Lie Symmetries and Invariants of General Time-Dependent Quadratic Hamiltonian System

Abstract

Eight Lie algebras of point-symmetric groups and corresponding generators are admitted by the equation of motion, which is obtained from a general time-dependent quadratic Hamiltonian. We show that invariant quantities obtained by eight algebraic generators are the Wronskian constant, three conserved quantities, which are time-dependent quadratic forms in position and momentum, and trivial, 0. All obtained invariant quantities are represented by auxiliary conditions, which are two linearly independent solutions of a homogeneous differential equation of the equations of motion. Invariant variables associated with an invariant consisting of the linearity of x and p are defined. It shows that, if the motion of the system is oscillatory, the Poisson bracket of the two invariant variables is obtained as i, and in the case of monotonic motion, it is obtained as 1.

1. Introduction

The Hamiltonian formalism not only enables us to obtain the system’s equations of motion, but also allows us to have knowledge of various mathematical structures related to the system [1,2,3]. Lie algebras facilitate the solution of differential equations when the symmetry of the system can be identified [4,5,6].

In general, one Hamiltonian gives one equation of motion, but one equation of motion is obtained from innumerable Hamiltonians. This is because there are infinitely many canonical pairs in which the position is fixed in a pair of canonical variables, and only the momentum is changed. Since one Hamiltonian in one dimension gives one pair of canonical variables, there are infinitely many Hamiltonians that give one equation of motion. In this paper, we select and deal with classical Hamiltonian systems in which the coefficients are time-dependent and a general quadratic form of position and momentum. This Hamiltonian system includes many systems that are basically handled, such as free particles, harmonic oscillators, damped harmonic oscillators, and forced harmonic oscillators [1].

We find the generators that form the Lie algebra admitted by the equation of motion, and find out whether the generators have an invariant or not. If the generator has the invariant, we will find out what the invariant is.

In Section 2, we choose a general time-dependent Hamiltonian system and find solutions of the equations of motion. Since the coefficients of the Hamiltonian are not fixed, in our expansion, the general solution is determined by assuming a possible solution.

Section 3 finds the invariant of the system. As we know, if we solve a linear differential equation, we can find the Wronskian, which is obtained by multiplying the coefficients of $p^2$ of the given Hamiltonian by the Wronskian constant. We also find that two invariant quantities given by a linear combination of position and momentum depend on the solution of the system as an auxiliary condition. Since the product of an invariant is also an invariant, we use these to find three invariant quantities given by the quadratic form of position and momentum.

In Section 4, we find eight generators of the point-symmetric Lie groups admitted by the equation of motion, which is a homogeneous differential equation. In these eight generators, we check whether the variation integral of the system is invariant or not. If it is invariant, we find its conserved quantities. We also examine how the conserved quantity is related to the invariant discussed in Section 3.

Section 5 defines two linear invariant variables by the conserved quantities obtained in Section 3. We examine the transformation between the set of two linear invariant variables and the set of position and momentum variables. Also, we find the Poisson brackets of the two invariant variables. It is also shown that, when two variables form a canonical transformation, the transformation becomes a Hamilton–Jacobi transformation. Poisson brackets between two linear invariant variables are calculated. It also calculates Poisson brackets for quadratic variables consisting of two linear invariant variables.

Section 6 shows the summary and conclusions, as well as a brief remark on the possibility of the extension of the techniques to quantum mechanics.

2. Time-Dependent General Quadratic Hamiltonian Systems

First, we briefly introduce the equations of motion of the time-dependent general quadratic Hamiltonian system and its solutions. The Hamiltonian of the system can be set as

$$H(x,p;t)={\displaystyle \frac{1}{2}}\left(\alpha \left(t\right)p^2+2\beta \left(t\right)xp+\zeta \left(t\right)x^2\right)+\mu \left(t\right)p+\nu \left(t\right)x+\sigma \left(t\right),$$

(1)

where the coefficients $\alpha \left(t\right),\beta \left(t\right),\zeta \left(t\right),\mu \left(t\right),\nu \left(t\right)$ and $\sigma \left(t\right)$ are some real functions of t and are differentiable. From now on, in all equations, the expression of the time function will be omitted to simplify the expression, except when it appears for the first time, in which case the time-dependence will be emphasized, and in special cases where omission may cause confusion. If the Hamilton equation is solved using the Hamiltonian of Equation (1), the equation of motion for the system is obtained as

$$\ddot{x}-{\displaystyle \frac{\dot{\alpha }}{\alpha }}\dot{x}+\left({\displaystyle \frac{\dot{\alpha }\beta }{\alpha }}-{\beta }^2+\alpha \delta -\dot{\beta }\right)x=\beta \mu -{\displaystyle \frac{\mu }{\alpha }}\dot{\alpha }+\dot{\mu }-\alpha \nu ,$$

(2)

where the dot represents time derivatives. From Equation (1), it can be seen that $\sigma \left(t\right)$ does not affect the equation of motion (2). Let the equation of motion (2) be simply written as

$$\ddot{x}+\gamma \left(t\right)\dot{x}+{\omega }^2\left(t\right)x=f\left(t\right),$$

(3)

$${\omega }^2\left(t\right)={\displaystyle \frac{\dot{\alpha }\beta }{\alpha }}-{\beta }^2+\alpha \delta -\dot{\beta },$$

(4)

$$f\left(t\right)=\beta \mu -{\displaystyle \frac{\mu }{\alpha }}\dot{\alpha }+\dot{\mu }-\alpha \nu .$$

(5)

As can be seen from Equations (1) and (2), there can be innumerable Hamiltonians giving one equation of motion. Among them, we will take and treat

$$H(x,p;t)={\displaystyle \frac{1}{2}}\left(\alpha \left(t\right)p^2+{\alpha }^{-1}\left(t\right){\omega }^2\left(t\right)x^2\right)-{\alpha }^{-1}\left(t\right)f\left(t\right)x,$$

(6)

as a special selection of Hamiltonian of Equation (1), giving the equation of motion (3). In Equation (6), $\alpha \left(t\right)$ is

$$\alpha \left(t\right)=e^{-{\mathit{\int }}^t\gamma \left(s\right)ds},$$

(7)

that is,

$$\gamma \left(t\right)=-{\displaystyle \frac{\dot{\alpha }\left(t\right)}{\alpha \left(t\right)}}.$$

(8)

Solving the Hamilton equation using Equation (6) gives

$$\dot{x}=\alpha \left(t\right)p,$$

(9)

and

$$\dot{p}=-{\alpha }^{-1}\left(t\right){\omega }^2\left(t\right)x+{\alpha }^{-1}\left(t\right)f\left(t\right).$$

(10)

Using Equations (6), (9) and (10), the Lagrangian of the system is obtained as

$$L(\dot{x},x;t)={\displaystyle \frac{{\dot{x}}^2}{2\alpha \left(t\right)}}-{\displaystyle \frac{{\omega }^2\left(t\right)}{2\alpha \left(t\right)}}{x}^2+{\displaystyle \frac{f\left(t\right)}{\alpha \left(t\right)}}x.$$

(11)

If $u_1\left(t\right)$ and $u_2\left(t\right)$ are the two independent solutions of the homogeneous form (in the case of $f\left(t\right)=0$) of the equation of motion Equation (3), the general solution of the differential Equation (3) is obtained as

$$x\left(t\right)=c_1{u}_1\left(t\right)+c_2{u}_2\left(t\right)+x_p\left(t\right),$$

(12)

where

$$x_p\left(t\right)={\mathit{\int }}_{t_0}^{t}G(t,t^{\prime })f\left(t^{\prime }\right)d{t}^{\prime },$$

(13)

$c_1$ and $c_2$ are arbitrary constants, and $t_0$ is the time when $x_p\left(t_0\right)=0$ at $t=t_0$. In Equation (13), the Green’s function, $G(t,t^{\prime })$, is

$$G(t,t^{\prime })=-\theta (t-t^{\prime }){\displaystyle \frac{u_1\left(t\right)u_2\left(t^{\prime }\right)-u_2\left(t\right)u_1\left(t^{\prime }\right)}{W\left(t^{\prime }\right)}},$$

(14)

where $\theta \left(t\right)$ is a Heaviside step function, and the Wronskian, $W\left(t\right)$ is

$$W\left(t\right)=u_1\left(t\right)\dot{u_2}\left(t\right)-u_2\left(t\right)\dot{u_1}\left(t\right)=W_0\alpha \left(t\right).$$

