# New Role of Null Lagrangians in Derivation of Equations of Motion for
Dynamical Systems

R. Das1 and Z.E. Musielak2 1Department of Physics and Earth Space Science, The
University of Indianapolis, Indianapolis, IN 46227, USA  2Department of
Physics, The University of Texas at Arlington, Arlington, TX 76019, USA
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

The space of Null Lagrangians is the least investigated territory in dynamics
since they are identically sent to zero by their Euler-Lagrange operator and
thereby having no effects on equations of motion. A humble effort to discover
the relevance of these Null Lagrangians in dynamics is made by introducing a
generalized procedure (with respect to the recent procedure introduced by the
authors of this paper) that takes advantage of the null-ness of these
Lagrangians to construct non-standard Lagrangians that represent a range of
interesting dynamical systems. By using the generalized procedure, derivation
of equations of motion for a harmonic oscillator as well as for the Bateman
and Duffing oscillators is presented. The obtained results demonstrate a new
role played by the null Lagrangians and their corresponding non-standard
Lagrangians in describing linear and nonlinear, and dissipative and non-
dissipative dynamical systems.

##  1 Introduction

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. An
additional structure, a tangent bundle $T\mathcal{Q}$, on $\mathcal{Q}$ is
necessary to define Lagrangians for dynamical systems. Then a Lagrangian
$\mathcal{L}$ for a system is commonly defined as a $C^{\infty}$ map from
$T\mathcal{Q}$ to the real line $\mathbf{R}$, i.e.,
$\mathcal{L}:T\mathcal{Q}\rightarrow\mathbf{R}$ [1, 41].

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 [1-4, 41].

There are two families of $\mathcal{L}(\dot{x},x,t)$ that are used to obtained
equations of motion, namely, standard and non-standard Lagrangians. The main
characteristic of standard Lagrangians is the presence of the kinetic and
potential energy-like terms [5], and these Lagrangians are commonly used in
Classical Mechanics (CM) to derive equations of motion for different dynamical
systems (e.g., [1-5]). On the other hand, in non-standard Lagrangians, neither
the kinetic nor the potential energy-like terms can be identified [6],
nevertheless, they may give the same equations of motions of motion as the
standard Lagrangians [7-10].

The third family of Lagrangians is significantly different from the two
families as it contains null Lagrangians (NLs), which identically satisfy the
E-L equation and can be expressed as the total derivative of any scalar
function, called here a gauge function. The NLs have been extensively studied
in mathematics (e.g., 11-18]); however, since they do not give equations of
motion, their physical applications have been least explored and limited to
only a few cases that involed elasticity [19,20] and forces in CM [21,22].

An interesting relationship was recently found between the non-standard and
null Lagrangians [23] and it was shown that

$\frac{d}{dt}\left[L_{null}(\dot{x},x,t)\right]=0,$ (1)

gives an equation of motion for any null Lagrangian $L_{null}(\dot{x},x,t)$.
In other words, the condtion plays the same role for NLs as the E-L equation
plays for standard and non-standard Lagrangians.

The main advantage of Eq. (1) is that its form is much simpler than the E-L
equation and thus it is easier to derive an equation of motion. However, its
disadvantage is that the resulting equation of motion must be in a form that
obeys a special relationship between the coefficients of this equation; the
relationship is unique for a given dynamical system but varies for different
systems [23]. Therefore, the mainpurpose of this paper is to identify
limitations of the method and Eq. (1), and find a way to remove these
limitations and develop a generalized procedure that allows deriving equations
of motion that are unrestricted by any relationship between its coefficients.
Our paper presents this generalized procedure and applies it to specific
examples of dynamical systems such as a harmonic oscillator as well as the
Bateman [24] and Duffing [25,26] oscillators, which shows a broad range of
applications of the developed procedure.

The paper is organized as follows: Section 2 gives an overview of previously
developed method and its limitation; in Section 3, construction of non-
standard Lagrangians from null Lagrangians is described; Section 4 presents
applications of the obtained results to different osillatory systems,
including a harmonic oscillator and the Bateman and Duffing oscillators; our
conclusions are given in Section 5.

##  2 Previously developed method and its limitations

Null Lagrangians, $L_{null}(\dot{x},x,t)$, are singular Lagrangians that
satisfy $\hat{EL}[L_{null}(\dot{x},x,t)]=0$. In other words, all null
Lagrangians are identically sent to zero by the Euler-Lagrange ($\hat{E}L$)
operator. The obvious ramification is that no unique trajectory of the time
evolution of the corresponding system is derivable through the extremization
of the action functional defined in terms of the NLs. However, it is possible
to exploit these NLs for constructing non-standard Lagrangians to represent
certain interesting dynamical systems. Recently, a procedure for deriving
equations of motion for a given NL has been developed [23], and in the
following we briefly summarize the procedure by presenting its main condition.
Then, we show its limitations and a need for its generalization.

Let
$L_{null}(\dot{x},x,t):\mathcal{U}\subset{T\mathcal{Q}}\rightarrow\mathbf{R}$
be an invertible and $C^{\infty}$ function. Then a non-standard Lagrangian
$\mathcal{L}:\mathcal{U}\subset{T\mathcal{Q}}\rightarrow\mathbf{R}$ can be
defined by using at least a $C^{2}$ function
$F:\mathbf{R}\rightarrow\mathbf{R}$ as $\mathcal{L}(\dot{x},x,t)=(F\circ
L_{null})(\dot{x},x,t))=F(L_{null}(\dot{x},x,t))$ such that, for any
permissible form of function $\mathcal{L}(\dot{x},x,t)=(F\circ
L_{null})(\dot{x},x,t)$ satisfying
$p_{{null}}\left[\frac{d^{2}F}{dL_{null}^{2}}\right]\neq 0$, the corresponding
equation of motion for a dynamical system represented by
$\mathcal{L}(\dot{x},x,t)$ is given by Eq. (1) with
$p_{null}:=\partial{L}_{null}/\partial\dot{x}$.

