1 Introduction

Consistent deformation of field theories is a powerful tool to explore in what sense or to what extent a given theory can be deformed by deforming its gauge symmetries and/or couplings in such a way the resulting theory is also consistent (removes the gauge degree of freedom in a consistent way). It is very useful to explore questions about if the theory is unique and/or how it can be extended to new field theories that could be interesting in physics. The deformations are regulated by symmetries and other constraints that we impose according to what questions we are addressing when deforming. All deformations preserve the field content of the given theory. A well-known example is a deformation of \(N^2-1\) free Maxwell fields where the gauge symmetry and the interaction, preserving global Poincaré symmetry, can be deformed to construct the Yang-Mills SU(N) theory. This deformation is also unique under the restrictions imposed [1].

We can also deform a free action in many other ways by imposing different restrictions. Maxwell’s theory can be deformed in different ways by preserving duality. Some celebrated examples are the Born-Infeld [2], Plebansky [3], and Biankii-Birula deformations [4, 5]. Recently a new deformation of Maxwell theory preserving conformal and duality invariance has been constructed [6, 7].Footnote 1 The result is quite interesting. It reduces to previously known deformations and gives us a new nonlinear electromagnetic theory (ModMax). Interestingly the theory preserves duality symmetry. As is well-known duality invariance is difficult to implement in interacting theories even if we know that the corresponding free theory is duality invariant. Another example of a theory with a duality invariance is the Fierz-Pauli action [9]. Still we do not know how to implement this duality in the full General Relativity. The duality symmetry plays a very important role in the formulation of string theory and many investigations have been developed to understand its implications. Understanding this fascinating symmetry is the focus of many studies in string theories.

The ModMax theory is a one-parameter family of nonlinear electrodynamics theories that was constructed in [6]. The Lagrangian derivation was also presented in [7]. The Legendre transformation between the Lagrangian and Hamiltonian theories is non-trivial. ModMax is constructed from the two basic Lorentz invariants of Maxwell theory \(S=-(1/4)F^{\mu \nu }F_{\mu \nu }\) and \(P=-(1/4)F_{\mu \nu }\tilde{F}^{\mu \nu }\), where \(\tilde{F}^{\mu \nu }\) is the Hodge dual of \(F_{\mu \nu }\), the usual Maxwell field strength. The conformal symmetry implies that the trace of the energy-momentum tensor is zero and this in turn implies that the Lagrangian is a homogeneous function of degree one in S and P. ModMax theory is the unique conformal U(1) gauge theory with a nonlinear constitutive relations, which preserves electric/magnetic duality invariance. In the phase space this means that ModMax is invariant under the S-duality transformation \((\textbf{D}\rightarrow -\textbf{B}, \textbf{B}\rightarrow \textbf{D})\). It is important to stress that, the duality symmetry is only manifest in Hamiltonian formalism. In Lagrangian formalism this invariance is not manifest [10]. The standard definition of \(\textbf{D}\) is \(\textbf{D}=\frac{\partial L}{\partial \textbf{E}}\) and in ModMax the vector \(\textbf{D}\) depends on \(\textbf{E}\) and \(\textbf{B}\) as in a generic nonlinear electrodynamics theory [11]. We review some of the basic properties of ModMax theory in appendices C, and D.

Although, there may be other Lorentz and duality invariant theories of electrodynamics corresponding to other deformations, there exist just two theories that are conformal invariant, Bialynicki-Birula electrodynamics and ModMax [6]. ModMax theory has been scrutinized and extended in recent works [12,13,14,15,16,17,18,19,20,21,22,23].

A very interesting question is if ModMax theory can be constructed from a deformation of Maxwell theory. Fortunately the answer to this question is affirmative and the deformation is know as a \(\sqrt{T\overline{T}}\) deformation. In 2D CFT a \(T\overline{T}\) operator is an irrelevant operator where T is the energy-momentum tensor. In holomorphic coordinates \(z,{\bar{z}}\) the components are denoted by \(T = T_{zz}\) and \(\overline{T} = T_ {\bar{z}\bar{z}}\) [26, 27]. Among the interesting properties of this irrelevant operator is that starting from an integrable QFT, the deformed theory preserves integrability [28, 29]. An example is \(T\overline{T}\) deformation of the massless scalar field action that leads to Nambu-Goto action [27]. Other examples and extensions can be found in [25, 30,31,32,33]. Another interesting property of this deformation is that it relates non/ultra-relativistic limits of the string sigma model [34, 35].

Denoting the deformed action as \(S_{QFT} = S_{CFT} + S_{\gamma }\) where \(S_\gamma = \gamma \int d^2x T\overline{T}\) and \(\gamma\) is the deformation parameter, the deformed action is a solution of the flow equation \(\frac{dS_{QF T}}{d\gamma } = \int d^2x T\overline{T}\) for finite \(\gamma\).

By applying these techniques, we can prove that ModMax is a deformation of Maxwell action [24, 25].

The role played by \(\sqrt{T\overline{T}}\)-deformations could have a relation with the realization of dualities in nonlinear theories [36].

Identifying \(\textbf{E}\rightarrow \varvec{\dot{\textbf{q}}},\textbf{D}\rightarrow \textbf{p},\textbf{B}\rightarrow \textbf{q}\), we present here a \(T{\bar{T}}\) deformation of a classical harmonic oscillator in 2D that shares many properties already present in ModMax theory. Duality symmetry is replaced by the transformation that rotates configuration variables with its conjugate momentum variables in the sense that \((\textbf{p} \rightarrow -\textbf{q},\textbf{q} \rightarrow \textbf{p})\). Lorentz symmetry is just represented by SO(2) space rotations and conformal symmetry restrict the form of the possible Hamiltonians (or Lagrangians) in a precise way that we will reveal later. The solution to these restrictions results in a dynamical system with a well-defined Hamiltonian (or Lagrangian), which also is parametrized by \(\gamma\) in the same way as in the ModMax theory.

Either in phase space or configuration space, the equations of motion and couplings are quite complicated and it seems to be very difficult to find an analytical solution to the system (see appendices A and B). Nevertheless, a numerical study reveals that the solution space is quite symmetric, developing beautiful curves, that encourage us to try to find an analytical solution. Moreover, as we know that the \(T\overline{T}\)-deformations preserve integrability we worked out the analytical solution.

Fig. 1
figure 1

This solution is numerical and plotted using \(\gamma =1\) and the initial conditions \(q^1(0)=0.7\), \(p^1(0)=0.2\), \(q^2(0)=1.0\), \(p^2(0)=0.5\), with time \(t\in \{0,100\}\)

Full size image

The Legendre transformation is also non-trivial as in the ModMax theory. In the classical mechanical system presented here the identification of two conserved quantities allows us to identify a relation between conjugate momenta and the configuration space variables \(q, \dot{q}\) that is invertible resolving the explicit Legendre transformation (Fig. 1).

In phase space, the generators of the infinitesimal duality rotations and the corresponding generator of space rotations are given, respectively, by

$$\begin{aligned} s=\frac{1}{2}(p^2+q^2), \quad j=\epsilon ^{ij}p^iq^j, \end{aligned}$$

and are constants of motion. Using them we can parametrize the space of solutions in terms of two functions A(s, j) and B(s, j). Notice that we are not reducing the degrees of freedom but just parametrizing, in a different way, the space of solutions. This was a key observation that allowed us to find the analytical solution. The oscillator frequencies emerge under this parametrization as the functions A and B.