(15)

Here, we will call the constant $W_0$ the Wronskian constant. The velocity $\dot{x}\left(t\right)$ is obtained as

$$\dot{x}\left(t\right)=c_1\dot{u_1}\left(t\right)+c_2\dot{u_2}\left(t\right)+{\dot{x}}_p\left(t\right),$$

(16)

from Equations (12) and (13), where

$${\dot{x}}_p\left(t\right)={\mathit{\int }}_{t_0}^t{G}_t(t,t^{\prime })f\left(t^{\prime }\right)d{t}^{\prime },$$

(17)

here, $G_t(t,t^{\prime })$ is

$$G_t(t,t^{\prime })=-\theta (t-t^{\prime }){\displaystyle \frac{{\dot{u}}_1\left(t\right)u_2\left(t^{\prime }\right)-{\dot{u}}_2\left(t\right)u_1\left(t^{\prime }\right)}{W\left(t^{\prime }\right)}}.$$

(18)

3. Invariant Quantities of the System

So far, in the general solution of the motion of the system, Equations (12) and (16), we know that there are time constants $c_1$ and $c_2$, and the Wronskian constant $W_0$. Considering Equations (9)–(15), we get

$${\alpha }^{-1}\left(t\right)W\left(t\right)=u_1\left(t\right)p_2\left(t\right)-u_2\left(t\right)p_1\left(t\right)=W_0.$$

(19)

If $W_0$ is an imaginary number, two independent solutions of the differential equation can be taken as $u_1\left(t\right)$ and $u_2\left(t\right)=u_1^{\ast }\left(t\right)$. If $W_0$ is a real number, we can take two independent solutions, $u_1\left(t\right)$ and $u_2\left(t\right)$, as real functions of t. In a conservative system, the mechanical energy E equals the Hamiltonian H and is a constant of time, but in a time-dependent system, it is not a constant of time. Let us find the new time constant of the time-dependent Hamiltonian system here. In general, if the quantity $I(x,p;t)$ satisfies

$$D\left[I\right(x,p;t\left)\right]=0,$$

(20)

it is a constant of time and is called an invariant quantity, where D is the total derivative operator,

$$D={\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }t}}+\dot{x}{\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }x}}+\ddot{x}{\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }\dot{x}}}+\dots .$$

(21)

If $I(x,p;t)$ is given by

$$I(x,p;t)=a\left(t\right)p+b\left(t\right)x+g\left(t\right),$$

(22)

which is a linear relation of x and p, substituting Equations (21) and (22) into Equation (20) and calculating, we can know that $a\left(t\right),b\left(t\right)$, and $g\left(t\right)$ in Equation (22) are satisfied

$$\dot{a}=-\alpha \left(t\right)b,$$

(23)

$$\dot{b}={\alpha }^{-1}\left(t\right)\zeta \left(t\right)a,$$

(24)

and

$$\dot{g}=-{\alpha }^{-1}f\left(t\right)a.$$

(25)

The homogeneous form (when $f\left(t\right)=0$ ) of the equation of motion (3) is

$$\ddot{x}+\gamma \left(t\right)\dot{x}+{\omega }^2\left(t\right)x=0.$$

(26)

In Equation (12), we assume that $u_1\left(t\right)$ and $u_2\left(t\right)$ are two linearly independent solutions of Equation (26). Then, solutions of Equations (23)–(25) are respectively obtained as

$$a\left(t\right)=c_1{u}_1+c_2{u}_2,$$

(27)

$$b\left(t\right)=-{\displaystyle \frac{1}{\alpha \left(t\right)}}\left(c_1{\dot{u}}_1+c_2{\dot{u}}_2\right),$$

(28)

and

$$g\left(t\right)=-{\mathit{\int }}_{t_0}^t{\alpha }^{-1}\left(t^{\prime }\right)f\left(t^{\prime }\right)\left(c_1{u}_1\left(t^{\prime }\right)+c_2{u}_2\left(t^{\prime }\right)\right)d{t}^{\prime }.$$

(29)

Substituting Equations (27)–(29) into Equation (22), we get two independent invariant quantities

$$I_1(x,p;t)=u_{1}p-{\displaystyle \frac{{\dot{u}}_1}{\alpha }}x+{\mathit{\int }}_{t_0}^t{\displaystyle \frac{u_1\left(t^{\prime }\right)}{\alpha \left(t^{\prime }\right)}}f\left(t^{\prime }\right)d{t}^{\prime },$$

(30)

and

$$I_2(x,p;t)=u_{2}p-{\displaystyle \frac{{\dot{u}}_2}{\alpha }}x+{\mathit{\int }}_{t_0}^t{\displaystyle \frac{u_2\left(t^{\prime }\right)}{\alpha \left(t^{\prime }\right)}}f\left(t^{\prime }\right)d{t}^{\prime }.$$

(31)

Since the product of two invariant quantities is also invariant, the three quadratic invariant quantities can be obtained as follows.

$$\begin{array}{c}\hfill I_1^2(x,p;t)=u_1^2{p}^2-{\displaystyle \frac{2u_1{\dot{u}}_1}{\alpha }}xp+{\displaystyle \frac{{\dot{u}}_1^2}{{\alpha }^2}}{x}^2-2x{\mathit{\int }}_{t_0}^t{\displaystyle \frac{u_1\left(t^{\prime }\right){\dot{u}}_1\left(t\right)}{\alpha \left(t\right)\alpha \left(t^{\prime }\right)}}f\left(t^{\prime }\right)d{t}^{\prime }\\ \hfill +2p{\mathit{\int }}_{t_0}^t{\displaystyle \frac{u_1\left(t\right)u_1\left(t^{\prime }\right)}{\alpha \left(t^{\prime }\right)}}f\left(t^{\prime }\right)d{t}^{\prime }+{\left({\mathit{\int }}_{t_0}^t{\displaystyle \frac{u_1\left(t^{\prime }\right)}{\alpha \left(t^{\prime }\right)}}f\left(t^{\prime }\right)d{t}^{\prime }\right)}^2,\end{array}$$

(32)

$$\begin{array}{c}\hfill I_2^2(x,p;t)=u_2^2{p}^2-{\displaystyle \frac{2u_2{\dot{u}}_2}{\alpha }}xp+{\displaystyle \frac{{\dot{u}}_2^2}{{\alpha }^2}}{x}^2-2x{\mathit{\int }}_{t_0}^t{\displaystyle \frac{u_2\left(t^{\prime }\right){\dot{u}}_2\left(t\right)}{\alpha \left(t\right)\alpha \left(t^{\prime }\right)}}f\left(t^{\prime }\right)d{t}^{\prime }\\ \hfill +2p{\mathit{\int }}_{t_0}^t{\displaystyle \frac{u_2\left(t\right)u_2\left(t^{\prime }\right)}{\alpha \left(t^{\prime }\right)}}f\left(t^{\prime }\right)d{t}^{\prime }+{\left({\mathit{\int }}_{t_0}^t{\displaystyle \frac{u_2\left(t^{\prime }\right)}{\alpha \left(t^{\prime }\right)}}f\left(t^{\prime }\right)d{t}^{\prime }\right)}^2,\end{array}$$

(33)

and

$$\begin{array}{ccc}& & I_1(x,p;t)I_2(x,p;t)=u_1{u}_2{p}^2-{\displaystyle \frac{u_1{\dot{u}}_2+{\dot{u}}_1{u}_2}{\alpha }}xp+{\displaystyle \frac{{\dot{u}}_1{\dot{u}}_2}{{\alpha }^2}}{x}^2\hfill \\ & & -x{\mathit{\int }}_{t_0}^t{\displaystyle \frac{u_1\left(t^{\prime }\right){\dot{u}}_2\left(t\right)+u_2\left(t^{\prime }\right){\dot{u}}_1\left(t\right)}{\alpha \left(t\right)\alpha \left(t^{\prime }\right)}}f\left(t^{\prime }\right)d{t}^{\prime }+p{\mathit{\int }}_{t_0}^t{\displaystyle \frac{u_1\left(t\right)u_2\left(t^{\prime }\right)+u_2\left(t\right)u_1\left(t^{\prime }\right)}{\alpha \left(t^{\prime }\right)}}f\left(t^{\prime }\right)d{t}^{\prime }\hfill \\ & & +\left({\mathit{\int }}_{t_0}^t{\displaystyle \frac{u_2\left(t^{\prime }\right)}{\alpha \left(t^{\prime }\right)}}f\left(t^{\prime }\right)d{t}^{\prime }\right)\left({\mathit{\int }}_{t_0}^t{\displaystyle \frac{u_1\left(t^{\prime }\right)}{\alpha \left(t^{\prime }\right)}}f\left(t^{\prime }\right)d{t}^{\prime }\right),\hfill \end{array}$$