The above construction can be carried out by using a null Lagrangian,
${L}_{null}(\dot{x},x,t)=d\Phi(x,t)/dt$, constructed from a smooth and
invertible gauge function $\Phi(x,t)$ defined over
${\Omega}\in\mathcal{Q}\times\mathbf{R}$. It is straightforward to show that
Eq. (1), upon inserting $L_{null}(\dot{x},x,t)=d\Phi(x,t)/dt$, takes the form:

$\Phi^{{}^{\prime}}(x,t)\ddot{x}+\left[\Phi^{{}^{\prime\prime}}(x,t)\dot{x}+2(\dot{\Phi})^{{}^{\prime}}(x,t)\right]\dot{x}+\ddot{\Phi}(x,t)=0,$
(2)

where primes and dots are defined respectively as
$\Phi^{{}^{\prime}}:=\partial\Phi(x,t)/\partial x$,
$\Phi^{{}^{\prime\prime}}:=\partial^{2}\Phi(x,t)/\partial x^{2}$,
$\dot{\Phi}:=\partial\Phi(x,t)/\partial t$ and
$\ddot{\Phi}:=\partial^{2}\Phi(x,t)/\partial t^{2}$.

To apply this result to dynamical systems, let us consider the following
general equation of motion

$\ddot{x}+\alpha(x,t)\dot{x}^{2}+\beta(x,t)\dot{x}+\gamma(x,t)x=0,$ (3)

where $\alpha(x,t)$, $\beta(x,t)$ and $\gamma(x,t)$ are the coefficients
determined by a given dynamical system. For specific forms of these
coefficients, this equation of motion describes a broad variety of dynamical
systems ranging from a harmonic oscillator and the Bateman and Duffing
oscillators to other dynamical systems, including the population dynamics
models, such as the Lotka-Volterra, Verhulst, Gompertz and Host-Parasite
models, and others [27].

By comparing Eq. (2) to Eq. (3), we find

$\Phi^{{}^{\prime\prime}}(x,t)=\alpha(x,t)\Phi^{{}^{\prime}}(x,t),$ (4)
$\dot{\Phi}^{{}^{\prime}}(x,t)=\frac{1}{2}\beta(x,t)\Phi^{{}^{\prime}}(x,t),$
(5)

and

$\ddot{\Phi}(x,t)=\gamma(x,t)x\Phi^{{}^{\prime}}(x,t),$ (6)

with $\Phi^{{}^{\prime}}(x,t)\neq 0$. These are the conditions that the gauge
function must satisfy in order to give the null Lagrangian for Eq. (3).
Clearly, the fact that different derivaties of $\Phi(x,t)$ depend on the
coefficients $\alpha(x,t)$, $\beta(x,t)$ and $\gamma(x,t)$ implies that the
resulting gauge function, which is the solution for these conditions, must
depend on a certain relationship between the coefficients. In other words,
$\Phi(x,t)$ that satisfy these conditions and the NL resulting from this gauge
function will give only the equation of motion for which the relationship is
valid.

After eliminating $\Phi(x,t)$ from the above conditions, we obtain the
following relationship between the coefficients

$\frac{1}{2}\left[\dot{\beta}(x,t)+\frac{1}{2}\beta^{2}(x,t)\right]=\gamma^{{}^{\prime}}(x,t)x+\gamma(x,t)\left[1+\alpha(x,t)x\right].$
(7)

If this condition is satisfied for a given equation of motion, then the gauge
function and the corresponding null Lagrangian can be constructed, and when
the latter is substituted into Eq. (1), the original equation of motion is
obtained. The condition significantly restricts forms of the equations of
motion that can be derived using this method. In the following, we first
consider some special restricted cases and then identify unrestricted cases.

In case $\alpha(x,t)=\alpha(x)$, $\beta(x,t)=\beta(x)$ and
$\gamma(x,t)=\gamma(x)$, the condition reduces to

$\frac{1}{4}\beta^{2}(x)=\gamma^{{}^{\prime}}(x)x+\left[1+\alpha(x)x\right]\gamma(x),$
(8)

which shows the dependence between all the coefficients.

If $\alpha(x,t)=\alpha(t)$, $\beta(x,t)=\beta(t)$ and $\gamma(x,t)=\gamma(t)$,
the condition becomes

$\frac{1}{2}\left[\dot{\beta}(t)+\frac{1}{2}\beta^{2}(t)\right]=\gamma(t),$
(9)

which demonstrates the relationship between $\beta(t)$ and $\gamma(t)$. In
this case, $\alpha(t)=0$ is required to eliminate the $x$-dependence from the
condition.

Now, the case of constant coefficients, $\alpha(x,t)=\alpha_{o}$ = const,
$\beta(x,t)=\beta_{o}$ = const and $\gamma(x,t)=\gamma_{o}$ = const, requires
also that $\alpha_{o}=0$, for the same reason as above, and gives the
relationship between $\beta_{o}$ and $\gamma_{o}$ to be
$\beta_{o}^{2}=4\gamma_{o}$.

Another interesting case is when $\alpha(x,t)=\gamma(x,t)=0$ and
$\beta(x,t)=\beta(t)$. Then, we find

$\dot{\beta}(t)+\frac{1}{2}\beta^{2}(t)=0,$ (10)

with the solution

$\beta(t)=\frac{2}{(c_{1}+t)^{2}},$ (11)

where $c_{1}$ is an integration constant. The obtained result means that the
gauge function and null Lagrangian exist only for the equation of motion with
$\beta$ coefficient having this time dependence; for some other special cases
see [23].

Let us now identify more interesting cases that are not restricted by the
condition given by Eq. (7). Obviously, the simplest such case is when
$\alpha(x,t)=\beta(x,t)=\gamma(x,t)=0$, which automatically satisfies the
condition. Then, the gauge function that is the solution to Eqs. (4), (5) and
(6) becomes

$\Phi(x,t)=c_{1}x+c_{2}t+c_{3},$ (12)

where $c_{1}$, $c_{2}$ and $c_{3}$ are integration constants, and the
corresponding null Lagrangian is

$L_{null}(x)=c_{1}\dot{x}+c_{2},$ (13)

and its substitution into Eq. (1) gives the law of inertia $\ddot{x}=0$.