Another interesting finding of this investigation is that we can construct a map (depending on \(q,p,\gamma\)) of the 2D harmonic oscillator to the full nonlinear oscillator at \(2\gamma\). We trace the existence of this map to the existence of two Lax pairs that we will present here. Starting from a Lax pair of the free harmonic oscillator in 2D the deformation map produces a new Lax pair and a constant of motion associated with the trace of the matrix \((L_2)^2\) that is precisely the deformed Hamiltonian in \(2\gamma\) (see below).

A reformulation of the problem in configuration space starting from the ModMax Lagrangian presented in [7] can be constructed in terms of two constants of motion \(C_1(\sigma , \rho )\), and \(C_2(\sigma , \rho )\) where the variables \(\sigma ,\rho\) are defined by

$$\begin{aligned} \sigma =\frac{1}{2}(\dot{q}^2+ q^2),\quad \rho =\epsilon ^{ij}\dot{q}^iq ^j, \end{aligned}$$

The analytical results suggests an interesting physical interpretation. The couplings between oscillators corresponds to an oscillator mounted in a non-inertial oscillating frame. The frequencies of the oscillators are parametrized by the constants of motion \(C_1\) and \(C_2\).

We reveal also a phenomenon of beats related to the transfer of energy between the coupled oscillators and we calculate the Hannay angle (that depends on the initial conditions through the numerical values of \(C_1\) and \(C_2\) evaluated on the given set of initial conditions).

All these interesting properties are rooted in the fact that we have a structure given by the duality symmetry implemented in the nonlinear problem.

The paper is organized as follows. In sect.2, we construct the nonlinear model in the Lagrangian and Hamiltonian formalism. Later, in sect. 3, we integrate the Hamiltonian system, additionally we introduce the necessary notation to implement Legendre’s transformation and integrate the Lagrangian system in sects. 4 and 5, respectively. In sect.6, we show how to construct the explicit map that transforms the 2D harmonic oscillator to the full nonlinear problem parametrized by \(2\gamma\) and we present two Lax pair formulations of this interesting property. In sect. 7, we show that our model presents an energy transfer phenomenon between the oscillators and compute the Hannay angle. Finally, in 8, we give our conclusions and the appendices A, B, C, D have complementary notes.

2 Nonlinear oscillator from a \(\sqrt{T\overline{T}}\)-deformation

Here we will construct the model using a \(\sqrt{T\overline{T}}\)-deformation. The model can be also constructed from scaling invariance (conformal invariance), see the appendices A and B. It consists of taking the homogeneous harmonic oscillator in 2D and deforming it as ModMax. Even though this is the simpler model, we consider that its theoretical details are relevant to the understanding the nonlinear systems that preserve duality and moreover improve the understanding of \(T\overline{T}\)-deformations.

2.1 \(\sqrt{T\overline{T}}\)-deformations in Lagrangian formalism

We perform a \(\sqrt{T\overline{T}}\)-deformations for the harmonic oscillator. First, we consider the Lagrangian of the harmonic oscillator in 2D with masses \(m_ i=1\) and frequencies \(\omega _ i=1\) for \(i=\{1,2\}\)

$$\begin{aligned} L_0=\frac{1}{2}\left( \dot{\varvec{\textbf{q}}}^2-\textbf{q}^2\right) , \end{aligned}$$
(1)

where \(\varvec{\dot{\textbf{q}}}^2=\dot{q}_1^2+\dot{q}_2^2\) and \(\textbf{q}^2=q_1^2+q_2^2\).

Due to the rotational and time translations invariance of the action, there exist two Noether conserved charges defined by

$$\begin{aligned} \delta _E q^i= & {} \dot{q}^i, E_0=\frac{1}{2}\left( \varvec{\dot{\textbf{q}}}^2+\textbf{q}^2\right) , \end{aligned}$$
(2)
$$\begin{aligned} \delta _J q^i= & {} \epsilon ^{ij}q^j,\quad J_0=\epsilon ^{ij}\dot{q}^iq^j. \end{aligned}$$
(3)

In order to preserve these symmetries at any order in \(\gamma\), the parameter of the deformation, we define the conserved quantities \(E_n\) and \(J_n\) whose are the energy and angular momentum of the deformed action at order n. Then we define the \(\sqrt{T\overline{T}}\) operator of order n

$$\begin{aligned} O^{\gamma }_{n}=\sqrt{E_n^2-J_n^2}. \end{aligned}$$
(4)

This is the analog to \(\sqrt{T\overline{T}}\) operator used in [24, 25, 30] to deform the Maxwell into ModMax theory.

The deformed Lagrangian of order \(n+1\) is defined by flow equation

$$\begin{aligned} L_{n+1}=L_0+\int d\gamma \ O_n^\gamma , \end{aligned}$$
(5)

then the first order deformation of the Lagrangian is

$$\begin{aligned} L_1=L_0+\gamma \sqrt{E_0^2-J_0^2}, \end{aligned}$$
(6)

which is invariant under \(\delta _E q^i\) and \(\delta _J q^i\) symmetries. Then the deformed conserved quantities are

$$\begin{aligned} E_1= & {} E_0\left (1+\frac{L_0\gamma }{\sqrt{E_0^2-J_0^2}}\right ), \end{aligned}$$
(7)
$$\begin{aligned} J_1= & {} J_0\left (1+\frac{L_0\gamma }{\sqrt{E_0^2-J_0^2}}\right ). \end{aligned}$$
(8)

Now, we are able to compute the operator \(O^\gamma _1\) as

$$\begin{aligned} O^{\gamma }_{1}=\sqrt{E_1^2-J_1^2}=\sqrt{E_0^2-J_0^2}+\gamma L_0, \end{aligned}$$
(9)

thus the second order Lagrangian is

$$\begin{aligned} L_2=L_0+\gamma \sqrt{E_0^2-J_0^2}+\frac{1}{2}\gamma ^2 L_0. \end{aligned}$$
(10)

Because \(\delta _E q^i\) and \(\delta _J q^i\) are still symmetries of the action, we obtain the deformed second order conserved quantities

$$\begin{aligned} E_2= & {} E_0\left (1+\frac{L_0\gamma }{\sqrt{E_0^2-J_0^2}}+\frac{1}{2}\gamma ^2\right ), \end{aligned}$$
(11)
$$\begin{aligned} J_2= & {} J_0\left (1+\frac{L_0\gamma }{\sqrt{E_0^2-J_0^2}}+\frac{1}{2}\gamma ^2\right ). \end{aligned}$$
(12)

and the second order \(T\overline{T}\)-like operator is

$$\begin{aligned} O_2^\gamma =\sqrt{E_0^2-J_0^2}+\gamma L_0+\frac{1}{2}\gamma ^2\sqrt{E_0^2-J_0^2}, \end{aligned}$$
(13)

then the third order Lagrangian is

$$\begin{aligned} L_3=L_0+\gamma \sqrt{E_0^2-J_0^2}+\frac{1}{2}\gamma ^2 L_0+\frac{1}{3!}\gamma ^3\sqrt{E_0^2-J_0^2}. \end{aligned}$$
(14)