(34)

Let ${\omega }_0^2$ be

$${\omega }_0^2\left(t\right)={\omega }^2\left(t\right)-{\displaystyle \frac{1}{2}}{\gamma }^{\prime }\left(t\right)-{\displaystyle \frac{1}{4}}{\gamma }^2\left(t\right).$$

(35)

If ${\omega }_0^2>0$, the solutions $u_1$ and $u_2$ of Equation (26) are oscillatory motions, and if ${\omega }_0^2<0$, monotonically increasing or monotonically decreasing motions. In the case of monotonic motion, if two independent solutions, $u_1$ and $u_2$, are respectively set to

$$u_1\left(t\right)=r\left(t\right)e^{\theta \left(t\right)},$$

(36)

and

$$u_2\left(t\right)=r\left(t\right)e^{-\theta \left(t\right)},$$

(37)

the general solution of Equation (12) becomes

$$x\left(t\right)=c_{1}r\left(t\right)e^{\theta \left(t\right)}\left(t\right)+c_{2}r\left(t\right)e^{-\theta \left(t\right)}\left(t\right)+x_p\left(t\right),$$

(38)

where $r\left(t\right)$ and $\theta \left(t\right)$ are real function of t. Equation (38) is not a specific solution, but a general form, which is a function obtained by solving Equation (12) when it is fixed. Applying Wronskian, Equation (15) becomes

$$W\left(t\right)=2r^2\dot{\theta }={\mathrm{\Omega }}_0\alpha \left(t\right),$$

(39)

where ${\mathrm{\Omega }}_0$ is a real constant. In Equation (39), it can be seen that r and $\theta$ are dependent on each other. In the case of oscillatory motion, if two independent solutions, $u_1$ and $u_2$, are respectively set to

$$u_1\left(t\right)=r\left(t\right)e^{i\theta \left(t\right)}\left(t\right),$$

(40)

and

$$u_2\left(t\right)=r\left(t\right)e^{-i\theta \left(t\right)}\left(t\right),$$

(41)

the general solution of Equation (12) is

$$x\left(t\right)=c_{1}r\left(t\right)e^{i\theta \left(t\right)}\left(t\right)+c_{2}r\left(t\right)e^{-i\theta \left(t\right)}\left(t\right)+x_p\left(t\right),$$

(42)

Substituting Equations (40) and (41) into Equation (15), Wronskian (15) becomes

$$W\left(t\right)=2i{r}^2\dot{\theta }=i{\mathrm{\Omega }}_0\alpha \left(t\right),$$

(43)

where ${\mathrm{\Omega }}_0$ is a real constant. In Equation (43), r and $\theta$ are also depend on each other. In the case of monotonic motion, from Equations (36) and (37), the Green function, Equations (14) and (18) are expressed as

$$G(t,t^{\prime })=-\theta (t-t^{\prime }){\displaystyle \frac{2r{r}^{\prime }sinh(\theta -{\theta }^{\prime })}{{\mathrm{\Omega }}_0{\alpha }^{\prime }}},$$

(44)

and

$$G_t(t,t^{\prime })=-\theta (t-t^{\prime }){\displaystyle \frac{2\dot{r}{r}^{\prime }sinh(\theta -{\theta }^{\prime })+2r{r}^{\prime }\dot{\theta }cosh(\theta -{\theta }^{\prime })}{{\mathrm{\Omega }}_0{\alpha }^{\prime }}},$$

(45)

by r and $\theta$. In the case of oscillatory motion, from Equations (40) and (41), the Green function Equations (14) and (18) are expressed as

$$G(t,t^{\prime })=-\theta (t-t^{\prime }){\displaystyle \frac{2r{r}^{\prime }sin(\theta -{\theta }^{\prime })}{{\mathrm{\Omega }}_0{\alpha }^{\prime }}},$$

(46)

and

$$G_t(t,t^{\prime })=-\theta (t-t^{\prime }){\displaystyle \frac{2\dot{r}{r}^{\prime }sin(\theta -{\theta }^{\prime })-2r{r}^{\prime }\dot{\theta }cos(\theta -{\theta }^{\prime })}{{\mathrm{\Omega }}_0{\alpha }^{\prime }}},$$

(47)

by r and $\theta$. In expressions (44)–(47), all variables expressed as $\prime$ are a function of $t^{\prime }$ of that variable. In the case of oscillatory motion with the above considerations, expressing r and $\theta$, Equation (34) becomes

$$\begin{array}{ccc}\hfill I_1{I}_2& =& r^2{p}^2-{\displaystyle \frac{r\dot{r}}{\alpha }}xp+\left({\displaystyle \frac{{\dot{r}}^2}{{\alpha }^2}}+{\displaystyle \frac{{\mathrm{\Omega }}_0^2}{4r^2}}\right)x^2\hfill \\ & & -x{\mathit{\int }}_{t_0}^t{\displaystyle \frac{r^{\prime }\dot{r}cos({\theta }^{\prime }-\theta )-r^{\prime }r\dot{\theta }sin({\theta }^{\prime }-\theta )}{\alpha {\alpha }^{\prime }}}f\left(t^{\prime }\right)d{t}^{\prime }\hfill \\ & & +p{\mathit{\int }}_{t_0}^t{\displaystyle \frac{r{r}^{\prime }cos({\theta }^{\prime }-\theta )}{{\alpha }^{\prime }}}f\left(t^{\prime }\right)d{t}^{\prime }+{\mathit{\int }}_{t_0}^t{\mathit{\int }}_{t_0}^t{\displaystyle \frac{r^{\prime }{r}^{\prime \prime }{e}^{i({\theta }^{\prime }+{\theta }^{\prime \prime })}}{{\alpha }^{\prime }{\alpha }^{\prime \prime }}}f\left(t^{\prime }\right)f\left(t^{\prime \prime }\right)d{t}^{\prime }d{t}^{\prime \prime },\hfill \end{array}$$

(48)

In the case of monotonic motion, the result of Equation (48) is obtained by replacing $+{\mathrm{\Omega }}_0^2$ with $-{\mathrm{\Omega }}_0^2$ and trigonometric functions with corresponding hyperbolic functions. For $f\left(t\right)=0$, Equation (46) is the Lewis–Riesenfeld invariant that has been dealt with in the literature [7,8,9,10,11,12]. So far, we have seen that the invariant quantities $c_1,c_2,{\mathrm{\Omega }}_0,I_1^2,I_2^2$, and $I_1{I}_2$ are related to the equations of motion and their solutions. The next section will discuss the relationship between invariant quantities obtained from the equation of motion of the system and the Lie algebra that makes the system invariant. Since it is so complicated to find the Lie algebraic generators of a non-homogeneous equation, let us find the Lie generators of a homogeneous differential equation to get an invariant and discuss the non-homogeneous case.