By taking $\alpha(x,t)=\beta(x,t)=0$ and $\gamma(x,t)\neq 0$, the condition
reduces to

$\gamma^{{}^{\prime}}(x,t)x+\gamma(x,t)=0,$ (14)

with the solution

$\gamma(x,t)=\frac{f_{0}(t)}{x},$ (15)

where $f_{0}(t)$ is an arbitrary function. Then, the gauge function is

$\Phi(x,t)=c_{1}x+c_{1}\int^{t}\left[\int^{\bar{t}}f_{0}(\tilde{t})d\tilde{t}\right]d\bar{t},$
(16)

and the null Lagrangian is given by

$L_{null}(x,t)=c_{1}\dot{x}+c_{1}\int^{t}f_{0}(\tilde{t})d\tilde{t}.$ (17)

Substitution of this Lagrangian into Eq. (1) gives the second law of dynamics
$\ddot{x}=F(t)$, where $F(t)=-f_{0}(t)$.

The last unrestricted case is when $\alpha(x,t)\neq 0$ and
$\beta(x,t)=\gamma(x,t)=0$. Then, the gauge function is

$\Phi(x)=c_{1}\int^{x}\rme^{I_{\alpha}(\tilde{x})}d\tilde{x}+c_{2}t+c_{3}\ ,$
(18)

and the null Lagrangian becomes

$L_{null}(\dot{x},x)=c_{1}\dot{x}\rme^{I_{\alpha}(x)}+c_{2}\ ,$ (19)

where $I_{\alpha}(x)=\int^{x}\alpha(\bar{x})d\bar{x}$.

By substituting $L_{null}(\dot{x},x)$ into Eq. (1), we obtain

$\ddot{x}+\alpha_{2}(x)\dot{x}^{2}=0\ ,$ (20)

which is the equation of motion for an oscillatory system with the quadratic
damping [23, 9].

The above results demonstrate that the previously developed method can be used
to derive the first and second law of Newtonian dynamics by using the
constructed gauge functions and their corresponding null Lagrangians.
Moreover, the method allows obtaining the equation of motion for an
oscillatory system with the quadratic damping term. The derivations of these
three equations are unrestricted by any constraints. However, any application
of the previous method to other dynamical systems including harmonic and
linearly damped oscillators is restricted by special relationships between the
system’s coefficients. Therefore, in order to apply to a broad variety of
dynamical systems, a generalization of the previous method is needed and this
is the main purpose of this paper. As shown in the following, the generalized
method allows constructing gauge functions and null Lagrangians and deriving
equations of motion for a harmonic oscillator as well as for the Bateman and
Duffing oscillators.

##  3 Construction of non-standard Lagrangians from null Lagrangians

After the above construction of the non-standard Lagrangians from null
Lagrangians, the following generalization is attempted to obtain new non-
standard Lagrangians to represent more dynamical systems of interest.

Proposition 1: Let $F(L_{null}(\dot{x},x,t))$, $G(L_{null}(\dot{x},x,t))$ and
$M(L_{null}(\dot{x},x,t))$ :
$\mathcal{U}\subset{T\mathcal{Q}}\times\mathbf{R}\rightarrow\mathbf{R}$ be at
least $C^{2}$ functions of a certain invertible $L_{null}\in\mathcal{F}$ along
with invertible and smooth maps $Q(\dot{x},x,t)$, $R(\dot{x},x,t)$ and
$U(\dot{x},x,t):$
$\mathcal{U}\subset{T\mathcal{Q}}\times\mathbf{R}\rightarrow\mathbf{R}$
satisfying constraints $Q(\dot{x},x,t)=R(\dot{x},x,t)L_{null}(\dot{x},x,t)$
and $Q\frac{dF}{dL_{null}}+R\frac{dG}{dL_{null}}=\lambda(constant)$. Then
there exists a non-standard Lagrangian of the form

$\mathcal{L}_{ns}(\dot{x},x,t)=Q(\dot{x},x,t)F(L_{null}(\dot{x},x,t))$
$+R(\dot{x},x,t)G(L_{null}(\dot{x},x,t))+U(\dot{x},x,t)M(L_{null}(\dot{x},x,t))$
(21)

such that the equation of motion of this non-standard Lagrangian is given by

$\hat{EL}[RL_{null}(\dot{x},x,t)]F(L_{null})+\hat{EL}[R(\dot{x},x,t)]G(L_{null})+\hat{EL}[U(\dot{x},x,t)]M(L_{null})$
$+\left[\lambda\left[\frac{p_{R}}{R}+\frac{p_{null}}{L_{null}}\right]+p_{U}\frac{dM}{dL_{null}}\right]\frac{dL_{null}}{dt}$
$+p_{null}\frac{d}{dt}\left[U\frac{dM}{dL_{null}}\right]-p_{{null}}\frac{R}{L_{null}}\frac{dG}{dt}=0,$
(22)

with $p_{U}:=\partial U/\partial\dot{x}$, $p_{null}=\partial
L_{null}/\partial\dot{x}$ and $p_{R}=\partial R/\partial\dot{x}$.

Proof: The derivation of Eq. (22) follows directly from
$\hat{EL}[\mathcal{L}_{ns}]=0$, and the constraints
$Q(\dot{x},x,t)=R(\dot{x},x,t)L_{null}(\dot{x},x,t)$ and
$Q\frac{dF}{dL_{null}}+R\frac{dG}{dL_{null}}=\lambda(constant)$. More
precisely, the action of the Euler-Lagrange operator on the non-standard
Lagrangian, Eq. (21), yields

$\left[p_{Q}\frac{dF}{dL_{null}}+p_{R}\frac{dG}{dL_{null}}+p_{U}\frac{dM}{dL_{null}}\right]\frac{dL_{null}}{dt}$
$+p_{{null}}\frac{d}{dt}\left[Q\frac{dF}{dL_{null}}+R\frac{dG}{dL_{null}}+U\frac{dM}{dL_{null}}\right]$
$+\hat{EL}[Q(\dot{x},x,t)]F(L_{null})+\hat{EL}[R(\dot{x},x,t)]G(L_{null})+\hat{EL}[U(\dot{x},x,t)]M(L_{null})=0.$
(23)

Then the application of the above constraints reduces the above equation,
Eq.(23), to Eq. (22). This completes the proof.

