General Null Lagrangians and Their Novel Role in Classical Dynamics

Dynamical systems with holonomic constraints can be analyzed using the Lagrangian formalism. The foundation of this formalism is the smooth configuration manifold $\mathcal{Q}$ constructed from the generalized coordinates of the system of interest with holonomic constraints and its tangent bundle $T\mathcal{Q}$. A Lagrangian $\mathcal{L}$ for this system is commonly defined as a $C^{\infty}$ map from $T\mathcal{Q}$ to the real line $\mathbf{R}$, i.e., $\mathcal{L}:T\mathcal{Q}\rightarrow\mathbf{R}$ [3].

In the case of one-dimensional dynamical systems, a Lagrangian can be symbolized as $\mathcal{L}=\mathcal{L}(x,\dot{x})$ in an open neighborhood $\mathcal{U}\subset T\mathcal{Q}$ containing a point $p\in T\mathcal{Q}$ in a local coordinate system defined by the coordinate function $x:\mathcal{Q}\rightarrow\mathbf{R}$ and its associated tangent vector $\dot{x}$ at the point $p$ using a diffeomorphism $\phi:p\in\mathcal{U}\subset{T{\mathcal{Q}}}\rightarrow\mathcal{Q}\times\mathbf{R}$ known as a local trivialization. Similarly, the explicit dependence of a Lagrangian on time can be defined as $\mathcal{L}:T\mathcal{Q}\times\mathbf{R}\rightarrow\mathbf{R}$ and thus expressing $\mathcal{L}=\mathcal{L}(x,\dot{x},t)$. After a suitable Lagrangian for an interested system is constructed or discovered, the time evolution of the system is analyzed by applying the Euler-Lagrange operator on the chosen Lagrangian, i.e., $\hat{EL}[\mathcal{L}(\dot{x},x,t)]=0$, with $\hat{EL}$ being the Euler-Lagrange (E-L) operator [2-4].

The Euler-Lagrange (EL) operator in each degree of freedom for a system is a linear operator acting on the vector space of $C^{\infty}$ functions defined on $T\mathcal{Q}$ and thus $\mathcal{L}\in\mathcal{F}(T\mathcal{Q})$ with $\mathcal{F}$ denoting the space of $C^{\infty}$ functions. Most commonly, a standard strategy for the construction of the most common Lagrangians (thus known as standard Lagrangians) are carried out from the difference of the kinetic and potential energy of the respective system under consideration, i.e., $\mathcal{L}(x,\dot{x})=\frac{1}{2}\dot{x}^{2}-V(x)$, with the potential energy $V(x)$ for a system with only one degree of freedom [1-3].

On the other hand, non-standard Lagrangians are dependent on functions of time in addition to the usual dynamical variables such as $x$ and $\dot{x}$ for systems with only one degree of freedom [7-11]. In general, the construction of these non-standard Lagrangians for the set of desired equations of motion requires solving nonlinear (Riccati-type) equations [9,12]. However, the problem can be simplified by replacing functions to be determined by constants, for instance, we may consider the following form of the non-standard Lagrangian

$$\mathcal{L}_{ns}(\dot{x},x,t)={1\over{a_{1}\dot{x}+a_{2}t+a_{3}}}\ ,$$

(1)

with constants $a_{1}$, $a_{2}$, and $a_{3}$ describes a system undergoing a constant acceleration. It should be emphasized that the above method for constructing non-standard Lagrangians $C^{\infty}$ functions demands an extra constraint that the denominators do not vanish, i.e., ${a_{1}\dot{x}+a_{2}t+a_{3}}\neq 0$ in the domain $\mathcal{U}\subset{T\mathcal{Q}}$. Therefore, non-standard Lagrangians in general represent dynamical systems with more constraints than the usual accompanying holonomic constraints. The construction of these Lagrangians can be carried out with the assumption that the corresponding Lagrangians are well-behaved on the domain $\mathcal{U}\subset{T\mathcal{Q}}$.