4. Generators and Their Invariant Quantities

If $f\left(t\right)=0$, the equation of motion Equation (3) becomes a homogeneous form of Equation (26), and its general solution can be put

$$x\left(t\right)=c_1{u}_1\left(t\right)+c_2{u}_2\left(t\right),$$

(49)

in Equation (10). From now on, we want to obtain generators of continuous transformation groups admitted by the equation of motion (Equation (26) of the system) and to obtain their invariant solutions using the group’s point symmetries. First, let us find infinitesimal symmetries, i.e., generators.

$$X=\xi (t,x){\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }t}}+\eta (t,x){\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }x}},$$

(50)

of symmetry groups G. The infinitesimal invariant discrimination has the form:

$$X_{\left(2\right)}F{\mid }_{F=0}=(\xi F_t+\eta F_x+{\zeta }_1{F}_{\dot{x}}+{\zeta }_2{F}_{\ddot{x}}){\mid }_{F=0}=0,$$

(51)

where $F=0$ is the differential Equation (26), F is a function of the left side of Equation (26), subscripts indicate derivatives of its subscripts, and

$$X_{\left(2\right)}=\xi {\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }t}}+\eta {\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }x}}+{\zeta }_1{\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }\dot{x}}}+{\zeta }_2{\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }\ddot{x}}},$$

(52)

where ${\zeta }_1$ and ${\zeta }_2$ are

$${\zeta }_1={\eta }_t+({\eta }_x-{\xi }_t)\dot{x}-{\dot{x}}^2{\xi }_x,$$

(53)

and

$${\zeta }_2={\eta }_{tt}+({\eta }_{tx}-{\xi }_{tt})\dot{x}+({\eta }_{xx}-2{\xi }_{tx}){\dot{x}}^2-{\dot{x}}^3{\xi }_{xx}+({\eta }_x-2{\xi }_t-3\dot{x}{\xi }_x)\ddot{x}.$$

(54)

Equation (51) is called the determining equation for the group G admitted by the ordinary differential Equation (26) [4,5,6,13]. Using the determining Equation (51), we can find generators of symmetry groups of Equation (26). It is well known that the symmetric Lie algebra $\mathcal{L}$ of linear second-order differential Equation (26) has the dimension 8 [4,5,6]. Using Equations (51)–(54), we can know that the vector space of infinitesimal generators is eight-dimensional and a basis is obtained as [4,5,6,13].

$$\begin{array}{ccc}\hfill X_1& =& x{\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }x}},\hfill \end{array}$$

(55)

$$\begin{array}{ccc}\hfill X_2& =& u_1{\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }x}},\hfill \end{array}$$

(56)

$$\begin{array}{ccc}\hfill X_3& =& u_2{\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }x}},\hfill \end{array}$$

(57)

$$\begin{array}{ccc}\hfill X_4& =& {\alpha }^{-1}\left(u_{1}x{\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }t}}+{\dot{u}}_1{x}^2{\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }x}}\right),\hfill \end{array}$$

(58)

$$\begin{array}{ccc}\hfill X_5& =& {\alpha }^{-1}\left(u_{2}x{\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }t}}+{\dot{u}}_2{x}^2{\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }x}}\right),\hfill \end{array}$$

(59)

$$\begin{array}{ccc}\hfill X_6& =& {\alpha }^{-1}\left(u_1^2{\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }t}}+u_1{\dot{u}}_{1}x{\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }x}}\right),\hfill \end{array}$$

(60)

$$\begin{array}{ccc}\hfill X_7& =& {\alpha }^{-1}\left(2u_1{u}_2{\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }t}}+({\dot{u}}_1{u}_2+u_1{\dot{u}}_2)x{\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }x}}\right),\hfill \end{array}$$

(61)

and

$$X_8={\alpha }^{-1}\left(u_2^2{\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }t}}+u_2{\dot{u}}_{2}x{\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }x}}\right).$$

(62)

Every generator except $X_1$ depends upon independent solutions, $u_1$ and $u_2$ of the differential Equation (26). In the case of an independent variable t and a differential variable x, the Lagrangian variational integral $L(x,\dot{x};t)$ is defined as

$$J={\mathit{\int }}_{V}L(x,\dot{x};t)dt.$$

(63)

Now we want to check that the variational integral (63) has an invariant under each group G corresponding to eight infinitesimal generators (55)–(62). This is a point symmetry of the Euler–Lagrange equation [4,5,6]. Using Noether’s theorem to get the symmetry, we show that the variational integral is invariant under the group G if and only if

$$X\left(L\right)+LD\left(\xi \right)=0,$$

(64)

where X is Equation (52), i.e.,

$$X\left(L\right)=\xi {\displaystyle \frac{\mathit{\partial }L}{\mathit{\partial }t}}+\eta {\displaystyle \frac{\mathit{\partial }L}{\mathit{\partial }x}}+{\zeta }_1{\displaystyle \frac{\mathit{\partial }L}{\mathit{\partial }\dot{x}}}.$$

(65)

If the variational integral (63) is invariant in the group, with generator (65), then T is defined by

$$T=L\xi +(\eta -\xi \dot{x}){\displaystyle \frac{\mathit{\partial }L}{\mathit{\partial }\dot{x}}},$$

(66)

is a conserved quantity for the Euler–Lagrange equation,

$${\displaystyle \frac{\mathit{\partial }L}{\mathit{\partial }x}}-D{\displaystyle \frac{\mathit{\partial }L}{\mathit{\partial }\dot{x}}}=0.$$

(67)

That is,

$$D\left(T\right)=0,$$

(68)

with Equation (68) [4,5].

Adding $D\left[B(x.\dot{x};t)\right]$ to $L(x.\dot{x};t)$ gives the same result. Adding $B(x.\dot{x};t)$ to the Lagrangian of the integral of variation (63), it gives the same result of the variation principle, so Equation (64) can be replaced with

$$X\left(L\right)+LD\left(\xi \right)=D\left(B\right).$$

(69)

Then the Lagrange equation is again invariant and has a conservation law $D\left[I(x,\dot{x};t)\right]=0$, where Equation (66) is replaced by

$$I(x,\dot{x};t)=\xi L+(\eta -\xi \dot{x}){\displaystyle \frac{\mathit{\partial }L}{\mathit{\partial }\dot{x}}}-B,$$

(70)

The group G corresponding to Lie’s algebra, Equations (55)–(62), makes the differential Equation (26) invariant, but we do not know if it makes the variational integral invariant. We check whether the generator Equations (55)–(62) are symmetric under the variational integral, and if so, find out what the corresponding invariant is. First, to simply reduce the calculation of various formulas, the two Formulas (69) and (70) are calculated by substituting the Lagrangian Equation (11) as follows,

$$\begin{array}{ccc}\hfill X\left(L\right)+LD\left(\xi \right)& =& \gamma \left({\displaystyle \frac{{\dot{x}}^2}{2\alpha }}-{\displaystyle \frac{{\omega }^2\left(t\right)}{2\alpha }}{x}^2\right)\xi -\xi {\displaystyle \frac{\omega \dot{\omega }}{\alpha }}{x}^2-\eta {\displaystyle \frac{{\omega }^2}{\alpha }}x\hfill \\ & & +{\zeta }_1{\displaystyle \frac{\dot{x}}{\alpha }}+\left({\displaystyle \frac{{\dot{x}}^2}{2\alpha }}-{\displaystyle \frac{{\omega }^2\left(t\right)}{2\alpha }}{x}^2\right)({\xi }_t+{\xi }_x\dot{x})=D\left[B(x,\dot{x},t)\right],\hfill \end{array}$$

(71)

and

$$I(x,\dot{x};t)=-\xi \left({\displaystyle \frac{{\dot{x}}^2}{2\alpha }}-{\displaystyle \frac{{\omega }^2\left(t\right)}{2\alpha }}{x}^2\right)+\eta {\displaystyle \frac{\dot{x}}{\alpha }}-B(x,\dot{x};t).$$

(72)

In generator $X_1$, Equation (55), $\xi =0$ and $\eta =x$; hence, Equation (53) gives the following.

$${\zeta }_1=\dot{x}.$$

(73)

Calculation of Equation (71) is

$$X\left(L\right)+LD\left(\xi \right)={\dot{x}}^2-{\omega }^2\left(t\right)x^2=D\left(x\dot{x}\right),$$

(74)

And Equation (72) becomes

$$I(x,\dot{x};t)=\eta \dot{x}-B=x\dot{x}-x\dot{x}=0.$$

(75)

Thus, for generator $X_1$, we only get a trivial conservation quantity 0. This group $G_1$ tells us that, if x is the solution of Equation (26), x multiplied by an arbitrary constant is also a solution of it. That is, it is a group $G_1$ that admits in any linear differential equations. However, group $G_1$ does not admit in the non-homogeneous differential equation of Equation (3). In the generators, equation $X_2$ (54), $\xi =0$ and $\eta =u_1$; hence, Equation (53) becomes

$${\zeta }_1={\dot{u}}_1.$$

(76)

Calculation of Equation (71) is

$$X\left(L\right)+LD\left(\xi \right)=-u_1{\omega }^{2}x+{\dot{u}}_1\dot{x}=D\left({\dot{u}}_{1}x\right),$$