Here, a few noteworthy remarks are in order. First, it follows directly from
the constraint $Q\frac{dF}{dL_{null}}+R\frac{dG}{dL_{null}}=\lambda$ that

$F(L_{null})=\int{\frac{\lambda}{RL_{null}}dL_{null}}-\int{\frac{1}{L_{null}}\left[\frac{dG}{dL_{null}}\right]dL_{null}}+constant,$
(24)

which implies that the knowledge of the form of $G$ and the constant $\lambda$
completely determines $F$ modulo a constant provided Eq. (24) is integrable.
Second, the constraint $Q(\dot{x},x,t)=R(\dot{x},x,t)L_{null}(\dot{x},x,t)$
ensures that the above construction of non-standard Lagrangians centers around
only null Lagrangians if the last term in Eq. (21) vanishes. In this case, the
equation of motion, Eq. (22) simplifies, for $\lambda=0$ to

$\hat{EL}[RL_{null}(\dot{x},x,t)]F(L_{null})+\hat{EL}[R(\dot{x},x,t)]G(L_{null})-p_{{null}}\frac{R}{L_{null}}\frac{dG}{dt}=0.$
(25)

The above simplified equation of motion, Eq. (25), turns out to be extremely
advantageous for deriving the non-standard Lagrangians for several well-known
dynamical systems, which will be presented in the next section. Last, even for
the case of a non-vanishing last term with $M(L_{null})=m(constant)$, the
above construction in Proposition 1 simplifies to the following corollary.

Corollary 1: Let the construction in Eq. (21) be simplified as follows

$\mathcal{L}_{ns1}(\dot{x},x,t)=Q(\dot{x},x,t)F(L_{null}(\dot{x},x,t))$
$+R(\dot{x},x,t)G(L_{null}(\dot{x},x,t))+mU(\dot{x},x,t),$ (26)

where $m$ is a constant and $Q$, $R$, $U$, $F(L_{null})$ and $G(l_{null})$ are
defined as above in Proposition 1 satisfying the constraints
$Q(\dot{x},x,t)=R(\dot{x},x,t)L_{null}(\dot{x},x,t)$ and
$Q\frac{dF}{dL_{null}}+R\frac{dG}{dL_{null}}=\lambda(constant)$. Then it
follows that the corresponding equation of motion for
$\mathcal{L}_{ns1}(\dot{x},x,t)$ turns out to be

$\hat{EL}[RL_{null}(\dot{x},x,t)]F(L_{null})+\hat{EL}[R(\dot{x},x,t)]G(L_{null})+m\hat{EL}[U(\dot{x},x,t)]$
$+\lambda\left[\frac{p_{R}}{R}+\frac{p_{null}}{L_{null}}\right]\frac{dL_{null}}{dt}-p_{{null}}\frac{R}{L_{null}}\frac{dG}{dt}=0,$
(27)

The proof follows directly from Eq. (22). Furthermore, if
$U(\dot{x},x,t)=U(x,t)$ is a function of position and time only, then
$\hat{EL}[U(\dot{x},x,t)]=-\partial U/\partial x$ behaves like a force term,
which will be of particular interest in our application in the following
section. Next, it is crucial to explore the scope of this novel construction
of non-standard Lagrangians from null-Lagrangians to represent interesting
dynamical systems.

##  4 Applications to dynamical systems

###  4.1 General approach

The application of the above construction in Proposition 1 to derive non-
standard Lagrangians for dynamical systems becomes convenient if a null-
Lagrangian is expressed as follows

$L_{null}=\frac{d\Phi(x,t)}{dt}=\Phi^{{}^{\prime}}\dot{x}+\dot{\Phi}=\left(\dot{x}+\frac{\dot{\Phi}}{\Phi^{{}^{\prime}}}\right)\Phi^{{}^{\prime}},$
(28)

where $\Phi^{{}^{\prime}}:=\partial\Phi/\partial x\neq 0$ and
$\dot{\Phi}:=\partial\Phi/\partial t$. Now, $Q=RL_{null}$ implies that
$Q=\left(\dot{x}+\frac{\dot{\Phi}}{\Phi^{{}^{\prime}}}\right)$ and
$R=1/\Phi^{{}^{\prime}}$. Then, with the vanishing last term in Proposition 1,
the equation of motion, Eq. (25), takes the following form

$\frac{\Phi^{{}^{\prime\prime}}}{(\Phi^{{}^{\prime}})^{2}}G(L_{null})+\left[\lambda\Phi^{{}^{\prime}}-\frac{dG}{dL_{null}}\right]\frac{1}{L_{null}}\frac{dL_{null}}{dt}+\left[\frac{\dot{\Phi}\Phi^{{}^{\prime\prime}}-(\dot{\Phi})^{{}^{\prime}}\Phi^{{}^{\prime}}}{(\Phi^{{}^{\prime}})^{2}}\right]F(L_{null})=0.$
(29)

It is easy to notice that the last term in the above equation, Eq. (29),
vanishes for a gauge function $\Phi$ independent of time explicitly, i.e.,
$\Phi(x,t)=\Phi(x)$.

At this juncture, the knowledge of the form of $G(L_{null})$ is necessary to
move further. The form of $G(L_{null})$ may be determined by casting the above
equation of motion, Eq. (29), into the form of Eq. (3). Then a comparison with
the equation of motion of a desired dynamical system such as an oscillatory
system may be advantageous to determine the form of $G(L_{null})$. Hence, it
follows from Eq. (2) and $\lambda\Phi^{{}^{\prime}}-\frac{dG}{dL_{null}}\neq
0$ that the equation of motion, Eq. (29), may be recast as

$\ddot{x}+\frac{\Phi^{{}^{\prime\prime}}}{\Phi^{{}^{\prime}}}\dot{x}^{2}+2\frac{(\dot{\Phi})^{{}^{\prime}}}{\Phi^{{}^{\prime}}}\dot{x}+\frac{\Phi^{{}^{\prime\prime}}}{(\Phi^{{}^{\prime}})^{3}}\left[\frac{L_{null}}{\lambda\Phi^{{}^{\prime}}-\frac{dG}{dL_{null}}}\right]G(L_{null})$
$+\left[\frac{\dot{\Phi}\Phi^{{}^{\prime\prime}}-(\dot{\Phi})^{{}^{\prime}}\Phi^{{}^{\prime}}}{(\Phi^{{}^{\prime}})^{3}}\right]\left[\frac{L_{null}}{\lambda\Phi^{{}^{\prime}}-\frac{dG}{dL_{null}}}\right]F(L_{null})+\frac{\ddot{\Phi}}{\Phi^{{}^{\prime}}}=0.$
(30)

Next, with the above modified equation of motion, Eq. (30), the construction
of non-standard Lagrangians for oscillatory systems is explored below.