By induction, it is straightforward to prove that the n order deformation to the Lagrangian function is

$$\begin{aligned} L=\sum _{n=0}^{\infty }\bigg [\bigg (\frac{\gamma ^{2n}}{(2n)!} \bigg )L_0+\bigg (\frac{\gamma ^{2n+1}}{(2n+1)!} \bigg )\sqrt{E_0^2-J_0^2}\bigg ]=L_ 0\cosh \gamma +\sinh \gamma \sqrt{E_0^2-J_0^2}, \end{aligned}$$
(15)

the \(\delta _E q^i\) and \(\delta _J q^i\) symmetries generate the conserved quantitiesFootnote 2

$$\begin{aligned} E= & {} E_ 0\sum _{n=0}^{\infty }\bigg [\bigg (\frac{\gamma ^{2n}}{(2n)!} \bigg )+\bigg (\frac{\gamma ^{2n+1}}{(2n+1)!} \bigg )\frac{L_0}{\sqrt{E_0^2-J_0^2}} \bigg ]=E_0\bigg (\cosh \gamma +\frac{L_0\sinh \gamma }{\sqrt{E_0^2-J_0^2}}\bigg ), \end{aligned}$$
(16)
$$\begin{aligned} J= & {} J_ 0\sum _{n=0}^{\infty }\bigg [\bigg (\frac{\gamma ^{2n}}{(2n)!} \bigg )+\bigg (\frac{\gamma ^{2n+1}}{(2n+1)!} \bigg )\frac{L_0}{\sqrt{E_0^2-J_0^2}} \bigg ]=J_0\bigg (\cosh \gamma +\frac{L_0\sinh \gamma }{\sqrt{E_0^2-J_0^2}}\bigg ), \end{aligned}$$
(17)

it is important to stress that, the energy and the angular momentum of the theory are E and J, respectively. The Hamiltonian function is NOT the energy of the dynamical system as we will see later.

To all orders, the \(\sqrt{T \overline{T}}\)-like operator is

$$\begin{aligned} O^\gamma =\sum _{k=0}^{\infty }\bigg [\bigg (\frac{\gamma ^{2n+1}}{(2n+1)!} \bigg )L_0+\bigg (\frac{\gamma ^{2n}}{(2n)!} \bigg )\sqrt{E_0^2-J_0^2}\bigg ]=L_ 0\sinh \gamma +\cosh \gamma \sqrt{E_0^2-J_0^2}, \end{aligned}$$
(18)

which defines \(T\overline{T}\) flow equation

$$\begin{aligned} \frac{d S_{NL}}{d \gamma }=\int dt\ O^\gamma , \end{aligned}$$
(19)

with \(S_{NL}=\int dt\ L\).

If we use the definitions

$$\begin{aligned} S=\frac{1}{2}\left( \dot{\bf q}^2-\textbf{q}^2\right) , P=\dot{\bf q}\cdot \textbf{q}, \end{aligned}$$
(20)

we note that

$$\begin{aligned} S=L_0, E_0^2-J_0^2=S^2+P^2. \end{aligned}$$
(21)

Thus the deformed Lagrangian function is

$$\begin{aligned} L=S\cosh \gamma +\sinh \gamma \sqrt{S^2+P^2}. \end{aligned}$$
(22)

In terms of \(E_0\) and \(J_0\) we can write down the Lagrangian function as

$$\begin{aligned} L=E_ 0\cosh \gamma +\sinh \gamma \sqrt{E_0^2-J_0^2}-\textbf{q}^2\cosh \gamma . \end{aligned}$$
(23)

This form of the Lagrangian will be useful later when we implement the Legendre transformation to construct the Hamiltonian formalism.

2.2 \(\sqrt{T\overline{T}}\)-deformations in Hamiltonian formalism

Now we will develop the \(\sqrt{T\overline{T}}\)-deformations in the Hamiltonian formalism. To implement the deformation in Hamiltonian formalism we start with conserved quantities and then compute the Noether symmetries, so we use the deformed conserved quantities just constructed. Notice that the deformed energy and angular momentum are (16) and (17). In order to construct the deformations order by order in \(\gamma\) as in the Lagrangian case, we start with the harmonic oscillator and its conserved quantities,

$$\begin{aligned} H_0= & {} \frac{1}{2}\left( \textbf{p}^2+\textbf{q}^2\right) , \end{aligned}$$
(24)
$$\begin{aligned} s_0= & {} \frac{1}{2}\left( \textbf{p}^2+\textbf{q}^2\right) ,\;j_0=\epsilon ^{ij}p^iq^j, \end{aligned}$$
(25)

where \(\textbf{p}^2=p_1^2+p_2^2\) and \(\textbf{q}^2=q_1^2+q_2^2\) and \(i=\{1,2\}\). The conserved quantities \(s_0\) and \(j_0\) are just the Hamiltonian generators of the duality and rotational symmetries,

$$\begin{aligned} \delta _{s_0}q^i= & {} \{q^i,s_0\}=p^i,\;\delta _{s_0}p^i=\{p^i,s_0\}=-q^i, \end{aligned}$$
(26)
$$\begin{aligned} \delta _{j_0}q^i= & {} \{q^i,j_0\}=\epsilon ^{ij}q^j,\;\delta _{j_0}p^i=\{p^i,j_0\}=-\epsilon ^{ij}p^j. \end{aligned}$$
(27)

It is important to notice that \(s_0\) coincides with the energy of the system at zero order in \(\gamma\). Nevertheless at order \(n>0\), \(s_0\) is not the energy but is still the conserved quantity associated with the duality symmetry.

In order to keep \(s_0\) and \(j_0\) as Hamiltonian generators of the duality and rotational symmetry, we define \(T\overline{T}\)-like Hamiltonian operator of order n as

$$\begin{aligned} O^\gamma _n=\sqrt{s_n^2-j_n^2}. \end{aligned}$$
(28)

The deformed Hamiltonian is defined as (flow equation)

$$\begin{aligned} H_{n+1}=s_0-\int d\gamma \ O^\gamma _n. \end{aligned}$$
(29)

The first order deformation of the Hamiltonian function is then

$$\begin{aligned} H_1=s_0-\gamma \sqrt{s_0^2-j_0^2}, \end{aligned}$$
(30)

As a second step we propose \(s_1\) and \(j_1\) in terms of \(s_0\) and \(j_0\) in the following way

$$\begin{aligned} s_1= & {} s_0\left (1-\frac{s_0\gamma }{\sqrt{s_0^2-j_0^2}}\right ), \end{aligned}$$
(31)
$$\begin{aligned} j_1= & {} j_0\left (1-\frac{s_0\gamma }{\sqrt{s_0^2-j_0^2}}\right ). \end{aligned}$$
(32)

\(s_1\) and \(j_1\) are conserved quantities of the system due to \(s_0\) and \(j_0\) are conserved. Using the Noether theorem, we compute the deformed symmetries that are generated by \(s_1\) and \(j_1\)

$$\begin{aligned} \delta _{s_1}q^i= & {} \frac{\partial }{\partial p^i}\left (s_0\left [1-\frac{s_0\gamma }{\sqrt{s_0^2-j_0^2}}\right ]\right ),\quad \delta _{s_1}p^i=-\frac{\partial }{\partial q^i}\left (s_0\left [1-\frac{s_0\gamma }{\sqrt{s_0^2-j_0^2}}\right ]\right ), \end{aligned}$$
(33)
$$\begin{aligned} \delta _{j_1}q^i= & {} \frac{\partial }{\partial p^i}\left (j_0\left [1-\frac{s_0\gamma }{\sqrt{s_0^2-j_0^2}}\right ]\right ),\quad \delta _{j_1}p^i=-\frac{\partial }{\partial q^i}\left (j_0\left [1-\frac{s_0\gamma }{\sqrt{s_0^2-j_0^2}}\right ]\right ), \end{aligned}$$
(34)

that correspond to duality and rotation deformed symmetries.