(77)

and the equation, Equation (72), becomes

$$I(x,\dot{x};t)=\eta \dot{x}-B=u_1\dot{x}-{\dot{u}}_{1}x.$$

(78)

By replacing $u_1$ with $u_2$ in results, Equations (76)–(78) in the case of generator $X_2$, we get corresponding results in the case of generator $X_3$, i.e.,

$${\zeta }_1={\dot{u}}_1,$$

(79)

$$X\left(L\right)+LD\left(\xi \right)=u_2{\omega }^{2}x+{\dot{u}}_2\dot{x}=D\left({\dot{u}}_{2}x\right),$$

(80)

and

$$I(x,\dot{x};t)=\eta \dot{x}-B=u_2\dot{x}-{\dot{u}}_{2}x.$$

(81)

In Equations (78) and (81), both conserved quantities are the Wronskian constant $W_0$ in Equation (15). Thus, it is known that $X_2$ and $X_3$ are generators that give the Wronskian constant $W_0$ of the solution of the equation of motion as an invariant. The group $G_2$($G_3$) associated with these $X_2$ ($X_3$) is the group admitted when x is a solution of a differential equation, and $x+c_1{u}_1\left(x\right)$ ($x+c_2{u}_2\left(x\right)$) is also a solution of the same differential equation. That is, $X_2$ and $X_3$ are generators of the group admitted by linear differential equations. Their group is also admitted by non-homogeneous linear differential Equation (3). In the generators $X_4$, Equation (58),

$$\xi (x,\dot{x};t)={\displaystyle \frac{1}{\alpha \left(t\right)}}{u}_1\left(t\right)x,$$

(82)

and

$$\eta (x,\dot{x};t)={\dot{u}}_1\left(t\right)x^2.$$

(83)

Hence, Equation (53) gives

$${\zeta }_1(x,\dot{x};t)=-{\displaystyle \frac{1}{\alpha }}{\omega }^2{u}_1{x}^2+{\displaystyle \frac{1}{\alpha }}({\dot{u}}_1-\gamma u_1)x\dot{x}-{\displaystyle \frac{u_1}{\alpha }}{\dot{x}}^2.$$

(84)

In this case, the calculation of Equation (71) can be obtained as follows:

$$\begin{array}{ccc}\hfill X\left(L\right)+LD\left(\xi \right)& =& {\displaystyle \frac{3}{2{\alpha }^2}}{\dot{u}}_{1}x{\dot{x}}^2+\left(-{\displaystyle \frac{\gamma {\omega }^2}{{\alpha }^2}}{u}_1-u_1{\displaystyle \frac{\omega \dot{\omega }}{{\alpha }^2}}-{\dot{u}}_1{\displaystyle \frac{3{\omega }^2}{2{\alpha }^2}}\right)x^3-{\displaystyle \frac{{\omega }^2{u}_1}{{\alpha }^2}}{\dot{x}}^2-{\displaystyle \frac{u_1}{2{\alpha }^2}}{\dot{x}}^3\hfill \\ & =& D\left(-{\displaystyle \frac{u_1}{2{\alpha }^2}}x{\dot{x}}^2+{\displaystyle \frac{{\dot{u}}_1}{{\alpha }^2}}{x}^2\dot{x}-{\displaystyle \frac{u_1}{2{\alpha }^2}}{\omega }^2{x}^3\right).\hfill \end{array}$$

(85)

Thus, Equation (72) becomes

$$I(x,\dot{x};t)=0.$$

(86)

By replacing $u_1$ with $u_2$ in results, Equations (84)–(86) in the case of generator $X_4$, we get corresponding results in the case of generator $X_5$, i.e.,

$${\zeta }_1(x,\dot{x};t)=-{\displaystyle \frac{1}{\alpha }}{\omega }^2{u}_2{x}^2+{\displaystyle \frac{1}{\alpha }}({\dot{u}}_2-\gamma u_2)x\dot{x}-{\displaystyle \frac{u_2}{\alpha }}{\dot{x}}^2,$$

(87)

$$X\left(L\right)+LD\left(\xi \right)=D\left[B(x,\dot{x};t)\right]=D\left(-{\displaystyle \frac{u_2}{2{\alpha }^2}}x{\dot{x}}^2+{\displaystyle \frac{{\dot{u}}_2}{{\alpha }^2}}{x}^2\dot{x}-{\displaystyle \frac{u_2}{2{\alpha }^2}}{\omega }^2{x}^3\right),$$

(88)

and

$$I(x,\dot{x};t)=-{\displaystyle \frac{u_2}{2{\alpha }^2}}x{\dot{x}}^2+{\displaystyle \frac{{\dot{u}}_2}{{\alpha }^2}}{x}^2\dot{x}-{\displaystyle \frac{u_2}{2{\alpha }^2}}{\omega }^2{x}^3-B(x,\dot{x};t)=0,$$

(89)

In Equations (86) and (89), both generators $X_4$ and $X_5$ give a trivial conservation quantity, 0. $X_6$, $X_7$, and $X_8$ are the generators that give us the invariant we are interested in. However, the three generators have a similar form, but are associated with two independent solutions of the equations of motion. That is, $X_6$ is composed of $u_1$, $X_7$ is composed of both $u_1$ and $u_2$, and $X_8$ is composed of $u_2$. We do not list the process of obtaining the results of three cases here, but the case of $X_7$ seems to be the most meaningful calculation, so only this case will be dealt with in more detail, and the cases of $X_6$ and $X_7$ will briefly state each result as in the previous case.

In generators $X_6$, Equation (60),

$$\xi (x,\dot{x};t)={\displaystyle \frac{1}{\alpha \left(t\right)}}{u}_1^2\left(t\right),$$

(90)

and

$$\eta (x,\dot{x};t)={\displaystyle \frac{1}{\alpha \left(t\right)}}{u}_1\left(t\right){\dot{u}}_1^2\left(t\right)x.$$

(91)

Hence, Equation (53) gives

$${\zeta }_1(x,\dot{x};t)={\displaystyle \frac{1}{\alpha }}\left[\left({\dot{u}}_1^2-{\omega }^2{u}_1^2\right)x-\left(u_1{\dot{u}}_1+\gamma u_1^2\right)\dot{x}\right].$$

(92)

The calculation process of Equation (69) in $X_6$ is somewhat laborious, but it is very similar to that in $X_7$, which will cover the process. In $X_6$ it can be obtained as follows: calculation of Equation (71) is

$$\begin{array}{ccc}\hfill X\left(L\right)+LD\left(\xi \right)& =& -u_1^2{\displaystyle \frac{\omega \dot{\omega }}{{\alpha }^2}}{x}^2-{\displaystyle \frac{{\omega }^2}{{\alpha }^2}}(2u_1{\dot{u}}_1+\gamma u_1^2)x^2+({\dot{u}}_1^2-{\omega }^2{u}_1^2){\displaystyle \frac{x\dot{x}}{{\alpha }^2}}\hfill \\ & =& D\left(-{\displaystyle \frac{1}{{\alpha }^2}}{u}_1^2{\dot{x}}^2+{\displaystyle \frac{2}{{\alpha }^2}}{u}_1{\dot{u}}_{1}x\dot{x}-{\displaystyle \frac{1}{2{\alpha }^2}}({\omega }^2{u}_1^2+{\dot{u}}_1^2)x^2\right)\hfill \\ & =& D\left[(B(x,\dot{x};t)\right].\hfill \end{array}$$

(93)

Thus, Equation (72) becomes

$$\begin{array}{ccc}\hfill I(x,\dot{x};t)& =& -\xi \left({\displaystyle \frac{{\dot{x}}^2}{2\alpha }}+{\displaystyle \frac{{\omega }^2}{2\alpha }}{x}^2\right)+\eta {\displaystyle \frac{\dot{x}}{\alpha }}-B(x,\dot{x};t)\hfill \\ & =& {\displaystyle \frac{1}{2{\alpha }^2}}{u}_1^2{\dot{x}}^2-{\displaystyle \frac{1}{{\alpha }^2}}{u}_1{\dot{u}}_{1}x\dot{x}+{\displaystyle \frac{1}{2{\alpha }^2}}{\dot{u}}_1^2{x}^2\hfill \\ & =& {\displaystyle \frac{I_1^2}{2}},\hfill \end{array}$$