###  4.2 Harmonic oscillator

As the first desirable application to a harmonic oscillator, a comparison
between Eq. (30) and the desirable equation of motion,
$\ddot{x}+\gamma(x,t)f(x)=0$ with $\gamma(x,t)=\gamma_{0}$ (a constant),
becomes easier if the choices $\Phi(x,t)=\Phi(x)$ and $\lambda=0$ are made.
Then Eq. (30) simplifies to

$\ddot{x}+\frac{\Phi^{{}^{\prime\prime}}}{\Phi^{{}^{\prime}}}\dot{x}^{2}-\frac{\Phi^{{}^{\prime\prime}}}{(\Phi^{{}^{\prime}})^{3}}\left[\frac{L_{null}}{\frac{dG}{dL_{null}}}\right]G(L_{null})=0,$
(31)

and then the comparison between Eq. (31) and $\ddot{x}+\gamma_{0}f(x)=0$
yields

$\frac{\Phi^{{}^{\prime\prime}}}{\Phi^{{}^{\prime}}}\dot{x}^{2}-\frac{\Phi^{{}^{\prime\prime}}}{(\Phi^{{}^{\prime}})^{3}}\left[\frac{L_{null}}{\frac{dG}{dL_{null}}}\right]G(L_{null})=\gamma_{0}f(x).$
(32)

Next, it is obvious that

$\frac{\Phi^{{}^{\prime\prime}}}{(\Phi^{{}^{\prime}})^{3}}\left[\frac{L_{null}}{\frac{dG}{dL_{null}}}\right]G(L_{null})=\pm\frac{\Phi^{{}^{\prime\prime}}}{(\Phi^{{}^{\prime}})^{3}}+\frac{\Phi^{{}^{\prime\prime}}}{\Phi^{{}^{\prime}}}\dot{x}^{2},$
(33)

will reduce Eq. (31) to
$\ddot{x}\mp\frac{\Phi^{{}^{\prime\prime}}}{(\Phi^{{}^{\prime}})^{3}}=0$,
which in turn produces the constraint on the gauge function,
$\Phi^{{}^{\prime\prime}}\pm\gamma_{0}f(x)(\Phi^{{}^{\prime}})^{3}=0$ to yield
the desirable equation of motion, $\ddot{x}+\gamma_{0}f(x)=0$. Now, from Eq.
(33), it easily follows that

$\frac{dG}{dL_{null}}=\frac{L_{null}}{L^{2}_{null}\pm 1}G,$ (34)

where $L_{null}=\Phi^{{}^{\prime}}\dot{x}$ is used. Next, for $+1$ and $-1$,
it is straightforward to obtain the form of G from the above equation Eq. (34)
as $G(L_{null})=\sqrt{1+L^{2}_{null}}+c_{0}$ and
$G(L_{null})=\sqrt{L^{2}_{null}-1}+c_{0}$ respectively with $c_{0}$ being a
constant of integration. However, since
$G(L_{null})=\sqrt{L^{2}_{null}-1}+c_{0}$ is constrained to be valid for only
$L_{null}>1$, $G(L_{null})=\sqrt{1+L^{2}_{null}}+c_{0}$ is a natural choice
for application and thus it is chosen in this paper.

After determining the form of $G(L_{null})$, it is necessary to determine the
form of the appropriate null-Lagrangian $L_{null}$ to obtain the equation of
motion for a dynamical system. For an oscillatory system, it follows from the
above constraint,
$\Phi^{{}^{\prime\prime}}+\gamma_{0}f(x)(\Phi^{{}^{\prime}})^{3}=0$, that any
integrable function $f(x)$ defined over a appropriate domain of the
configuration space $\mathcal{Q}$ of the oscillator satisfying
$\Phi^{{}^{\prime}}=\left[c_{1}+2\gamma_{0}\int^{x}{f(\xi)d\xi}\right]^{-1/2}$,
with $c_{1}$ being the constant of integration, will give the respective null-
Lagrangian, $L_{null}=\Phi^{{}^{\prime}}\dot{x}$.

As a simple check, it is easy to see that $f(x)=x$ after inserted in
$L_{null}=\dot{x}\Phi^{{}^{\prime}}=\dot{x}\cdot\left[c_{1}+2\gamma_{0}\int^{x}{f(\xi)d\xi}\right]^{-1/2}$
will generate the null-Lagrangian
$L_{null}=\frac{\dot{x}}{\sqrt{\gamma_{0}x^{2}+c_{1}}}$ with $c_{1}$ being a
constant of integration. Next, with the form of
$\Phi^{{}^{\prime}}=\frac{1}{\sqrt{\gamma_{0}x^{2}+c_{1}}}$, the equation of
motion,
$\ddot{x}-\frac{\Phi^{{}^{\prime\prime}}}{(\Phi^{{}^{\prime}})^{3}}=0$,
reduces to the expected equation of motion for a simple harmonic oscillator,
$\ddot{x}+\gamma_{0}x=0$. Moreover, the non-standard Lagrangian for a simple
harmonic oscillator, from Proposition 1, is determined to be

$L^{HO}_{ns}(\dot{x},x,t)=\dot{x}\cdot{sinh}^{-1}\left(\frac{\dot{x}}{\sqrt{\gamma_{0}x^{2}+c_{1}}}\right)-\sqrt{\gamma_{0}x^{2}+c_{1}}\cdot\sqrt{1+\left(\frac{\dot{x}}{\sqrt{\gamma_{0}x^{2}+c_{1}}}\right)^{2}},$
(35)

where $F(L_{null})=sinh^{-1}(L_{null})+\ constant$ is used after obtaining
from Eq. (24). It should be noted here that
$G(L_{null})=-\sqrt{1+L^{2}_{null}}+c_{0}$ is necessary to obtain
$F(L_{null})=sinh^{-1}(L_{null})+\ constant$ from Eq. (24) to have Eq. (35).
This is compatible with Proposition 1 since an overall extra negative sign
appears naturally in the transition from Eq. (29) to Eq. (31). Finally, as a
cross check, it can be easily shown that
$\hat{EL}\left[L^{HO}_{ns}(\dot{x},x,t)\right]=\ddot{x}+\gamma_{0}x=0$.