Since the null space of the Euler-Lagrange (EL) operator is itself a vector subspace, the members of this null space, rightly known as the null Lagrangians, are nullified by the E-L operator. More explicitly, an action functional defined in terms of null Lagrangians as

$$A[x;t_{e},t_{o}]=\int^{t_{e}}_{t_{o}}L_{\rm null}(t,x,\dot{x})dt\ ,$$

(2)

satisfies the property $A[x;t_{e},t_{o}]=A[x+\eta;t_{e},t_{o}]$ for continuous functions $x(t)$ and $\eta(t)$ of time on the same domain. This implies through approximations [13,14] that $A[x;t_{e},t_{o}]=A[\eta;t_{e},t_{o}]$ for all permissible continuous functions $x(t)$ and $\eta(t)$ on the domain such that $x(t_{e})=\eta(t_{e})$ and $x(t_{0})=\eta(t_{o})$. In other words, the above action integral produces the same real number for all admissible trajectories $x(t)$’s between $t_{e}$ and $t_{o}$ in the same domain, which in turn implies that the variation of the action integral is rendered identically to zero. This also implies that no preferred trajectory that extremizes the action integral exists for the relevant null Lagrangians. Therefore, the E-L operator renders all the null Lagrangians defined on the same domain identically to zero, i.e., no equation of motion can be obtained directly from the application of the E-L operator on null Lagrangians.

However, the study of null space of the E-L operator has been of particular interest due to its applicability to the exploration of symmetries in field theories. An effort to construct these null Lagrangians has been an active practice for a while [13,14]. The most prevalent constructions centered around the observation that the E-L operator nullifies the total time derivative of an arbitrary $C^{1}$ (continuous first derivatives) function $\Phi(x,t)$, known as the gauge function, defined on the domain of interest $\mathcal{U}\subset{T\mathcal{Q}}$. More explicitly, a Lagrangian is null if, and only if, it is the total time derivative of a scalar function $\Phi(x,t)$ [13] that can be expressed as

$$L_{null}(\dot{x},x,t)=\frac{d\Phi(x,t)}{dt}=\frac{\partial\Phi}{\partial x}\dot{x}+\frac{\partial\Phi}{\partial t}.$$

(3)

Again, it should be emphasized that the above form of a null Lagrangian is rendered identically to zero by the E-L operator, i.e., $\hat{EL}[L_{null}:=\frac{d\Phi(x,t)}{dt}]=0$. Then a natural strategy to construct a standard null Lagrangian is to cast it in the following form:

$$L_{null}(\dot{x},x,t)=B(x,t)\dot{x}+C(x,t)$$

(4)

such that $B(x,t):=\frac{\partial\Phi}{\partial x}$ and $C(x,t):=\frac{\partial\Phi}{\partial t}$. A similar construction has been presented in [14]. A simple example of a standard null Lagrangian can be obtained from a given gauge function $\Phi(x,t):=f_{1}(t)x^{2}+f_{2}(t)$ as $L_{null}=2f_{1}(t)x\dot{x}+\dot{f}_{1}(t)x^{2}+\dot{f}_{2}(t)$ with the obvious identifications $B(x,t)=2f_{1}(t)x$ and $C(x,t)=\dot{f}_{1}(t)x^{2}+\dot{f}_{2}(t)$. It is also possible to construct non-standard null Lagrangians following the same method as above. For example, the following non-standard null Lagrangian

$$L_{ns,test1}(\dot{x},x,t)={{a_{1}\dot{x}}\over{a_{2}x+a_{4}}}\ .$$

(5)

can be obtained from $\Phi(x,t)={{a_{1}}\over{a_{2}}}\ln|a_{2}x+a_{4}|$ [28]. Therefore, it is always possible to construct a standard as well as a non-standard null Lagrangian in the above form in Eq.(4) if a corresponding gauge function $\Phi(x,t)$ is known.

On the contrary, it is also possible to construct some standard null Lagrangians in the above form in Eq. (4) without any reference to the corresponding gauge functions. In the following section, an earnest humble effort is made to construct a set of both standard and non-standard null Lagrangians and their higher harmonics in the similar form as in Eq. (4) without explicitly resorting to any gauge function.

The construction of null Lagrangians is motivated by the recognition of the null-condition $\hat{EL}[\mathcal{L}_{null}(\dot{x},x,t)]=0$, i.e., the action of the E-L operator on these smooth functions nullifies them identically to zero. In other words, these null Lagrangians must satisfy the following relation:

$$\frac{dp_{null}}{dt}=\frac{\partial L_{null}}{\partial x}\ ,$$

(6)

with the generalized momentum $p_{null}=\partial{L_{null}}/\partial{\dot{x}}$.