(94)

where $I_1$ is Equation (30). Equation (94) is an invariant, Equation (32). Now, in the case of $X_7$, which we are interested in, we will cover the process in some detail. In these cases,

$$\xi (x,\dot{x};t)={\displaystyle \frac{2}{\alpha \left(t\right)}}{u}_1\left(t\right)u_2\left(t\right),$$

(95)

and

$$\eta (x,\dot{x};t)={\displaystyle \frac{1}{\alpha \left(t\right)}}\left(u_1\left(t\right){\dot{u}}_2\left(t\right)+{\dot{u}}_1\left(t\right)u_2\left(t\right)\right)x.$$

(96)

Hence, Equation (53) gives

$${\zeta }_1(x,\dot{x};t)={\displaystyle \frac{1}{\alpha }}\left[2\left({\dot{u}}_1{\dot{u}}_2-{\omega }^2{u}_1{u}_2\right)x-\left(u_1{\dot{u}}_2+{\dot{u}}_1{u}_2+2\gamma u_1{u}_2\right)\dot{x}\right],$$

(97)

Substituting Equations (95)–(97) into Equation (71) and rearranging, we get

$$X\left(L\right)+LD\left(\xi \right)=-2u_1{u}_2{\displaystyle \frac{\omega \dot{\omega }}{{\alpha }^2}}{x}^2-2{\displaystyle \frac{{\omega }^2}{{\alpha }^2}}(u_1{\dot{u}}_2+{\dot{u}}_1{u}_2+\gamma u_1{u}_2)x^2+{\displaystyle \frac{2}{{\alpha }^2}}({\dot{u}}_1{\dot{u}}_2-{\omega }^2{u}_1{u}_2)x\dot{x}.$$

(98)

We do not know whether Equation (98) can be expressed in the form $D\left(B\right)$ or not. Since Equation (98) is a quadratic expression of both x and $\dot{x}$, let us assume that it has the form $D\left(B\right)$ and B is a quadratic form of x and $\dot{x}$. And calculating and rearranging $D\left(B\right)$ gives

$$\begin{array}{ccc}\hfill D\left(B\right)& =& D\left(f{\dot{x}}^2+gx\dot{x}+h{x}^2\right)\hfill \\ \hfill & =& (\dot{f}-2f\gamma +g){\dot{x}}^2+(-2f{\omega }^2+\dot{g}-\gamma g+2h)x\dot{x}+(\dot{h}-{\omega }^{2}g)x^2.\hfill \end{array}$$

(99)

In Equation (99), f, g, and h are time-dependent functions and functions to be obtained. To find it, we compare Equations (98) and (99), and if we set the coefficients of each quadratic form to be equal, we get the following differential equations:

$$\left\{\begin{array}{c}\dot{f}-2f\gamma +g=0,\hfill \\ \dot{g}-\gamma g-2f{\omega }^2+2h={\displaystyle \frac{1}{{\alpha }^2}}(2{\dot{u}}_1{\dot{u}}_2-2{\omega }^2{u}_1{u}_2),\hfill \\ \dot{h}-{\omega }^{2}g=-{\displaystyle \frac{2}{{\alpha }^2}}{u}_1{u}_2\omega \dot{\omega }-{\displaystyle \frac{2{\omega }^2}{{\alpha }^2}}(u_1{\dot{u}}_2+{\dot{u}}_1{u}_2)-{\displaystyle \frac{2\gamma {\omega }^2}{{\alpha }^2}}{u}_1{u}_2.\hfill \end{array}\right.$$

(100)

The solution of Equation (100) is obtained as

$$\left\{\begin{array}{c}g\left(t\right)={\displaystyle \frac{2}{{\alpha }^2\left(t\right)}}(2u_1\left(t\right){\dot{u}}_2\left(t\right)+{\dot{u}}_1\left(t\right)u_2\left(t\right)),\hfill \\ f\left(t\right)=-{\displaystyle \frac{2}{{\alpha }^2\left(t\right)}}{u}_1\left(t\right)u_2\left(t\right),\hfill \\ h\left(t\right)=-{\displaystyle \frac{{\omega }^2\left(t\right)}{{\alpha }^2\left(t\right)}}{u}_1\left(t\right)u_2\left(t\right)-{\displaystyle \frac{1}{{\alpha }^2\left(t\right)}}{\dot{u}}_1\left(t\right){\dot{u}}_2\left(t\right).\hfill \end{array}\right.$$

(101)

Thus, with Equation (101), Equation (99) becomes

$$D\left(B\right)=D\left[-{\displaystyle \frac{2}{{\alpha }^2}}{u}_1{u}_2{\dot{x}}^2+{\displaystyle \frac{2}{{\alpha }^2}}(u_1{\dot{u}}_2+{\dot{u}}_1{u}_2)x\dot{x}-{\displaystyle \frac{1}{{\alpha }^2}}({\omega }^2{u}_1{u}_2+{\dot{u}}_1{\dot{u}}_2)x^2\right].$$

(102)

From Equation (102), Equation (98) becomes

$$XL+LD\left(\xi \right)=D\left[-{\displaystyle \frac{2}{{\alpha }^2}}{u}_1{u}_2{\dot{x}}^2+{\displaystyle \frac{2}{{\alpha }^2}}(u_1{\dot{u}}_2+{\dot{u}}_1{u}_2)x\dot{x}-{\displaystyle \frac{1}{{\alpha }^2}}({\omega }^2{u}_1{u}_2+{\dot{u}}_1{\dot{u}}_2)x^2\right].$$

(103)

Thus, Equation (72) is obtained as

$$\begin{array}{ccc}\hfill I(x,\dot{x};t)& =& -\xi \left({\displaystyle \frac{{\dot{x}}^2}{2\alpha }}+{\displaystyle \frac{{\omega }^2}{2\alpha }}{x}^2\right)+\eta {\displaystyle \frac{\dot{x}}{\alpha }}-B\hfill \\ & =& {\displaystyle \frac{1}{{\alpha }^2}}{u}_1{u}_2{\dot{x}}^2-{\displaystyle \frac{1}{{\alpha }^2}}(u_1{\dot{u}}_2+{\dot{u}}_1{u}_2)x\dot{x}+{\displaystyle \frac{1}{{\alpha }^2}}\left({\dot{u}}_1{\dot{u}}_2\right)x^2=I_1{I}_2.\hfill \end{array}$$

(104)

Equation (104) is the same invariant quantity as the result of setting $f\left(t\right)=0$ in Equation (34). In the case of $X_8$, the procedure for finding the result is the same as in the case of $X_7$, and the results are the same as replacing $u_1$ with $u_2$ in the results of $X_6$. In $X_8$, Equation (62),

$$\xi (x,\dot{x};t)={\displaystyle \frac{1}{\alpha \left(t\right)}}{u}_1^2\left(t\right),$$

(105)

and

$$\eta (x,\dot{x};t)={\displaystyle \frac{1}{\alpha \left(t\right)}}{u}_1\left(t\right){\dot{u}}_2\left(t\right)x.$$

(106)

Hence, Equation (53) gives

$${\zeta }_1(x,\dot{x};t)={\displaystyle \frac{1}{\alpha \left(t\right)}}\left[({\dot{u}}_2^2-{\omega }^2{u}_2^2)x-(u_2{\dot{u}}_2+\gamma u_2^2)\dot{x}\right].$$

(107)

Equation (93) gives

$$\begin{array}{ccc}\hfill X\left(L\right)+LD\left(\xi \right)& =& -u_2^2{\displaystyle \frac{\omega \dot{\omega }}{{\alpha }^2}}{x}^2-{\displaystyle \frac{{\omega }^2}{{\alpha }^2}}(2u_2{\dot{u}}_2+\gamma u_2^2)x^2+({\dot{u}}_2^2-{\omega }^2{u}_2^2){\displaystyle \frac{x\dot{x}}{{\alpha }^2}}\hfill \\ & =& D\left(-{\displaystyle \frac{1}{{\alpha }^2}}{u}_2^2{\dot{x}}^2+{\displaystyle \frac{2}{{\alpha }^2}}{u}_2{\dot{u}}_{2}x\dot{x}-{\displaystyle \frac{1}{2{\alpha }^2}}({\omega }^2{u}_2^2+{\dot{u}}_2^2)x^2\right)\hfill \\ & =& D\left[B(x,\dot{x};t)\right].\hfill \end{array}$$