Similarly, it follows that after inserting $f(x)=sin(ax)$, with $a\in\bf R$,
in
$L_{null}=\dot{x}\Phi^{{}^{\prime}}=\dot{x}\cdot\left[c_{1}+2\gamma_{0}\int^{x}{f(\xi)d\xi}\right]^{-1/2}$
generates the null-Lagrangian
$L^{f}_{null}=\frac{\sqrt{a}\dot{x}}{\sqrt{2}\sqrt{ac_{2}-\gamma_{0}cos(ax)}}$
and the corresponding non-standard Lagrangian turns out to be

$L^{f}_{ns}(\dot{x},x,t)=\sqrt{a}\dot{x}\cdot{sinh}^{-1}(L^{f}_{null})-\sqrt{2}\sqrt{ac_{2}-\gamma_{0}cos(ax)}\cdot\sqrt{1+(L^{f}_{null})^{2}},$
(36)

with $c_{2}$ being a constant of integration and obviously the respective
equation of motion is given by
$\hat{EL}\left[L^{f}_{ns}(\dot{x},x,t)\right]=\ddot{x}+\gamma_{0}sin(ax)=0$.
It should be noted here that it is the form of the null Lagrangian by
construction that determines the dynamical system of interest once the
structure of a non-standard Lagrangian such as Eq. (35) and Eq. (36) is
chosen.

###  4.3 Linear and non-linear Bateman oscillator

Once a non-standard Lagrangian for an oscillator is obtained, it is possible
to derive the related non-standard Lagrangian for a damped oscillator by
multiplying the non-standard Lagrangian with a damping function $e^{bt}$ for a
damping parameter $b$ as in [42,43]. However, in order to be consistent with
the construction of non-standard Lagrangian above, i.e., Proposition 1, it is
necessary to express the null-Lagrangian as

$L_{null}=e^{bt}\left(\dot{x}+\frac{\dot{\Phi}}{\Phi^{{}^{\prime}}}\right)\Phi^{{}^{\prime}}e^{-bt}$
(37)

so that $L_{null}$ does not depend on time explicitly with transferring the
explicit time dependence to $Q$ and $R$ as
$Q=e^{bt}\left(\dot{x}+\frac{\dot{\Phi}}{\Phi^{{}^{\prime}}}\right)$ and
$R=e^{bt}/\Phi^{{}^{\prime}}$.

With the above identifications along with $F=sinh^{-1}(L_{null})$ and
$G=-\sqrt{1+L_{null}^{2}}$, a non-standard Lagrangian for a damped oscillator
can be expressed as

$L^{damped}_{ns}(\dot{x},x,t)=e^{bt}\dot{x}\cdot{sinh}^{-1}(L^{f}_{null})-\frac{1}{\Phi^{{}^{\prime}}e^{-bt}}\cdot\sqrt{1+(L^{f}_{null})^{2}}.$
(38)

Then it is straightforward to obtain the corresponding equation of motion for
a damped oscillator as

$\ddot{x}+\frac{b}{\Phi^{{}^{\prime}}}\cdot
sinh^{-1}(L^{f}_{null})\cdot\sqrt{1+(L^{f}_{null})^{2}}+\frac{\Phi^{{}^{\prime\prime}}}{\Phi^{{}^{\prime}}}\dot{x}^{2}-\frac{\Phi^{{}^{\prime\prime}}}{(\Phi^{{}^{\prime}})^{3}}\left(1+(L^{f}_{null})^{2}\right)=0.$
(39)

However, in order to extract the desired equation of motion for a damped
oscillator, it is necessary to express the second term in Eq. (39) using
Taylor expansion around $L^{f}_{null}=0$ as

$\frac{b}{\Phi^{{}^{\prime}}}\cdot
sinh^{-1}(L^{f}_{null})\cdot\sqrt{1+(L^{f}_{null})^{2}}\approx
b\dot{x}+\frac{1}{3}b(\Phi^{{}^{\prime}})^{2}{\dot{x}^{3}}-\frac{2}{15}b(\Phi^{{}^{\prime}})^{4}\dot{x}^{5}+\cdots,$
(40)

where $L^{f}_{null}=\phi^{{}^{\prime}}\dot{x}$ is used. Inserting Eq. (40)
into Eq. (39) gives

$\ddot{x}+\left(b\dot{x}+\frac{1}{3}b(\Phi^{{}^{\prime}})^{2}{\dot{x}^{3}}-\frac{2}{15}b(\Phi^{{}^{\prime}})^{4}\dot{x}^{5}+\cdots\right)-\frac{\Phi^{{}^{\prime\prime}}}{(\Phi^{{}^{\prime}})^{3}}=0.$
(41)

It is obvious that Eq. (41) reduces to the equation of motion for a damped
oscillator in the zeroth order approximation, $L^{f}_{null}\ll 1$. In
particular, with
$-\frac{\Phi^{{}^{\prime\prime}}}{(\Phi^{{}^{\prime}})^{3}}=\gamma_{0}x$, Eq.
(41) takes the form of the linear Bateman oscillator, i.e.,
$\ddot{x}+b\dot{x}+\gamma_{0}x=0$ in the zeroth order approximation and the
corresponding non-standard Lagrangian is given by Eq. (38) with
$L^{f}_{null}=\frac{\dot{x}}{\sqrt{\gamma_{0}x^{2}+c_{1}}}$.