Now we construct the first order \(T\overline{T}\) operator

$$\begin{aligned} O_1^\gamma =\sqrt{s_0^2-j_0^2}-\gamma s_0, \end{aligned}$$
(35)

then the second order Hamiltonian function is

$$\begin{aligned} H_2=s_0-\gamma \sqrt{s_0^2-j_0^2}+\frac{1}{2}\gamma ^2 s_0. \end{aligned}$$
(36)

In order to construct the second order \(T\overline{T}\)-like operator, we propose \(s_2\) and \(j_2\) conserved quantities in terms of \(s_0\) and \(j_0\) as follows

$$\begin{aligned} s_2= & {} s_0\left (1-\frac{s_0\gamma }{\sqrt{s_0^2-j_0^2}}+\frac{1}{2}\gamma ^2\right ), \end{aligned}$$
(37)
$$\begin{aligned} j_2= & {} j_0\left (1-\frac{s_0\gamma }{\sqrt{s_0^2-j_0^2}}+\frac{1}{2}\gamma ^2\right ), \end{aligned}$$
(38)

these conserved quantities generate the deformed symmetries

$$\begin{aligned} \delta _{s_1}q^i= & {} \frac{\partial }{\partial p^i}\left (s_0\left [1-\frac{s_0\gamma }{\sqrt{s_0^2-j_0^2}}+\frac{1}{2}\gamma ^2\right ]\right ),\quad \delta _{s_1}p^i=-\frac{\partial }{\partial q^i}\left (s_0\left [1-\frac{s_0\gamma }{\sqrt{s_0^2-j_0^2}}+\frac{1}{2}\gamma ^2\right ]\right ), \end{aligned}$$
(39)
$$\begin{aligned} \delta _{j_1}q^i= & {} \frac{\partial }{\partial p^i}\left (j_0\left [1-\frac{s_0\gamma }{\sqrt{s_0^2-j_0^2}}+\frac{1}{2}\gamma ^2\right ]\right ),\quad \delta _{j_1}p^i=-\frac{\partial }{\partial q^i}\left (j_0\left [1-\frac{s_0\gamma }{\sqrt{s_0^2-j_0^2}}+\frac{1}{2}\gamma ^2\right ]\right ). \end{aligned}$$
(40)

which in turn are the deformed duality and rotation symmetries.

The second order \(T\overline{T}\)-like operator is

$$\begin{aligned} O_2^\gamma =\sqrt{s_0^2-j_0^2}-\gamma s_0+\frac{1}{2}\gamma ^2 \sqrt{s_0^2-j_0^2}, \end{aligned}$$
(41)

Continuing with the same steps, the third order Hamiltonian function is

$$\begin{aligned} H_3=s_0-\gamma \sqrt{s_0^2-j_0^2}+\frac{1}{2}\gamma ^2 s_0-\frac{1}{3!}\gamma ^3\sqrt{s_0^2-j_0^2}. \end{aligned}$$
(42)

By induction we prove that the Hamiltonian function at all orders in \(\gamma\) is

$$\begin{aligned} H=\sum _{n=0}^{\infty }\bigg [\bigg (\frac{\gamma ^{2n}}{(2n)!} \bigg )s_0-\bigg (\frac{\gamma ^{2n+1}}{(2n+1)!} \bigg )\sqrt{s_0^2-j_0^2}\bigg ]=s_ 0\cosh \gamma -\sinh \gamma \sqrt{s_0^2-j_0^2}, \end{aligned}$$
(43)

which was constructed using the deformed s and j variables

$$\begin{aligned} s= & {} s_0\left (\cosh \gamma -\frac{s_0\sinh \gamma }{\sqrt{s_0^2-j_0^2}}\right ), \end{aligned}$$
(44)
$$\begin{aligned} j= & {} j_0\left (\cosh \gamma -\frac{s_0\sinh \gamma }{\sqrt{s_0^2-j_0^2}}\right ). \end{aligned}$$
(45)

As a consequence of the underlaying scaling symmetry we found

$$\begin{aligned} \frac{s}{j}=\frac{s_0}{j_0}. \end{aligned}$$
(46)

These conserved quantities generate the deformed symmetries

$$\begin{aligned} \delta _{s}q^i= & {} \frac{\partial }{\partial p^i}\left (s_0\left [\cosh \gamma -\frac{s_0\sinh \gamma }{\sqrt{s_0^2-j_0^2}}\right ]\right ), \delta _{s}p^i=-\frac{\partial }{\partial q^i}\left (s_0\left [\cosh \gamma -\frac{s_0\sinh \gamma }{\sqrt{s_0^2-j_0^2}}\right ]\right ), \end{aligned}$$
(47)
$$\begin{aligned} \delta _{j}q^i= & {} \frac{\partial }{\partial p^i}\left (j_0\left [\cosh \gamma -\frac{s_0\sinh \gamma }{\sqrt{s_0^2-j_0^2}}\right ]\right ), \delta _{j}p^i=-\frac{\partial }{\partial q^i}\left (j_0\left [\cosh \gamma -\frac{s_0\sinh \gamma }{\sqrt{s_0^2-j_0^2}}\right ]\right ). \end{aligned}$$
(48)

The conserved quantities s and j can be used to construct \(T\overline{T}\)-like operator that deforms the harmonic oscillators to the nonlinear oscillators using the flow equation

$$\begin{aligned} H=-\int d\gamma \ O^\gamma \end{aligned}$$

where

$$\begin{aligned} O^\gamma =-s_ 0\sinh \gamma +\cosh \gamma \sqrt{s_0^2-j_0^2}, \end{aligned}$$
(49)

In the remainder of the article we will use instead of \(s_0, j_0\) just s, j so the Hamiltonian function is

$$\begin{aligned} H=s\cosh \gamma -\sqrt{s^2-j^2}\sinh \gamma . \end{aligned}$$
(50)

Notice that this procedure generates the correct deformed Hamiltonian that can be constructed in the same way as the ModMax Hamiltonian as presented in appendix C. In contrast with the Lagrangian case here the symmetries and the constants of motion are deformed.

3 Integration of the Hamiltonian system

Now we discuss how to use the conserved quantities s and j to integrate the Hamiltonian system. As is well-known when we have n degrees of freedom the Hamiltonian system needs n conserved quantities in involution to be integrable [37]. So from the knowledge of s and j we were able to integrate the system.

It is easy to prove that our nonlinear oscillators has two conserved quantities in involution that can be used to find an explicit analytic integration of the Hamiltonian equations of motion.