(108)

Thus, Equation (72) is obtained as

$$\begin{array}{ccc}\hfill I(x,\dot{x};t)& =& -\xi \left({\displaystyle \frac{{\dot{x}}^2}{2\alpha }}+{\displaystyle \frac{{\omega }^2}{2\alpha }}{x}^2\right)+\eta {\displaystyle \frac{\dot{x}}{\alpha }}-B\hfill \\ & =& {\displaystyle \frac{1}{2{\alpha }^2}}{u}_2^2{\dot{x}}^2-{\displaystyle \frac{1}{{\alpha }^2}}{u}_2{\dot{u}}_{2}x\dot{x}+{\displaystyle \frac{1}{2{\alpha }^2}}{\dot{u}}_2^2{x}^2=I_2/2.\hfill \end{array}$$

(109)

where $I_2$ is Equation (31). Equation (109) is an invariant, Equation (34). Summarizing the above results, there are eight generators corresponding to the groups that admit the equation of motion, Equation (26), which is a second-order linear differential equation. There exists a $B(x,\dot{x};t)$ that satisfies Equation (93) in all eight generators. Thus, the variational integral formed by the Lagrangian of a system is invariant for each group corresponding to eight generators (55)–(62). The invariants obtained from each generator are the Wronskian constant ${\mathrm{\Omega }}_0$, the quadratic invariant $I_1^2,I_1{I}_2,I_2^2$ for the equation of motion (26), and the trivial invariant 0. We discuss, in this section, the invariants in groups and generators admitted by the homogeneous second-order linear differential Equation (39) for $f\left(t\right)=0$. For non-homogeneous differential equations, Equation (3) ($f\left(t\right)\ne 0$), finding admitted generators is too complicated. In the case of a relatively simple example of Equation (3), for a forced free particle whose equation of motion is

$$\ddot{x}=f\left(t\right),$$

(110)

Calculating the generators shows that all eight generators depend on $f\left(t\right)$. Although generators and invariants cannot be directly dealt with in the case of non-homogeneous differential equations, Equation (3), they can be predicted from the results of homogeneous differential equations. $X_1$ is a generator that does not depend on the solution of the system. It is obtained from any linear differential equation. In the case of the non-homogeneous differential equation, Equation (3), it is a generator obtained by replacing x with $x-x_p$, where the particular solution, $x_p$, is Equation (13). The generators $X_2$ and $X_3$ are the same for both homogeneous and non-homogeneous forms in differential equations (Equation (3)). This means that adding the general solution of a homogeneous differential equation to the solution of a non-homogeneous differential equation results in a general solution of the non-homogeneous differential equation. The invariant is the Wronskian constant ${\mathrm{\Omega }}_0$ in both homogeneous and non-homogeneous cases. The results of $X_4$ and $X_5$ depend on the cube of the two independent solutions $u_1$ and $u_2$ of the system. The invariant obtained a trivial 0. In the case of inhomogeneity, we can only predict that it will depend on the cube of $u_1$, $u_2$, and f. The results of $X_6$, $X_7$, and $X_8$ are related to the square of $u_1$ and $u_2$, and the invariant shows that there exists a quadratic invariant of x and $\dot{x}$. In the non-homogeneous case, we can predict that the invariants of the generator associated with $X_6$, $X_7$, and $X_8$ will be Equations (32), (33) and (34), respectively.

5. Invariant Variable Space

We regard invariant quantities (30) and (31) as new variables and examine the system in the new variable space. We also cover variable transformations between the two spaces. If Equation (26) has a solution of the oscillatory motion, two independent solutions of Equation (26) can be set as $u_1\left(t\right)$ and $u_2\left(t\right)=u_1^{\ast }\left(t\right)$. If so, Equation (15) becomes

$$W\left(t\right)=u_1\left(t\right){\dot{u}}_1^{\ast }\left(t\right)-u_1^{\ast }\left(t\right){\dot{u}}_1\left(t\right)=i{\mathrm{\Omega }}_0\alpha \left(t\right),$$

(111)

where we can put ${\mathrm{\Omega }}_0>0$. (If ${\mathrm{\Omega }}_0<0$, we can select $u_1\left(t\right)$ and $u_2\left(t\right)=u_1^{\ast }\left(t\right)$ interchangeably.) In this case, the new variable associated with the invariant that can be inferred from Equations (30) and (31) can be defined as follows.

$$I_1={\displaystyle \frac{1}{\sqrt{{\mathrm{\Omega }}_0}}}\left(u_1\left(t\right)p-{\displaystyle \frac{{\dot{u}}_1\left(t\right)}{\alpha \left(t\right)}}x+g_1\left(t\right)\right),$$

(112)

and

$$I_2={\displaystyle \frac{1}{\sqrt{{\mathrm{\Omega }}_0}}}\left(u_1^{\ast }\left(t\right)p-{\displaystyle \frac{{\dot{u}}_1^{\ast }\left(t\right)}{\alpha \left(t\right)}}x+g_1^{\ast }\left(t\right)\right)=I_1^{\ast },$$

(113)

where

$$g_1\left(t\right)={\mathit{\int }}_{t_0}^t{\displaystyle \frac{u_1\left(t^{\prime }\right)}{\alpha \left(t^{\prime }\right)}}f\left(t^{\prime }\right)d{t}^{\prime }.$$

(114)

Calculating the Poisson square brackets of $I_1$ of Equation (112) and $I_2$ of Equation (113) gives

$$\{I_1,I_2\}={\displaystyle \frac{\mathit{\partial }{I}_1}{\mathit{\partial }x}}{\displaystyle \frac{\mathit{\partial }{I}_2}{\mathit{\partial }p}}-{\displaystyle \frac{\mathit{\partial }{I}_1}{\mathit{\partial }p}}{\displaystyle \frac{\mathit{\partial }{I}_2}{\mathit{\partial }x}}=i$$

(115)

Thus, the transformation consisting of Equations (112) and (113) is not canonical. Let two independent solutions of Equation (26), $u_1$ and $u_2$, be in monotonic motion. If so, $u_1$ and $u_2$ can be chosen so that the Wronskian Equation (15) can be put as

$$W\left(t\right)=u_1\left(t\right){\dot{u}}_1\left(t\right)-u_2\left(t\right){\dot{u}}_2\left(t\right)={\mathrm{\Omega }}_0\alpha \left(t\right),$$

(116)

where ${\mathrm{\Omega }}_0>0$. In this case, the new variable associated with the invariant that can be inferred from Equations (30) and (31) can be defined as

$$I_1={\displaystyle \frac{1}{\sqrt{{\mathrm{\Omega }}_0}}}\left(u_1\left(t\right)p-{\displaystyle \frac{{\dot{u}}_1\left(t\right)}{\alpha \left(t\right)}}x+g_1\left(t\right)\right),$$

(117)

and

$$I_2={\displaystyle \frac{1}{\sqrt{{\mathrm{\Omega }}_0}}}\left(u_2\left(t\right)p-{\displaystyle \frac{{\dot{u}}_2\left(t\right)}{\alpha \left(t\right)}}x+g_2\left(t\right)\right),$$

(118)

where

$$g_2\left(t\right)={\mathit{\int }}_{t_0}^t{\displaystyle \frac{u_2\left(t^{\prime }\right)}{\alpha \left(t^{\prime }\right)}}f\left(t^{\prime }\right)d{t}^{\prime }.$$

(119)

Here, $I_1^{\ast }\ne I_2$. Calculating the Poisson square brackets of $I_1$ and $I_2$ gives

$$\{I_1,I_2\}={\displaystyle \frac{\mathit{\partial }{I}_1}{\mathit{\partial }x}}{\displaystyle \frac{\mathit{\partial }{I}_2}{\mathit{\partial }p}}-{\displaystyle \frac{\mathit{\partial }{I}_1}{\mathit{\partial }p}}{\displaystyle \frac{\mathit{\partial }{I}_2}{\mathit{\partial }x}}=1$$

(120)