On the other hand,
$-\frac{\Phi^{{}^{\prime\prime}}}{(\Phi^{{}^{\prime}})^{3}}=\gamma_{0}sin(ax)$,
in Eq. (41) generates the non-linear form of Bateman oscillator, i.e.,
$\ddot{x}+b\dot{x}+\gamma_{0}sin(ax)=0$ in the zeroth order approximation, and
therefore the corresponding non-standard Lagrangian is also given by Eq. (38)
with
$L^{f}_{null}=\frac{\sqrt{a}\dot{x}}{\sqrt{2}\sqrt{ac_{2}-\gamma_{0}cos(ax)}}$.

###  4.4 Duffing Oscillator:

The above non-standard Lagrangian can easily be extended to a non-standard
Lagrangian for the Duffing oscillator by setting
$-\frac{\Phi^{{}^{\prime\prime}}}{(\Phi^{{}^{\prime}})^{3}}=\gamma_{0}x+\beta
x^{3}$ in Eq. (41) with $\beta$ representing the non-linearity in the
restoring force. Then, again, it follows that
$\Phi^{{}^{\prime}}=\left[c_{1}+2\gamma_{0}\int^{x}{f(\xi)d\xi}\right]^{-1/2}=\frac{\sqrt{2}}{\sqrt{2\gamma_{0}x^{2}+\beta
x^{4}+c_{1}}}$ and the corresponding non-standard Lagrangian is given by Eq.
(38) with $L^{f}_{null}=\frac{\sqrt{2}\dot{x}}{\sqrt{2\gamma_{0}x^{2}+\beta
x^{4}+c_{1}}}$ in the zeroth order approximation, i.e., $L^{f}_{null}\ll 1$.
With the above null Lagrangian, $L^{f}_{null}$, the equation of motion for a
damped oscillator, Eq. (41), reduces to the unforced Duffing equation,
$\ddot{x}+b\dot{x}+\gamma_{0}x+\beta x^{3}=0$ in the zeroth order
approximation.

Furthermore, the identification of
$-\frac{\Phi^{{}^{\prime\prime}}}{(\Phi^{{}^{\prime}})^{3}}=\gamma_{0}sin(ax)+\beta
x^{3}$ gives
$L^{f}_{null}=\frac{\sqrt{2a}\dot{x}}{\sqrt{-4\gamma_{0}cos(ax)+\gamma_{0}\beta
x^{4}+ac_{1}}}$ and the corresponding equation of motion is given by
$\ddot{x}+b\dot{x}+\gamma_{0}sin(ax)+\beta x^{3}=0$ in the zeroth order
approximation.

Until now, the scope of applications was restricted by the assumptions
$Q=RL_{null}$ and $\lambda=0$. There was no a priori assumption made on the
forms of $Q$ and $R$. Hence, it would be curious to explore the effects of
further restrictions on both $Q$ and $R$ such that they still satisfy
$Q=RL_{null}$, which is presented below.

###  4.5 Null Lagrangians and potential functions:

A convenient assumption would be $Q=L_{null}$ so that $Q=RL_{null}$ implies
$R=1$. It is obvious that this assumption will make the non-standard
Lagrangians constructed so far depend solely on $L_{null}$. However, as shown
in Section 2 and in [23], this further restriction on $Q$ will rather render
this construction inexpedient for the generalization to the applications
presented above. Nevertheless, it turns out that the application of Corollary
1 does generalize the construction in [23] and furthermore elicits interesting
physics.

For instance, with $Q=L_{null}$, $\lambda=0$, $U=U(x,t)$ and
$M(L_{null})=m=constant$, the equation of motion in Corollary 1, Eq. (27),
reduces to

$\ddot{x}+\frac{\Phi^{{}^{\prime\prime}}}{\Phi^{{}^{\prime}}}\dot{x}^{2}+\frac{mU^{{}^{\prime}}}{(\Phi^{{}^{\prime}})^{2}}\left[\frac{L_{null}}{dG/dL_{null}}\right]=0,$
(42)

where $U^{{}^{\prime}}=\partial U/\partial x$. Then, in order to derive an
equation of motion for an oscillator, it is necessary to express
$\frac{L_{null}}{dG/dL_{null}}$ in Eq. (42) as follows

$\frac{dG}{dL_{null}}=-\frac{L_{null}}{1+L^{2}_{null}}.$ (43)

Upon integration, it follows that $G=c_{1}-ln\sqrt{1+L^{2}_{null}}$ with the
constant of integration $c_{1}$ and the corresponding $F$ is obtained from Eq.
(24) to be $F=tan^{-1}(L_{null})+constant$. Now, inserting Eq. (43) into Eq.
(42), the equation of motion takes the following form

$\ddot{x}+\left(\frac{\Phi^{{}^{\prime\prime}}}{\Phi^{{}^{\prime}}}-mU^{{}^{\prime}}\right)\dot{x}^{2}-\frac{mU^{{}^{\prime}}}{(\Phi^{{}^{\prime}})^{2}}=0,$
(44)

where again $L_{null}=\Phi^{{}^{\prime}}\dot{x}$ is used. Next, following the
similar steps in Section 4.2, the exactly the same constraint on the gauge
function, $\Phi^{{}^{\prime\prime}}+\gamma_{0}f(x)(\Phi^{{}^{\prime}})^{3}=0$,
has to be satisfied to obtain an equation of motion for an oscillatory system
and therefore null-Lagrangians are obtained from
$L^{f}_{null}=\dot{x}\Phi^{{}^{\prime}}=\dot{x}\left[c_{1}+2\gamma_{0}\int^{x}{f(\xi)d\xi}\right]^{-1/2}$.