First notice that the Hamilton equations are

$$\begin{aligned} \begin{aligned} \dot{p}^i&=-\frac{\partial H}{\partial q^i}=-\left (\cosh (\gamma )-\frac{\sinh (\gamma )s}{\sqrt{s^2-j^2}}\right ) q^i+\frac{\sinh (\gamma )j}{\sqrt{s^2-j^2}}\epsilon ^{ij}p^j,\\ \dot{q}^i&=\frac{\partial H}{\partial p^i}=\left (\cosh (\gamma )-\frac{\sinh (\gamma )s}{\sqrt{s^2-j^2}}\right ) p^i+\frac{\sinh (\gamma )j}{\sqrt{s^2-j^2}}\epsilon ^{ij}q^j. \end{aligned} \end{aligned}$$
(51)

It is straightforward to prove that s and j are conserved quantities in involution, i.e

$$\begin{aligned} \{s,j\}=\{s,H\}=\{j,H\}=0, \end{aligned}$$
(52)

where \(\{F,G\}\) denotes the Poisson bracket.

Using this fact we can define two conserved quantities

$$\begin{aligned} A\equiv \cosh (\gamma )-\frac{\sinh (\gamma )s}{\sqrt{s^2-j^2}},B\equiv \frac{\sinh (\gamma )j}{\sqrt{s^2-j^2}}, \end{aligned}$$
(53)

that are also in involution

$$\begin{aligned} \{A,B\}=\{A,H\}=\{B,H\}=0. \end{aligned}$$
(54)

In terms of these quantities the Hamilton equations can be rewritten as

$$\begin{aligned} \begin{aligned} \dot{p}^i&=-A q^i+B\epsilon ^{ij}p^j,\\ \dot{q}^i&=A p^i+B\epsilon ^{ij}q^j, \end{aligned} \end{aligned}$$
(55)

Therefore, over the surface defined by A and B constants, the nonlinear Hamiltonian equations are just linear equations! Moreover this reduction is consistent with the variational principle as we show in the appendices A and B.

We can rewrite the differential equations in a matrix notation as

$$\begin{aligned} \begin{pmatrix} \dot{p}^i \\ \dot{q}^i \end{pmatrix}=\begin{pmatrix} B\epsilon ^{ij} &{} -A\delta ^{ij}\\ A\delta ^{ij} &{} B\epsilon ^{ij} \end{pmatrix}\begin{pmatrix} p^j \\ q^j \end{pmatrix},C^{ab}\equiv \begin{pmatrix} B\epsilon ^{ij} &{} -A\delta ^{ij}\\ A\delta ^{ij} &{} B\epsilon ^{ij} \end{pmatrix},z^a\equiv \begin{pmatrix} p^j \\ q^j \end{pmatrix}, \end{aligned}$$
(56)

with \(a,b=\{1,2,3,4\}\). Over the surface defined by A and B constants, the solutions of the system are obtained by taking the exponential of the matrix C

$$\begin{aligned} \dot{z}^a=D^{ab}z_0^b,D^{ab}=\big (e^{C t}\big )^{ab} \end{aligned}$$
(57)

where \(z_0^b\) are the initial conditions.

A remarkable property of these dynamics is the fact the matrix \(C^{ab}\) admits a decomposition in term of two matrices \(C_A\) and \(C_B\), respectively,

$$\begin{aligned} C^{ab}=\begin{pmatrix} B\epsilon ^{ij} &{} -A\delta ^{ij}\\ A\delta ^{ij} &{} B\epsilon ^{ij} \end{pmatrix}=\begin{pmatrix} 0 &{} -A\delta ^{ij}\\ A\delta ^{ij} &{}0 \end{pmatrix}+\begin{pmatrix} B\epsilon ^{ij} &{} 0\\ 0 &{} B\epsilon ^{ij} \end{pmatrix}, \end{aligned}$$
(58)

These matrices commute between themselves and the exponential of C is the exponential of \(C_A\) times the exponential of \(C_B\),

$$\begin{aligned} D^{ab}=\big (e^{C_A t}e^{C_B t}\big )^{ab}=\big (e^{C_B t}e^{C_A t}\big )^{ab}, D_A\equiv e^{C_A t},D_B\equiv e^{C_B t}. \end{aligned}$$
(59)

Then we observe that the dynamics of our system is the same as two coupled oscillators with frequencies A and B (that now we fix as numbers corresponding with the initial data). The explicit construction is then

$$\begin{aligned} D^{ab}_A z_0^b= \begin{pmatrix} p^i_0\cos (At)-q^i_0\sin (At) \\ q^i_0\cos (At)+p^i_0\sin (At) \end{pmatrix}, \end{aligned}$$
(60)

and

$$\begin{aligned} D^{ab}_B z_0^b= \begin{pmatrix} p^i_0\cos (Bt)+\epsilon ^{ij}p^j_0\sin (Bt) \\ q^i_0\cos (Bt)+\epsilon ^{ij}q^j_0\sin (Bt) \end{pmatrix}. \end{aligned}$$
(61)

Notice that the oscillations are not the same in phase-space.

The solutions of the system are

$$\begin{aligned} \begin{aligned} p^i(t)=&\cos (At)(p_0^i\cos (Bt)+\epsilon ^{ij}p_0^j\sin (Bt))-\sin (At)(q_0^i\cos (Bt)+\epsilon ^{ij}q_0^j\sin (Bt)),\\ q^i(t)=&\sin (At)(p_0^i\cos (Bt)+\epsilon ^{ij}p_0^j\sin (Bt))+\cos (At)(q_0^i\cos (Bt)+\epsilon ^{ij}q_0^j\sin (Bt)), \end{aligned} \end{aligned}$$
(62)

where \(p^i_0\) and \(q^i_0\) are the initial conditions. These relations are the general solution of the system (51) and correspond to two coupled oscillators with frequencies A and B, respectively.

4 Legendre’s transform

As has been shown in [6], Legendre’s transformation in ModMax theory must be implemented carefully. Due to the nonlinear behavior of the theory, the task of inverting the velocities in terms of momentum variables is cumbersome. A more easy task is to implement Legendre’s transformation over the surface A and B constants because there the system is linear (cf. eq. (55)). Moreover we know that a deduction of the corresponding Lagrangian [7] can be implemented from first principles (appendix D).

First, we consider the Hamiltonian function,

$$\begin{aligned} H(q,p)=\frac{1}{2}A(q,p)(\textbf{p}^2+\textbf{q}^2)+B(q,p)\epsilon ^{ij}p^iq^j, \end{aligned}$$
(63)

and we take A and B as constants. The Hamiltonian equations are

$$\begin{aligned} \begin{aligned} \dot{p}^i&=-A q^i+B\epsilon ^{ij}p^j,\\ \dot{q}^i&=A p^i+B\epsilon ^{ij}q^j, \end{aligned} \end{aligned}$$
(64)

by definition the substitution of A and B in terms of phase space variables in (63) leads to the nonlinear Hamiltonian function (50). A remarkable property of this Hamiltonian is that we can recover from it the complete equations of motion. Now we can invert the momentum variables in terms of the configuration space

$$\begin{aligned} p^i=\frac{1}{A}\dot{q}^i-\frac{B}{A}\epsilon ^{ij}q^j, \end{aligned}$$
(65)

and over the surface defined by A and B constants, compute the Lagrangian function

$$\begin{aligned} L=\frac{1}{2A}\left( \dot{\textbf{q}}^2-\left( A^2-B^2\right) \textbf{q}^2\right) -\frac{B}{A}\epsilon ^{ij}\dot{q}^iq^j, \end{aligned}$$
(66)

of course, this Lagrangian functions is consistent with the momentum definition (65).