Therefore, the transformation of monotonic motion consisting of Equations (118) and (119) is a canonical transformation. Even in the case of oscillatory motion, if the variables are set to $I_1/\sqrt{i}$ and $I_2/\sqrt{i}$, respectively, the transformation consisting of Equations (112) and (113) is a canonical transformation. In this case, of course ${(I_1/\sqrt{i})}^{\ast }\ne I_2/\sqrt{i}$. If the canonical variables x and p of a system whose Hamiltonian is Equation (6), $H(x,p;t)$, and the new canonical variables $I_1$ and $I_2$ are connected by a point transformation, the Hamiltonian, $K(I_1,I_2;t)$, giving $I_1$ and $I_2$ is

$$K(I_1,I_2;t)=H(x,p;t)-p{\displaystyle \frac{\mathit{\partial }x}{\mathit{\partial }t}}-{\displaystyle \frac{\mathit{\partial }F}{\mathit{\partial }t}}.$$

(121)

Here, the generating function $F(I_1,I_2;t)$ can be obtained from the following relational expression [2].

$$\begin{array}{ccc}\hfill I_2-p{\displaystyle \frac{\mathit{\partial }x}{\mathit{\partial }{I}_1}}& =& {\displaystyle \frac{\mathit{\partial }F}{\mathit{\partial }{I}_1}},\hfill \end{array}$$

(122)

$$\begin{array}{ccc}\hfill -p{\displaystyle \frac{\mathit{\partial }x}{\mathit{\partial }{I}_1}}& =& {\displaystyle \frac{\mathit{\partial }F}{\mathit{\partial }{I}_2}}.\hfill \end{array}$$

(123)

Using Equations (117), (118), (122) and (123), Equation (121) is obtained by

$$K(I_1,I_2;t)=0.$$

(124)

Equation (124) means that the new variables $I_1$ and $I_2$ can be obtained from the calculation results of the Hamilton–Jacobi theory. The inverse transformation of Equations (117) and (118) is

$$\begin{array}{ccc}\hfill x& =& {\displaystyle \frac{1}{{\mathrm{\Omega }}_0}}(u_2{I}_1-u_1{I}_2)+x_p,\hfill \end{array}$$

(125)

$$\begin{array}{ccc}\hfill p& =& {\displaystyle \frac{1}{\alpha {\mathrm{\Omega }}_0}}({\dot{u}}_2{I}_1-{\dot{u}}_1{I}_2)+{\displaystyle \frac{{\dot{x}}_p}{\alpha }},\hfill \end{array}$$

(126)

where $x_p$ is Equation (13) and ${\dot{x}}_p$ is Equation (17). The inverse transformation of Equations (112) and (113) is

$$\begin{array}{ccc}\hfill x& =& {\displaystyle \frac{1}{i{\mathrm{\Omega }}_0}}(u_1^{\ast }{I}_1-u_1{I}_2)+x_p,\hfill \end{array}$$

(127)

$$\begin{array}{ccc}\hfill p& =& {\displaystyle \frac{1}{i\alpha {\mathrm{\Omega }}_0}}({\dot{u}}_1^{\ast }{I}_1-{\dot{u}}_1{I}_2)+{\displaystyle \frac{{\dot{x}}_p}{\alpha }}.\hfill \end{array}$$

(128)

Simplifying Equations (32)–(34) with Equations (114) and (119), we get

$$I_1^2=u_1^2{p}^2-{\displaystyle \frac{2u_1{\dot{u}}_1}{\alpha }}xp+{\displaystyle \frac{{\dot{u}}_1^2}{{\alpha }^2}}{x}^2-{\displaystyle \frac{2}{\alpha }}{\dot{u}}_1{g}_1\left(t\right)x+2u_1{g}_1\left(t\right)p+g_1^2\left(t\right),$$

(129)

$$I_2^2=u_2^2{p}^2-{\displaystyle \frac{2u_2{\dot{u}}_2}{\alpha }}xp+{\displaystyle \frac{{\dot{u}}_2^2}{{\alpha }^2}}{x}^2-{\displaystyle \frac{2}{\alpha }}{\dot{u}}_2{g}_2\left(t\right)x+2u_2{g}_2\left(t\right)p+g_2^2\left(t\right),$$

(130)

and

$$\begin{array}{ccc}\hfill I_1{I}_2& =& u_1{u}_2{p}^2-{\displaystyle \frac{u_1{\dot{u}}_2+{\dot{u}}_1{u}_2}{\alpha }}xp+{\displaystyle \frac{{\dot{u}}_1{\dot{u}}_2}{{\alpha }^2}}{x}^2-{\displaystyle \frac{1}{\alpha }}({\dot{u}}_2{g}_1\left(t\right)+{\dot{u}}_1{g}_2\left(t\right))x,\hfill \\ & & +(u_1{g}_2\left(t\right)+u_2{g}_1\left(t\right))p+g_1\left(t\right)g_2\left(t\right).\hfill \end{array}$$

(131)

Calculating the Poisson brackets between $I_1{I}_2$ and $I_1^2$, $I_1{I}_2$ and $I_2^2$, and $I_1^2$ and $I_2^2$, respectively,

$$\{I_1{I}_2,I_1^2\}=-2{\mathrm{\Omega }}_0{I}_1^2,$$

(132)

$$\{I_1{I}_2,I_2^2\}=2{\mathrm{\Omega }}_0{I}_2^2,$$

(133)

and

$$\{I_1^2,I_2^2\}=4{\mathrm{\Omega }}_0{I}_1{I}_2.$$

(134)

In the case of oscillatory motion, the result calculated by Equations (112) and (113) is obtained by replacing ${\mathrm{\Omega }}_0$ with i in Equations (132)–(134), and in the case of monotonic motion, the result is obtained by replacing ${\mathrm{\Omega }}_0$ with 1.

6. Summary and Conclusions

We chose a time-dependent system given by the general quadratic Hamiltonian of position and momentum and showed that the invariant quantities of the system are expressed by two independent solutions of homogeneous equations. The Wronskian constant belongs to one of the invariants of the system obtained from the equations of motion. In Section 3, two linear invariants given by the linear equations of x and p were also obtained, and each of them has independent solutions of the homogeneous equations of the system as auxiliary conditions. We also obtained three quadratic invariants of x and p consisting of the product of the two linear invariants. The eight generators, i.e., Lie algebras, associated with the point symmetry group of the system are obtained from the homogeneous differential equation of the equation of motion of the system. It was proven that all of these generators have invariants, and their respective invariants were derived in Section 4. The invariants obtained by the eight generators are the Wronskian constant of the system, the three invariants given by the quadratic form of x and p, and the trivial invariants 0, respectively. In Section 4, we defined new variables as two invariants consisting of a linear combination of x and p. Treating our time-dependent system with new variables makes our system more convenient to handle. When the solution to the equation of motion of a system is an oscillatory function, that is, a bounded system, the two new variables are complex conjugates of each other. And if the solution of the system is a monotonic function, that is, the unbound system, a canonical transformation relation is formed between the set of two new variables and the set of x and p. Our system is time-dependent. Mechanical energy, which is conserved in conservative systems, is not conserved in the time-dependent system. If a quantity that is not conserved in time is defined as a physical quantity, it becomes difficult to interpret it physically. Among the three physical quantities that consist of quadratic form of x and p, one that depends on the two homogeneous solutions can be made into the energy dimension and defined as a new energy quantity. Our system includes the conservation system. For example, if $\gamma \left(t\right)=0$, and ${\omega }^2\left(t\right)=\mathrm{const}.$ in Equation (3), our system becomes a harmonic oscillator. In this case, the new energy, which is made up of invariant quantities, is reduced to mechanical energy. We can expect that, in classical mechanics as well as quantum mechanics, the physical quantities of time-dependent systems should be expressed as invariant quantities so that they can be easily handled physically. The results of this paper show that new physical quantities can be defined and treated as invariant. Through this paper, we have found the conservation of a system using Lie algebra, but the main purpose of this paper is to find a more understandable way to treat time-dependent systems quantum mechanically in the next paper. The two invariants of the system, which are linearly given by the canonical position and canonical momentum, can be assumed to correspond to the creation and destruction operators in quantum mechanics. It can be predicted that the operator set corresponding to the three invariant sets, which are the quadratic forms of the canonical position and canonical momentum, forms the SU(2) Lie algebra. We can predict that an easier perturbation theory can be developed by comparing the $f\left(t\right)=0$ system with the $f\left(t\right)\ne 0$ system. And we can also discuss the coherent state of time-dependent systems. We will subsequently attempt to use our method to study other time-dependent systems.