_{\mathrm{1}},a^{\mathrm{2}}=q_{\mathrm{2}},a^{\mathrm{3}}=q_{\mathrm{1}}+p_{\mathrm{1}},a^{\mathrm{4}}=q_{\mathrm{2}}+p_{\mathrm{2}}.$(51)

Then Eq. (50) can be rewritten as

${\dot{a}}^{\mathrm{1}}=-a^{\mathrm{1}}+a^{\mathrm{3}},{\dot{a}}^{\mathrm{2}}=-a^{\mathrm{2}}+a^{\mathrm{4}},{\dot{a}}^{\mathrm{3}}=a^{\mathrm{1}}-\mathrm{2}{a}^{\mathrm{3}},{\dot{a}}^{\mathrm{4}}=a^{\mathrm{2}}-\mathrm{2}{a}^{\mathrm{4}}.$(52)

Take

$V=\frac{\mathrm{1}}{\mathrm{2}}{\left(a^{\mathrm{1}}\right)}^{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{2}}{\left(a^{\mathrm{2}}\right)}^{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{2}}{\left(a^{\mathrm{3}}\right)}^{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{2}}{\left(a^{\mathrm{4}}\right)}^{\mathrm{2}},$(53)

$\left({\gamma }_{\mu \rho }\right)=\left(\begin{array}{cccc}-\mathrm{1}& \mathrm{0}& \mathrm{1}& \mathrm{0}\\ \mathrm{0}& -\mathrm{1}& \mathrm{0}& \mathrm{1}\\ \mathrm{1}& \mathrm{0}& -\mathrm{2}& \mathrm{0}\\ \mathrm{0}& \mathrm{1}& \mathrm{0}& -\mathrm{2}\end{array}\right),$(54)

then the condition (16) is satisfied, and Eq.(52) can be reduced to the third kind of basic gradient system (11). From Eq.(22), the equilibrium position can be solved as

$a_{\mathrm{0}}^{\mathrm{1}}=a_{\mathrm{0}}^{\mathrm{2}}=a_{\mathrm{0}}^{\mathrm{3}}=a_{\mathrm{0}}^{\mathrm{4}}=\mathrm{0}.$(55)

Since $V$ is positive definite, the solution (55) is asymptotically stable.

Example 4 The Lagrangian function of Herglotztype for a non-conservative single-degree-of-freedom system is

$\mathrm{ℒ}=\frac{\mathrm{1}}{\mathrm{2}}m{\dot{q}}^{\mathrm{2}}-\frac{\mathrm{1}}{\mathrm{2}}k{q}^{\mathrm{2}}\mathrm{e}\mathrm{x}\mathrm{p}q-cz,$(56)

where $m, k, c$ are positive constants, and $z$ is defined as

$\frac{\mathrm{d}z}{\mathrm{d}t}=\frac{\mathrm{1}}{\mathrm{2}}m{\dot{q}}^{\mathrm{2}}-\frac{\mathrm{1}}{\mathrm{2}}k{q}^{\mathrm{2}}\mathrm{e}\mathrm{x}\mathrm{p}q-cz.$(57)

Try to transform it into the fourth kind of basic gradient system and study its stability.

According to formulas (3), we have

$\mathrm{\mathscr{H}}=\frac{\mathrm{1}}{\mathrm{2}m}{p}^{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{2}}k{q}^{\mathrm{2}}\mathrm{e}\mathrm{x}\mathrm{p}q+cz,p=\frac{\partial \mathrm{ℒ}}{\partial \dot{q}}=m\dot{q}.$(58)

Thus, Eq.(4) give

$\dot{q}=\frac{p}{m},\dot{p}=-kq\mathrm{e}\mathrm{x}\mathrm{p}q-\frac{\mathrm{1}}{\mathrm{2}}k{q}^{\mathrm{2}}\mathrm{e}\mathrm{x}\mathrm{p}q-cp.$(59)

Let $a^{\mathrm{1}}=q$, $a^{\mathrm{2}}=p$, then Eq.(59) can be rewritten as

${\dot{a}}^{\mathrm{1}}=\frac{a^{\mathrm{2}}}{m},{\dot{a}}^{\mathrm{2}}=-k{a}^{\mathrm{1}}\mathrm{e}\mathrm{x}\mathrm{p}{a}^{\mathrm{1}}-\frac{\mathrm{1}}{\mathrm{2}}k{\left(a^{\mathrm{1}}\right)}^{\mathrm{2}}\mathrm{e}\mathrm{x}\mathrm{p}{a}^{\mathrm{1}}-c{a}^{\mathrm{2}}.$(60)

If we take

$V=\frac{\mathrm{1}}{\mathrm{2}}{\left(a^{\mathrm{1}}\right)}^{\mathrm{2}}\mathrm{e}\mathrm{x}\mathrm{p}{a}^{\mathrm{1}}+\frac{\mathrm{1}}{\mathrm{2}}{\left(a^{\mathrm{2}}\right)}^{\mathrm{2}},$(61)

then Eq. (60) can be expressed as

$\left(\begin{array}{c}{\dot{a}}^{\mathrm{1}}\\ {\dot{a}}^{\mathrm{2}}\end{array}\right)=\left(\begin{array}{cc}\mathrm{0}& \frac{\mathrm{1}}{m}\\ -k& -c\end{array}\right)\left(\begin{array}{c}\frac{\partial V}{\partial a^{\mathrm{1}}}\\ \frac{\partial V}{\partial a^{\mathrm{2}}}\end{array}\right),$(62)

where the matrix

$\left({\vartheta }_{\mu \rho }\right)=\left(\begin{array}{cc}\mathrm{0}& \frac{\mathrm{1}}{m}\\ -k& -c\end{array}\right)$(63)

is a semi-negative definite matrix. Therefore, Eq.(60) can be reduced to a fourth kind of basic gradient system. According to the equations (24), the equilibrium position can be solved as

$a_{\mathrm{0}}^{\mathrm{1}}=\mathrm{0}, a_{\mathrm{0}}^{\mathrm{2}}=\mathrm{0}.$(64)

Since $V$ is positive definite, the solution (64) is stable.

6 Conclusion

The gradient system has many good properties and is especially suitable for studying with Lyapunov function. If a constrained mechanical system can be reduced to a gradient system, then its dynamical behavior can be analyzed by using the characteristics of the gradient system. The novelty of this paper is that the gradientization of Herglotz-type equations for non-conservative systems are proposed and studied, and then the stability of their solutions is analyzed. The conditions and forms of transformation of Herglotz-type equations into four kinds of basic gradient systems are given. Four examples are given to demonstrate the calculation process for the four kinds of cases, and the results verify the feasibility of the proposed method. Future studies may consider the gradientization of Herglotz equations for nonholonomic systems or Birkhoff systems, and generalized gradientization for the systems with rheonomic constraints.

References