Now the question is how to write the corresponding expressions for A and B in terms of the configuration space. With the notation

$$\begin{aligned} \sigma \equiv \frac{1}{2}(\varvec{\dot{q}}^2+\textbf{q}^2)=E_0,\rho \equiv \epsilon ^{ij}\dot{q}^iq^j=J_0, \end{aligned}$$
(67)

where \(E_0\) and \(J_0\) are in terms of \(q,\dot{q}\) (eqs. (2),(3)), we can define the two functions \(C_1\) and \(C_2\)

$$\begin{aligned} C_1\equiv \cosh \gamma +\frac{\sigma \sinh \gamma }{\sqrt{\sigma ^2-\rho ^2}},C_2\equiv -\frac{\rho \sinh \gamma }{\sqrt{\sigma ^2-\rho ^2}}. \end{aligned}$$
(68)

Now the Lagrangian function (22) is

$$\begin{aligned} L(\dot{\textbf{q}},\textbf{q})=C_1(\dot{\textbf{q}},\textbf{q})\sigma (\dot{\textbf{q}},\textbf{q})+C_2(\dot{\textbf{q}},\textbf{q})\rho (\dot{\textbf{q}},\textbf{q})-\textbf{q}^2\cosh \gamma . \end{aligned}$$
(69)

Now we define the momentum by

$$\begin{aligned} p^i=C_1(\dot{\textbf{q}},\textbf{q})\dot{q}^i+C_2(\dot{\textbf{q}},\textbf{q})\epsilon ^{ij}q^j. \end{aligned}$$
(70)

This definition of the momentum is consistent with the starting Lagrangian in terms of \((q,\dot{q})\). This property comes from the fact that the Lagrangian is a homogeneus function of degree one in \(\sigma\) and \(\rho\) (see appendix A for details). Demanding the consistency between the Hamiltonian momentum (65) and the Lagrangian momentum (70) we conclude that the implicit Legendre’s transformation is

$$\begin{aligned} C_1(\dot{\textbf{q}},\textbf{q})=\frac{1}{A(\textbf{p},\textbf{q})}, C_2(\dot{\textbf{q}},\textbf{q})=-\frac{B(\textbf{p},\textbf{q})}{A(\textbf{p},\textbf{q})}, \end{aligned}$$
(71)

and using these identities, we can prove that the Lagrangian function (66) matches the Lagrangian function (69), the complete nonlinear Lagrangian function.

Therefore we have implemented Legendre’s transformation over the surface defined by A and B constants. Due to A and B being constants of motion in phase space, \(C_1\) and \(C_2\) are also conserved in configuration space. This last statement can also be proved using the Lagrangian formalism. We want to stress that although \(C_1\) and \(C_2\) are conserved, \(\sigma\) and \(\rho\) are NOT conserved.

5 Integration of the Lagrangian system

Now we have all the necessary tools to show the complete integration of the Lagrangian equations of motion. As in Hamiltonian formalism, we use the conserved quantities at the level of the equation of motion to integrate the system.

The Lagrangian equations of motion are

$$\begin{aligned} \ddot{q}^i=\frac{C_2^2-1}{C_1^2}q^i-2\frac{C_2}{C_1}\epsilon ^{ij}\dot{q}^j, \end{aligned}$$
(72)

where we are using that \(C_1\) and \(C_2\) are constants of motion.

The substitution of the expressions (68) leads to the nonlinear equation of motion of the Lagrangian function (22). More details about how the equations of motion can be reduced to this form are given in the appendix A. In particular, these differential equations are linear over the surface defined by \(C_1\) and \(C_2\) as constants, and the solution is

$$\begin{aligned} q^{i} (t) &= \cos \left( {\frac{1}{{C_{1} }}t} \right)\left[ {q_{0}^{i} \cos \left( {\frac{{C2}}{{C1}}t} \right) - \varepsilon ^{{ij}} q_{0}^{j} \sin \left( {\frac{{C2}}{{C1}}t} \right)} \right] \hfill \\ &\quad + \sin \left( {\frac{1}{{C_{1} }}t} \right)\left[ {\left( {C_{1} \dot{q}_{0}^{i} + C_{2} \varepsilon ^{{ij}} q_{0}^{j} } \right)\cos \left( {\frac{{C2}}{{C1}}t} \right) - \varepsilon ^{{ij}} \left( {C_{1} \dot{q}_{0}^{j} + C_{2} \varepsilon ^{{jk}} q_{0}^{k} } \right)\sin \left( {\frac{{C2}}{{C1}}t} \right)} \right]. \hfill \\ \end{aligned}$$
(73)

We notice that using the momentum’s definition (65), the Hamiltonian solutions (62) are consistent with (73). This fact is another check that our Legendre’s transformation over the surface defined by A and B constants, or equivalently \(C_1\) and \(C_2\) as constants, is consistent.

Interestingly the functional form of the Hamiltonian and Lagrangian functions in terms of (s, j) and (S, P), respectively, are the same as the Hamiltonian and Lagrangian densities of ModMax [6, 7].

6 A deformation map

In this section we will show how to construct a map from the usual oscillator in 2D to the nonlinear coupled oscillator presented here.

Starting from a 2D oscillator with mass and frequency equal to one

$$\begin{aligned} H_0=\frac{1}{2}\left( {\widehat{p}}^2+{\widehat{q}}^2\right) \end{aligned}$$
(74)

we define the variables \({\widehat{p}}_i, {\widehat{q}}^i\) , using the matrix notation

$$\begin{aligned} \begin{pmatrix} {\widehat{p}}_i \\ {\widehat{q}}^i \end{pmatrix} ={{\varvec{M}}}(\gamma )\begin{pmatrix} p^j \\ q^j \end{pmatrix}, \end{aligned}$$
(75)

where

$$\begin{aligned} {{\varvec{M}}}(\gamma )= \begin{pmatrix} B(q,p,\gamma )\epsilon ^{ij}&{} -A(q,p,\gamma )\delta ^{ij} \\ A(q,p,\gamma )\delta ^{ij}&{} B(q,p,\gamma )\epsilon ^{ij} \end{pmatrix}. \end{aligned}$$
(76)

As A and B are functions of \(q,p,\gamma\) the map is quite nonlinear. But on the surface A and B constants the map looks linear. This map is very powerful because, in just one step, implements the deformation of the harmonic oscillator (74) into the full nonlinear oscillator in \(2\gamma\)

$$\begin{aligned} H(q,p,2\gamma )=\frac{1}{2}\Big (A^2(q,p,\gamma )+B^2(q,p,\gamma )\Big )(\textbf{p}^2+\textbf{q}^2)+2A(q,p,\gamma )B(q,p,\gamma )\epsilon ^{ij}p^iq^j. \end{aligned}$$
(77)

Doing the explicit calculation taking into account the functional dependence of A and B in terms of \(q,p,\gamma\) we find

$$\begin{aligned} H(q,p,2\gamma )=\frac{1}{2}A(q,p,2\gamma )(\textbf{p}^2+\textbf{q}^2)+B(q,p,2\gamma )\epsilon ^{ij}p^iq^j, \end{aligned}$$
(78)

thus this map does the job of the \(T{\bar{T}}\) deformation in just one step. But notice that here the deformation is obtained at \(2\gamma\).