Proposition 1: Let $B(x,t)$, $C(x,t)$ and $f(t)$ be locally $C^{1}$ functions on $\mathcal{U}\subset{T\mathcal{Q}}$ such that they satisfy the following condition:

$$\frac{dB(x,t)}{dt}=\left[\frac{\partial B(x,t)}{\partial x}\right]\dot{x}+\frac{\partial[xC(x,t)]}{\partial x}.$$

(7)

Then $B(x,t)$ generates a family of null Lagrangians $L_{null}(\dot{x},x,t)$ of the form

$$L_{null}(\dot{x},x,t)=B(x,t)\dot{x}+C(x,t)x+f(t).$$

(8)

Proof: The proof follows from the null-condition $\hat{EL}[{L}_{null}(\dot{x},x,t)]=0$ and Eq. (6).

It is interesting to note that the only restriction on $B(x,t)$ and $C(x,t)$ is that they be continuously differentiable functions on $\mathcal{U}\subset{T\mathcal{Q}}$ and they satisfy the null condition, i.e., Eq. (6). A similar construction is presented in [14] albeit with a different constraint.

However, the application of the null condition, Eq. (7), on a Lagrangian of the form in Eq. (8) implies that there exists a gauge function $\Phi(x,t)$ such that the gauge condition, $L_{\rm null}(\dot{x},x,t)=d\Phi(x,t)/dt$, is automatically satisfied. Therefore, the null condition on the function $B(x,t)$ is equivalent to the gauge condition on the gauge function $\Phi(x,t)$ related to the same null Lagrangian. In this paper, no direct reference to a gauge function $\Phi(x,t)$, i.e., no a priori assumption of the gauge condition on the gauge function is made to construct the corresponding null Lagrangian. Instead, a general function $B(x,t)$, called a generating function, is chosen along with its null condition to construct the appropriate null Lagrangian. Now, some choices of $B(x,t)$ generating different null Lagrangians would clarify the applications of the above proposition.

1. 1.

   For the simplest choice of a constant function $B(x,t)$, Eq. (7) forces both $xC(x,t)$ and $L_{null}(\dot{x},x,t)$ to be functions of $t$ only, implying that $\hat{EL}[{L}_{null}(\dot{x},x,t)]=0$.

2. 2.

   If $B(x,t)$ is a linear function of $x$, then the null-Lagrangian presented in [27] is reproduced. Let $B(x,t)=f_{1}(t)x+f_{2}(t)t+f_{3}(t)$ such that $f_{1}(t)$, $f_{2}(t)$, $f_{3}(t)$ and $f_{4}(t)$ are assumed to be arbitrary but at least twice differentiable functions of the independent variable, then the following null Lagrangian [26] is obtained

   $${L}_{null1}(\dot{x},x,t)=\left[f_{1}(t)x+f_{2}(t)t+f_{3}(t)\right]\dot{x}$$

   $$\hskip 18.06749pt+\left[\frac{1}{2}\dot{f}_{1}(t)x+\dot{f}_{2}(t)t+f_{2}(t)+\dot{f}_{3}(t)\right]x+f_{4}(t)\ .$$

   (9)

3. 3.

   Now, let $B(x,t)=f_{1}(t)x^{2}+f_{2}(t)t+f_{3}(t)$. Then the corresponding null Lagrangian turns out to be

   $${L}_{null2}(\dot{x},x,t)=\left[f_{1}(t)x^{2}+f_{2}(t)t+f_{3}(t)\right]\dot{x}$$

   $$\hskip 18.06749pt+\left[\frac{1}{3}\dot{f}_{1}(t)x^{2}+\dot{f}_{2}(t)t+f_{2}(t)+\dot{f}_{3}(t)\right]x+f_{4}(t)\ .$$

   (10)

4. 4.

   It follows from the above special choices of $B(x,t)$ that it can be any polynomial of $x$. However, in general, $B(x,t)$ can be any continuously differentiable function of any order such as a $C^{k}$ function to obtain a null Lagrangian. To show that, let $B(x,t)=f_{1}(t)\sin(x)+f_{2}(t)e^{x}t+f_{3}(t)$. Then, the corresponding null Lagrangian turns out to be

   $${L}_{null3}(\dot{x},x,t)=\left[f_{1}(t)sin(x)+f_{2}(t)e^{x}t+f_{3}(t)\right]\dot{x}$$

   $$\hskip 18.06749pt-\dot{f}_{1}(t)cos(x)+\dot{f}_{2}(t)e^{x}t+f_{2}(t)e^{x}+\dot{f}_{3}(t)x+f_{4}(t)\ .$$

   (11)

Furthermore, since the addition of a total time derivative of a function whose variation vanishes at the both ends of the trajectory does not affect the dynamics of the system, the total time derivative of $B(x,t)$ can be added to the null-Lagrangian in Eq. (8) to generate its higher harmonics, which is presented below in Proposition 2.