However, the appearance of a potential function $U(x,t)$ in the non-standard
Lagrangian is the new phenomenon in this construction and $U(x,t)$ must
satisfy

$\frac{\partial U}{\partial
x}+\frac{\gamma_{0}}{m}f(x)\left[c_{1}+2\gamma_{0}\int^{x}{f(\xi)d\xi}\right]^{-1}=0.$
(45)

In particular, for a simple harmonic oscillator, the above equation for
potential, Eq. (45), takes the following form

$\frac{\partial U}{\partial x}+\frac{\gamma_{0}x}{m(\gamma_{0}x^{2}+c_{1})}=0$

and the potential function is solved to be $U(x)=k-\frac{1}{m}\cdot
ln\sqrt{\gamma_{0}x^{2}+c_{1}}$ with $k$ and $c_{1}$ being constants of
integration. Finally, the corresponding non-standard Lagrangian is given by

$L_{ns}^{potential}=L_{null}^{f}\cdot
tan^{-1}(L_{null}^{f})-ln\sqrt{1+(L_{null}^{f})^{2}}+U(x),$ (46)

where again $L_{null}=\frac{\dot{x}}{\sqrt{\gamma_{0}x^{2}+c_{1}}}$.
Similarly, the potentials for other oscillators discussed above may be
obtained, which is avoided here for the sake of abstaining from repetition.

##  5 Discussion

Though the Null Lagrangians appear irrelevant in the derivation of equation of
motion for dynamical systems, the procedure above demonstrates that these
innocuous Lagrangians may have applications to the construction of non-
standard Lagrangians for important dynamical systems. Moreover, it follows
that the appropriate Null Lagrangian for a given dynamical system may be
determined from the above construction, which was not possible in the previous
construction. For example, the Null Lagrangian for a harmonic oscillator in
the above construction is given by
$L_{null}=\frac{\dot{x}}{\sqrt{\gamma_{0}x^{2}+c_{1}}}$, whereas that for the
Bateman oscillator turns out to be
$L^{f}_{null}=\frac{\sqrt{a}\dot{x}}{\sqrt{2}\sqrt{ac_{2}-\gamma_{0}cos(ax)}}$
with $c_{1}$ and $c_{2}$ being the constants of integration. Therefore, this
new generalized procedure enables us to determine a Null Lagrangian for a
dynamical system.

Next, the non-standard Lagrangians (NSLs) constructed in this paper from the
null Lagrangians are of special forms that must be now compared to the
previously found NSLs. The form of NSLs originally introduced by Arnold [6]
was extensively explored in many papers (e.g., [9,10,29-32]). These NSLs are
typically written in the following form [30,31]

$L_{ns1}(\dot{x},x,t)=\frac{1}{F(x)t)\dot{x}+G(x,t)x+H(x,t)}\ ,$ (47)

where $F(t,x)$, $G(t,x)$ and $H(t,x)$ are arbitrary, but at least twice
differentiable, scalar functions of time $t$ and $x$. The Lagrangian
$L_{ns1}(\dot{x},x,t)$ may also be given in a more general form

$L_{ns2}(\dot{x},x,t)=\frac{1}{B(x,t)\dot{x}+C(x,t)}\ ,$ (48)

where $B(x,t)=F(x,t)$ and $C(x,t)=G(x,t)x+H(x,t)$. An interesting result is
that the denominator of $L_{ns2}(\dot{x},x,t)$ is the general null Lagrangian
if, and only if, the following null condition is satisfied [23]:

$\left(\frac{\partial B(x,t)}{\partial
t}\right)=\frac{\partial[xC(x,t)]}{\partial x}\ .$ (49)

There are many different NSLs of the form given by Eq. (48) that were obtained
before for different dynamical systems (e.g., [9,10,29-32]). However, for most
of these NSLs the null condition is not satisfied, which means that the
relationship between the NSLs and NLs is not valid and, as a result, most
previously obtained NSLs do not have their corresponding NLs - see examples
presented in Section 2 of this paper as well as examples given in [23]. These
examples clearly show the limitations of the method developed in [23]; these
limitations are overcome in this paper as the presented approach allows
finding more general relationships between NSLs and NLs, and construct the
NSLs for some well-known oscillators (see Section 4).

The NSLs derived in this paper by using the newly developed generalized
procedure described in Section 3 have distinct forms from those obtained
before [9,10,29-32]. But they are also different from the NSLs obtained by
Nucci and Leach [8,33,34] and others [35], who constructed them by using the
Jacobi Last Multiplier method, and those NSLs proposed by El-Nabulsi
[7,36-38], who applied them to many different physics and astronomy problems.
One exception is the NSL originally found by Havas [39] for a harmonic
oscillator and recently explored in [40], which is a special case of Eq. (46),
when $\gamma_{0}=\omega^{2}$ and $c_{1}=0$, but it is different than the NSL
for a harmonic oscillator derived in this paper and given by Eq. (35); see
also Eq. (36).

Let us point out that the NSLs derived in this paper are significantly
different in their forms than those presented before in [23]. Moreover, as the
results of Section 2 show, the previous method based on the relationship
$L_{ns}(\dot{x},x,t)=1/L_{null}(\dot{x},x,t)$ allows only finding NSLs whose
forms are strongly limited. The generalized method developed in this paper
removes these limitations by postulating more general relationshisps between
$L_{null}(\dot{x},x,t)$ and $L_{ns}(\dot{x},x,t)$ (for details, see Section 3
and Section 4) and allows deriving a new family of NSLs for some of the best
known and most commonly used linear and nonlinear oscillators in dynamical
systems.

##  6 Conclusions

The exploration in this paper is focused on the generalization of the
applicability of Null Lagrangians in dynamics. The previous method to
construct non-standard Lagrangians from null Lagrangians [23] is reviewed and
its limitations are identified. A new generalized procedure that allows
constructing non-standard Lagrangians from null Lagrangians surpasses those
limitations and its validity is demonstrated by deriving the non-standard
Lagrangians and the equations of motion resulting from them for a harmonic
oscillator, the linear and nonlinear Bateman oscillators, and the Duffing
oscillator. In addition, the new generalized method allows the Null
Lagrangians for their respective dynamical systems to be determined uniquely.
The derived non-standard Lagrangians form a new family of Lagrangians in
Classical Mechanics. The obtained results show a new role played by the null
Lagrangians and their corresponding non-standard Lagrangians in describing
linear and nonlinear, and dissipative and non-dissipative dynamical systems.

In summary, a humble effort is made to explore the least investigated
territory of differential equations and dynamics, the space of Null-
Lagrangians, to discover the novel role of these null Lagrangians in dynamics.
We believe that this exploration is important because these null Lagrangians
will shed more light on our investigation into the fundamental nature of
dynamics.

##  7 References

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