Analogous properties can be found for ModMax theory. Footnote 3See [6, 7] for details.

In Lagrangian formalism the same idea can be implemented as follows. First we notice that the Lagrangian (66) in terms of \(C_1\) and \(C_2\) can be written as

$$\begin{aligned} L=\frac{1}{2}C_1\bigg (\varvec{\dot{q}}^2+\frac{C_2^2-1}{C_1^2}\textbf{q}^2 \bigg )+C_2\epsilon ^{ij}\dot{q}^iq^j. \end{aligned}$$
(81)

Using the matrix notation

$$\begin{aligned} L=\frac{1}{2} \begin{pmatrix} \dot{q}^i \\ q^i \end{pmatrix}^T \begin{pmatrix} C_1\delta ^{ij} &{} C_2\epsilon ^{ij}\\ -C_2\epsilon ^{ij} &{} \frac{C_2^2-1}{C_1}\delta ^{ij} \end{pmatrix}\begin{pmatrix} \dot{q}^j \\ q^j \end{pmatrix}. \end{aligned}$$
(82)

Now, we define the matrices and the vector

$$\begin{aligned} M^{ab}=\begin{pmatrix} C_1\delta ^{ij} &{} C_2\epsilon ^{ij}\\ C_2\epsilon ^{ij} &{} -\frac{C_2^2-1}{C_1}\delta ^{ij} \end{pmatrix},g^{ab}=\begin{pmatrix} \delta ^{ij} &{} 0\\ 0&{} -\delta ^{ij} \end{pmatrix},v^a=\begin{pmatrix} \dot{q}^i \\ q^i \end{pmatrix}, \end{aligned}$$
(83)

\(i={1,2}\) and \(a,b={1,2,3,4}\). Using these definitions we have

$$\begin{aligned} L[v;2\gamma ]=\frac{1}{2}(v^{T})^a(M^T(\gamma ))^{ab}g^{bc}M^{cd}(\gamma )v^d, \end{aligned}$$
(84)

where \(M(\gamma )\) is the matrix M evaluated on \(\gamma\). Then we identify the variables

$$\begin{aligned} \hat{v}^a=M(\gamma )^{ab}v^b, \end{aligned}$$
(85)

Using these variables (85) the Lagrangian (84) is written as

$$\begin{aligned} L[\hat{v};2\gamma ]=\frac{1}{2}(\hat{v}^T)^ag^{ab}\hat{v}^b, \end{aligned}$$
(86)

This last result is the Lagrangian of the harmonic oscillator in 2D. So we started with the full nonlinear Lagrangian and obtain through the definition (85) the Lagrangian of the harmonic oscillator 2D.

Returning to the Hamiltonian description of the map that relates the 2D free harmonic oscillator (74) with the full deformed oscillator in \(2\gamma\) (78) we will present here two Lax pairs. These two Lax pairs can be obtained from the Lax pair of the harmonic oscillator in 2D

$$\begin{aligned} L_o=\begin{pmatrix} p_1 &{} q_1 &{} p_2 &{} q_2\\ q_1 &{} -p_1 &{} q_2 &{} -p_2\\ p_2 &{} q_2 &{} -p_1 &{} -q_1\\ q_2 &{} -p_2 &{} -q_1 &{} p_1 \end{pmatrix}, N_o=\begin{pmatrix} 0 &{} \frac{1}{2} &{} 0 &{} 0\\ -\frac{1}{2} &{} 0 &{} 0 &{} 0\\ 0 &{} 0 &{} 0 &{} \frac{1}{2}\\ 0 &{} 0 &{} -\frac{1}{2} &{} 0 \end{pmatrix}. \end{aligned}$$
(87)

The first Lax pair can be obtained by just deforming the matrix N and with the same \(L_o\)

$$\begin{aligned} N=\begin{pmatrix} 0 &{} \frac{A}{2} &{} -\frac{B}{2} &{} 0\\ -\frac{A}{2} &{} 0 &{} 0 &{} -\frac{B}{2}\\ \frac{B}{2} &{} 0 &{} 0 &{} \frac{A}{2}\\ 0 &{} \frac{B}{2} &{} -\frac{A}{2} &{} 0 \end{pmatrix}. \end{aligned}$$
(88)

The resulting Lax pair is (with the identification \(L_1=L_o\))

$$\begin{aligned} L_1=\begin{pmatrix} p_1 &{} q_1 &{} p_2 &{} q_2\\ q_1 &{} -p_1 &{} q_2 &{} -p_2\\ p_2 &{} q_2 &{} -p_1 &{} -q_1\\ q_2 &{} -p_2 &{} -q_1 &{} p_1 \end{pmatrix}, N=\begin{pmatrix} 0 &{} \frac{A}{2} &{} -\frac{B}{2} &{} 0\\ -\frac{A}{2} &{} 0 &{} 0 &{} -\frac{B}{2}\\ \frac{B}{2} &{} 0 &{} 0 &{} \frac{A}{2}\\ 0 &{} \frac{B}{2} &{} -\frac{A}{2} &{} 0 \end{pmatrix}, \end{aligned}$$
(89)

and the equations of motion are

$$\begin{aligned} \dot{L}_1=[L_1,N]=L_1N-N L_1=\begin{pmatrix} B p_2 - A q_1 &{} A p_1 + B q_2&{} -B p_1 - A q_2 &{} A p_2 - B q_1\\ A p_1 + B q_2&{} -B p_2 + A q_1&{} A p_2 - B q_1&{} B p_1 + A q_2\\ -B p_1 - A q_2&{} A p_2 - B q_1&{} -B p_2 + A q_1&{} -A p_1 - B q_2\\ A p_2 - B q_1&{} B p_1 + A q_2&{} -A p_1 - B q_2&{} B p_2 - A q_1 \end{pmatrix}. \end{aligned}$$
(90)

The equation (90) reproduces exactly the Hamiltonian equations of the system. In addition, the trace of \(L_1^2\) satisfies

$$\begin{aligned} s=\frac{1}{8}\text {Tr}(L_1^2). \end{aligned}$$
(91)

The other deformation of the harmonic oscillator Lax pair (87) can be obtained by our nonlinear mapping (75) applied to \(L_o\) and with the same deformed matrix N (88). The deformed matrix \(L_2\) is

$$\begin{aligned} L_2=\begin{pmatrix} A p_1 + B q_2&{} -B p_2 + A q_1&{} A p_2 - B q_1&{} B p_1 + A q_2\\ -B p_2 + A q_1&{} -A p_1 - B q_2&{} B p_1 + A q_2&{} -A p_2 + B q_1\\ A p_2 - B q_1&{} B p_1 + A q_2&{} -A p_1 - B q_2&{} B p_2 - A q_1\\ B p_1 + A q_2&{} -A p_2 + B q_1&{} B p_2 - A q_1&{} A p_1 + B q_2 \end{pmatrix}. \end{aligned}$$
(92)

The Lax pair \((L_2,N)\) gives then an equivalent set of equations of motion of the original deformed equations of motion. Explicitly

$$\begin{aligned} \begin{aligned} \dot{L}_2 & =[L_2,N]=L_2 N-N L_2\\ &=\begin{pmatrix} 2 A B p_2 - (A^2 + B^2) q_1&{} (A^2 + B^2) p_1 + 2 A B q_2&{} -2 A B p_1 - (A^2 + B^2) q_2&{} (A^2 + B^2) p_2 - 2 A B q_1\\ (A^2 + B^2) p_1 + 2 A B q_2&{} -2 A B p_2 + (A^2 + B^2) q_1 &{}(A^2 + B^2) p_2 - 2 A B q_1&{} 2 A B p_1 + (A^2 + B^2) q_2\\ -2 A B p_1 - (A^2 + B^2) q_2&{} (A^2 + B^2) p_2 - 2 A B q_1&{} -2 A B p_2 + (A^2 + B^2) q_1&{} -(A^2 + B^2) p_1 - 2 A B q_2\\ (A^2 + B^2) p_2 - 2 A B q_1&{} 2 A B p_1 + (A^2 + B^2) q_2&{} -(A^2 + B^2) p_1 - 2 A B q_2&{} 2 A B p_2 - (A^2 + B^2) q_1 \end{pmatrix}. \end{aligned} \end{aligned}$$
(93)

To obtain the above result we need to take into account that A and B are constants of motion of the deformed system.