Proposition 2: Let $B(x,t)$, and $C(x,t)$ be locally $C^{\infty}$ functions on $\mathcal{U}\subset{T\mathcal{Q}}$ such that they are differentiable up to order $n$ to satisfy the following condition:

$$\frac{dB_{n}(x,t)}{dt}=\left[\frac{\partial B_{n}(x,t)}{\partial x}\right]\dot{x}+\frac{\partial[xC(x,t)]_{n}}{\partial x},$$

(12)

with

$$B_{n}(x,t):=\Sigma^{n}_{i=0}\mathcal{C}^{n}_{n-i}B^{(i)}(x,t)$$

(13)

and

$$[xC(x,t)]_{n}:=\Sigma^{n}_{i=0}\mathcal{C}^{n}_{n-i}\left[xC(x,t)\right]^{(i)}.$$

(14)

Here, $(i)$ represents the $i$-th spatial derivative and $\mathcal{C}^{n}_{n-i}=n!/[(n-i)!i!]$. Then $B(x,t)$ generates a family of higher harmonics of null Lagrangians $L_{null}(\dot{x},x,t)$ of the form

$$L^{(n)}_{null}(\dot{x},x,t)=B_{n}(x,t)\dot{x}+\left[xC(x,t)\right]_{n}+f(t).$$

(15)

Proof: Starting from the base case, i.e., the first harmonic, $L^{(1)}_{null}=L_{null}+\frac{dB(x,t)}{dt}$, it follows by induction that $L^{(n)}_{null}=L^{(n-1)}_{null}+\frac{dB_{n-1}(x,t)}{dt}$. Then the application of the null condition for the higher harmonics, i.e., Eq. (12), proves equation Eq. (15).

Again, if $B(x,t)$ is a linear function of $x$, then the null-Lagrangian presented in [26,27] is reproduced. Let $B(x,t)=f_{1}(t)x+f_{2}(t)t+f_{3}(t)$ and the following harmonics of the null-Lagrangian in Eq. (9) are obtained

$${L}^{(1)}_{null1}(\dot{x},x,t)=\left[f_{1}(t)(x+1)+f_{2}(t)t+f_{3}(t)\right]\dot{x}$$

$$\hskip 18.06749pt+\dot{f}_{1}(t)(\frac{x}{2}+1)x+\left[\dot{f}_{2}(t)t+f_{2}(t)+\dot{f}_{3}(t)\right](x+1)+f_{4}(t)\ ,$$

(16)

and the highest harmonics

$${L}^{(2)}_{null1}(\dot{x},x,t)=\left[f_{1}(t)(x+2)+f_{2}(t)t+f_{3}(t)\right]\dot{x}$$

$$\hskip 18.06749pt+\dot{f}_{1}(t)(\frac{x^{2}}{2}+2x+1)+\left[\dot{f}_{2}(t)t+f_{2}(t)+\dot{f}_{3}(t)\right](x+2)+f_{4}(t).$$

(17)

Let $B(x,t)=f_{1}(t)sin(x)+f_{2}(t)e^{x}t+f_{3}(t)$. Then, the corresponding null Lagrangian turns out to be

$${L}^{(1)}_{null3}(\dot{x},x,t)=\left[f_{1}(t)[sin(x)+cos(x)]+2f_{2}(t)e^{x}t+f_{3}(t)\right]\dot{x}$$

$$\hskip 18.06749pt\dot{f}_{1}(t)[sin(x)-cos(x)]+2\dot{f}_{2}(t)e^{x}t+2f_{2}(t)e^{x}+\dot{f}_{3}(t)(x+1)+f_{4}(t),\$$

(18)

and the higher harmonics can be computed according to Eq. (15). It turns out that the above strategy can also be employed to generate a family of non-standard null Lagrangians as presented below.

The key idea to generate a non-standard null-Lagrangian is to make a judicious choice of $B(x,t)$ as a fraction defined as $B(x,t)=f(t)/h(x,t)$ with $h(t)$ being a non-vanishing differentiable and integrable function. Once a choice on $B(x,t)$ is made, the corresponding non-standard null Lagrangian can be obtained by following the procedure described above in Proposition 1. Let a generating function $B(x,t)$ be expressed as

$$B(x,t)=\frac{f_{1}(t)}{f_{2}(t)x+f_{3}(t)t+f_{4}(t)},$$

(19)

where $f(t)$’s are differential functions of time and $f_{2}(t)x+f_{3}(t)t+f_{4}(t)\neq 0$. Then, the corresponding non-standard null Lagrangian can be obtained by using Eqs. (7) and (8) as

$$L_{nsn}(\dot{x},x,t)=\frac{f_{1}(t)\dot{x}}{f_{2}(t)x+f_{3}(t)t+f_{4}(t)}$$

$$\hskip 18.06749pt+\frac{h_{2}(t)}{f_{2}^{2}}\left[ln|f_{2}x+f_{3}t+f_{4}|+\frac{f_{3}t+f_{4}}{{f_{3}(t)x+f_{3}(t)t+f_{4}(t)}}\right]$$

$$\hskip 18.06749pt-\frac{h_{3}(t)t+h_{4}(t)}{f_{2}\left[f_{2}(t)x+f_{3}(t)t+f_{4}(t)\right]}+f(t),$$

(20)

where $h_{2}(t):=[\dot{f}_{1}(t)f_{2}(t)-f_{1}(t)\dot{f}_{2}(t)]$, $h_{3}(t):=[\dot{f}_{1}(t)f_{3}(t)-f_{1}(t)\dot{f}_{3}(t)]-f_{1}(t)f_{3}(t)$, and $h_{4}(t):=[\dot{f}_{1}(t)f_{4}(t)-f_{1}(t)\dot{f}_{4}(t)]$. It is worth mentioning that any other suitably chosen differentiable and integrable function $B(x,t)$ of the form in Eq. (19) can be used as a generating function to construct its corrensponding non-standard null Lagrangian.

Again, the harmonics of any non-standard null-Lagrangian can be obtained using the procedure presented in Proposition 2. For instance, the first harmonics of the above non-standard null Lagrangian in Eq. (20) can be calculated as

$$L^{(1)}_{nsn}(\dot{x},x,t)=\left[\frac{f_{1}(t)}{f_{2}(t)x+f_{3}(t)t+f_{4}(t)}-\frac{f_{1}(t)f_{2}(t)}{[f_{2}(t)x+f_{3}(t)t+f_{4}(t)]^{2}}\right]\dot{x}$$

$$\hskip 18.06749pt+\frac{h_{2}(t)}{f_{2}^{2}}\left[ln|f_{2}x+f_{3}t+f_{4}|+\frac{f_{3}t+f_{4}}{{f_{3}(t)x+f_{3}(t)t+f_{4}(t)}}\right]$$

$$\hskip 18.06749pt-\frac{h_{3}(t)t+h_{4}(t)}{f_{2}\left[f_{2}(t)x+f_{3}(t)t+f_{4}(t)\right]}+\frac{h_{2}(t)x+h_{3}(t)t+h_{4}(t)}{\left[f_{2}(t)x+f_{3}(t)t+f_{4}(t)\right]^{2}}+f(t).$$

(21)

After the exploration of the above strategy for constructing both standard and non-standard null Lagrangians, it is natural to seek for similar strategy for constructing non-standard Lagrangians. In the following section, it is shown that indeed a simple strategy for constructing non-standard Lagrangians from the above null-Lagrangians can be adopted.

The general form of the equation of motion (see Eq. 24) resulting from Eq. (23) for the null Lagrangian $L_{null}(\dot{x},x,t)=B(x,t)\dot{x}+C(x,t)x+f(t)$ may imply that the equation represents a broad variety of dynamical systems. However, it must be kept in mind that there is also the null condition that can be written as

$$\left(\frac{\partial B(x,t)}{\partial t}\right)=\frac{\partial[xC(x,t)]}{\partial x}\ ,$$

(26)

and that this condition imposes stringent constraints on admissible functions $B(x,t)$ and $C(x,t)$. As a result, only certain dynamical systems may have null Lagrangians that allow deriving their equations of motion directly from Eq. (23), which is a necessary condition for obtaining the equation of motion directly from a given null Lagrangian. The existence of this condition is a new phenomenon in the calculus of variations.