After using the explicit expressions for A and B in terms of \((p_i,q_i)\), the trace of \(L_2^2\) satisfies

$$\begin{aligned} H(s,j;2\gamma )=\frac{1}{8}\text {Tr}(L_2^2). \end{aligned}$$
(94)

This is the resulting deformed Hamiltonian of the one step deformation obtained above (78). So this Lax pair can be used to explain why the deformed oscillator is in \(2\gamma\) and not in \(\gamma\) as we could expect. We will return to this interesting point elsewhere, but now in context of field theory and ModMax theory where the Lax pairs can be used study the integrability of the deformed Maxwell theory.

7 Some dynamical properties of the system

In this section, we present two properties of the dynamical system. First, we show how these two coupled oscillators have an energy transfer phenomena between oscillators. Then we compute the Hannay angle, which is the classical analog of the Berry phase for the corresponding quantum system. So the Hannay angle captures geometrical information (holonomy, parallel transport) in the space of solutions.

7.1 Energy transfer

A well-known property of two coupled systems that oscillate and conserve energy, is the energy transfer phenomenon [38]. It consists in that the amplitudes of the two oscillators also oscillate in such a way that when one of the systems oscillates with a large amplitude, the second one oscillates with a small amplitude and vice versa.

If we consider the amplitudes

$$\begin{aligned} \alpha ^i= q_0^i\cos \bigg (\frac{C_2}{C_1}t\bigg )-\epsilon ^{ij}q_0^j\sin \bigg (\frac{C_2}{C_1}t\bigg ), \end{aligned}$$
(95)

and

$$\begin{aligned} \beta ^i=\big (C_1\dot{q}_0^i+C_2\epsilon ^{ij}q_0^j\big )\cos \bigg (\frac{C_2}{C_1}t\bigg )-\epsilon ^{ij}\big (C_1\dot{q}^j_0+C_2\epsilon ^{jk}q_0^k\big )\sin \bigg (\frac{C_2}{C_1}t\bigg ), \end{aligned}$$
(96)

the solutions of the Lagrangian (73) system have the form

$$\begin{aligned} q(t)^i=\alpha ^i \cos \bigg (\frac{1}{C_1}t\bigg )+\beta ^i \sin \bigg (\frac{1}{C_1}t\bigg ) \end{aligned}$$
(97)

which are oscillation functions that have amplitudes that also oscillate.

We can observe the transfer phenomenon if we plot some solutions of the system.(Fig. 2)

Fig. 2
figure 2

These solutions are plotted using \(\gamma =1\) and the initial conditions \(q^1(0)=1.5\), \(\dot{q}^1(0)=-5.1\), \(q^2(0)=0.3\), \(\dot{q}^2(0)=1\)

Full size image

7.2 Hannay angle

As we have observed, our system could be interpreted as an oscillator with frequency A mounted in a non-inertial reference frame that also oscillates but with frequency B. The Hannay angle [39] could be interpreted as a phase shift between the two oscillators.

We compute the Hannay angle considering the shift \(2\pi +Bt'\), where \(t'\) is the time that takes one period of the first oscillator with frequency A. Then the Hannay angle in our system is

$$\begin{aligned} \Theta _H=2\pi +Bt'=2\pi \left( 1+\frac{B}{A}\right) , t'=\frac{2\pi }{A}. \end{aligned}$$
(98)

Using the second equation in (71), we can write the Hannay angle as

$$\begin{aligned} \Theta _H=2\pi \bigg (1-\frac{\rho \sinh \gamma }{\sqrt{\sigma ^2-\rho ^2}}\bigg ). \end{aligned}$$
(99)

Of course, when \(\gamma =0\), then \(\Theta _H=2\pi\), it means that the two oscillators are in phase. This geometrical angle depends on the initial conditions through the definitions A and B evaluated at the corresponding initial condition vector. It also is the consequence of a coupling that can be interpreted as a covariant derivative

$$\begin{aligned} Dv^i=\frac{d v^i}{dt}+\frac{C_2}{C_1}\epsilon ^{ij}q^j, \end{aligned}$$
(100)

then we can write the Lagrangian function (66) as

$$\begin{aligned} L=C_1Dq^i Dq^i-\frac{1}{C_1}q^2. \end{aligned}$$
(101)

8 Conclusions

We have constructed a nonlinear classical system, which consists of two coupled oscillators. The construction can be performed by a \(\sqrt{T\overline{T}}\)-deformation (in Lagrangian and Hamiltonian formalisms) of the 2D homogeneous free harmonic oscillators. The system has a duality symmetry and conformal (scaling) symmetries. It could be interesting to study our system in the context of T-duality for point particles as [40].

The system is integrable, it has two conserved charges in involution s and j, which are associated with duality and rotational symmetry, respectively. It is very interesting that considering the constants of motion A and B (defined in terms of s and j), it is straightforward to integrate the equation of motion as a linear system. Moreover, A and B are the frequencies of the two coupled oscillators, respectively. Because we can use the conserved quantities in the action, we found how to perform the Legendre transformation in a simple way. It could be interesting to investigate the possibility of thinking of our model as a non-relativistic limit of some action, this research area has been addressed by [34, 35, 41] in the context of \(T{\bar{T}}\) deformations. Many of the dynamical properties of the system studied here can be translated to ModMax theory. This step is a work in progress that we will publish elsewhere.

We construct a map of the 2D harmonic oscillator with mass and frequency equal to 1 to the nonlinear system at \(2\gamma\). This map performs, in just one step, the complete deformation of the harmonic oscillator produced by \(T{\bar{T}}\) mechanism.

We studied the mechanical properties of our system and found that the system presents an energy transfer phenomenon. Finally, we compute the Hannay angle, which is a geometrical property of the system and space of solutions.

We have many advances understanding the quantum theory and the resulting quantum system. But our advances are not completely well understood and we need to work more to try to understand better the quantum system. In particular we think that upon quantization the frequencies of our system are quantized and not just the Hamiltonian (energy levels). The result can be interesting to study of the quantum metric tensor, entanglement, mutual entropy, quantum energy transfer, Berry phase and perhaps time-crystals. We are working along these lines and we will report our results elsewhere when we have a better understanding of the physical and conceptual properties of the system.

Other applications and extensions like the supersymmetric case and the relativistic version are also worth to study in a future development of the ideas presented here.