In the simplest case when $B(x,t)=c_{1}$ = const, $C(x,t)=c_{2}$ = const, and $f(t)=c_{3}$ = const, Eq. (24) reduces to $\ddot{x}=0$, which is the Newton law of inertia. Then, $L_{null}(\dot{x})=c_{1}\dot{x}$, which is the simplest null Lagrangian. It is easy to see that substitution of this Lagrangian into Eq. (23) gives the required equation of motion. Thus, the law of inertia can be derived by using the null Lagrangian, which is a new result of this paper.

To indentify dynamical systems, whose equations of motion can be obtained from the null Lgrangians by using Eq. (23), we consider the following equation

$$\ddot{x}+\alpha_{o}\dot{x}^{2}+\beta_{o}\dot{x}+\gamma_{o}x=0\ ,$$

(27)

where $\alpha_{o}$ = const, $\beta_{o}$ = const and $\gamma_{o}$ = const. Comparison of this equation to Eq. (24) allows finding the functions $B(x,t)$ and $C(x,t)$ and obtaining the corresponding null Lagrangian.

However, if the coefficients in Eq. (27) are constant, we obtain

$$B(x,t)=B_{o}e^{\alpha_{o}x+\beta_{o}t/2}\ ,$$

(28)

and

$$C(x,t)=2B_{o}\left(\frac{\gamma_{o}}{\beta_{o}}\right)e^{\alpha_{o}x+\beta_{o}t/2}\ .$$

(29)

Moreover, the null condition gives $(1+\alpha_{o}x)\gamma_{o}=\beta_{o}^{2}/4$. Since the coefficients are constant, the dependence on $x$ must be eliminated by either $\alpha_{o}=0$ or $\alpha_{o}\neq 0$ but $\beta_{o}=\gamma_{o}=0$. This shows that there is no null Lagrangian for the equation of motion $\ddot{x}+\gamma_{o}x=0$, which for $\gamma_{o}$ being a spring constant represents a harmonic oscillator. Our method of deriving equations of motion from null Lagrangians seems to be restricted to dissipative systems, with the law of inertia being an exception.

We now present the resulting ODEs for the cases when null Lagrangians exist and can be used used to derived them. With $\alpha_{o}=0$ and $\gamma_{o}=\beta_{o}^{2}/4$, Eq. (27) reduces to

$$\ddot{x}+\beta_{o}\dot{x}+\frac{1}{4}\beta_{o}^{2}x=0\ ,$$

(30)

and its null Lagrangian is

$$L_{null}(\dot{x},x,t)=\left(\dot{x}+\frac{1}{2}x\right)B_{o}e^{\beta_{o}t/2}\ .$$

(31)

It is easy to verify that $L_{null}(\dot{x},x,t)$ is the null Lagrangian and that this Lagrangian gives the equation of motion (see Eq. 29) after it is substituted into Eq. (23). The obtained equation of motion describes a damped harmonic oscillator in which the damping term and the natural frequency of this system are related to each other.

In case $\alpha_{o}\neq 0$ and $\beta_{o}=\gamma_{o}=0$, the resulting equation of motion is

$$\ddot{x}+\alpha_{o}\dot{x}^{2}=0\ ,$$

(32)

and the null Lagrangian that gives this equation is

$$L_{null}(\dot{x},x,t)=B_{o}\dot{x}e^{\alpha_{o}x}\ .$$

(33)

In this case the equation of motion describes a dynamical system with the quadratic damping and its coefficient $\alpha_{o}$ being arbitrary.

Let us generalize Eq. (27) and consider the following equation

$$\ddot{x}+\alpha_{1}(t)\dot{x}^{2}+\beta_{1}(t)\dot{x}+\gamma_{1}(t)x=0\ ,$$

(34)

where $\alpha_{1}(t)$, $\beta_{1}(t)$ and $\gamma_{1}(t)$ are given functions of time. By comparing this equation to Eq. (24), the functions $B(x,t)$ and $C(x,t)$ are obtained

$$B(x,t)=B_{o}e^{\alpha_{1}(t)x+I_{\beta}(t)}\ ,$$

(35)

where

$$I_{\beta}(t)=\frac{1}{2}\int^{t}\beta_{1}({\tilde{t}})d\tilde{t}\ ,$$

(36)

and

$$C(x,t)=B_{o}\int^{t}\gamma_{1}({\tilde{t}})e^{\alpha_{1}({\tilde{t}})x+I_{\beta}({\tilde{t}})}d\tilde{t}\ .$$

(37)