A Reappraisal of Lagrangians with Non-Quadratic Velocity Dependence and Branched Hamiltonians

Abstract

Time and again, non-conventional forms of Lagrangians with non-quadratic velocity dependence have received attention in the literature. For one thing, such Lagrangians have deep connections with several aspects of nonlinear dynamics including specifically the types of the Liénard class; for another, very often, the problem of their quantization opens up multiple branches of the corresponding Hamiltonians, ending up with the presence of singularities in the associated eigenfunctions. In this article, we furnish a brief review of the classical theory of such Lagrangians and the associated branched Hamiltonians, starting with the example of Liénard-type systems. We then take up other cases where the Lagrangians depend on velocity with powers greater than two while still having a tractable mathematical structure, while also describing the associated branched Hamiltonians for such systems. For various examples, we emphasize the emergence of the notion of momentum-dependent mass in the theory of branched Hamiltonians.

1. Introduction

During the past few decades, the study of non-conventional types of dynamical systems, in particular those that are controlled by Lagrangians that are not quadratic in velocity, has entered a new phase of intense development [1,2,3,4]. Such Lagrangians lead to certain exotic Hamiltonians, commonly termed as branched Hamiltonians, which have relevance in their applicability to problems of nonlinear dynamics pertaining to autonomous differential equations [5,6] and to certain exotic quantum mechanical models, especially in the context of non-Hermitian parity-time ($\mathcal{PT}$)-symmetric schemes [7], along with their relativistic counterparts [8].

A simple way to see how Lagrangians that are not quadratic in velocity can lead to meaningful dynamical systems is to consider the following toy model [9,10] (see also Refs. [11,12]):

$$L(x,\dot{x})={(\alpha x+\beta \dot{x})}^{-1},$$

(1)

where $\alpha$ and $\beta$ are real numbers satisfying $\alpha \beta >0$. We may also require that $\alpha x+\beta \dot{x}\ne 0$, i.e., that the velocity phase space accessible to the system is defined as a subset of ${\mathbb{R}}^2$.

Notice that the Lagrangian cannot be expressed as the difference between the kinetic and potential energies; such Lagrangians shall be referred to as nonstandard, i.e., in this paper, we will be adopting such nomenclature in which the term ‘nonstandard Lagrangian’ would refer to a Lagrangian with a non-quadratic velocity dependence. (A linear dependence on velocity makes the Hessian matrix singular, resulting in a singular Legendre transform while passing from the Lagrangian to the Hamiltonian formalism (see, for example, Ref. [13]). We do not address such cases here and deal with Lagrangians that have velocity dependence either in excess of quadratic powers or in inverse powers).

A direct computation reveals that the Euler–Lagrange equation is

$$\ddot{x}+\gamma \dot{x}+{\omega }_0^{2}x=0,$$

(2)

where $\gamma =\frac{3\alpha }{2\beta }$ and ${\omega }_0=\frac{\alpha }{\sqrt{2}\beta }$. (2) is just the harmonic oscillator in the presence of linear damping. We remind the reader that there is no time-independent Lagrangian of the ‘standard’ kind from which one can reproduce (2) upon invoking the Euler–Lagrange equation. (One could recover the damped oscillator from a standard Lagrangian by using a Rayleigh dissipation function [14]. Alternatively, one can consider the modified Euler–Lagrange equations from the Herglotz variational problem to describe the damped oscillator [15]. We do not consider such situations here.) There exist various other families of nonstandard Lagrangians (giving rise to different dynamical systems), which look quite different from (1); each family is endowed with its own intriguing features. However, the common theme is the existence of Lagrangians that are not quadratic in velocity, thereby leading to a nonlinear relationship between the velocity and the momentum. It may be emphasized that a Lagrangian $L=L(x,\dot{x})$ defined for a system whose configuration space is a subset of $\mathbb{R}$ is called regular if its Hessian with respect to the velocity is non-vanishing, i.e.,

$$\frac{{\partial }^{2}L(x,\dot{x})}{\partial {\dot{x}}^2}\ne 0,$$

(3)

and is of constant sign, allowing us to solve for the velocity $\dot{x}$ in favor of the momentum $p(x,\dot{x})=\frac{\partial L(x,\dot{x})}{\partial \dot{x}}$, i.e., we can write $\dot{x}(x,p)$. Thus, Lagrangians with a quadratic velocity dependence are regular, and one can formulate a Hamiltonian description by means of a Legendre transform. The condition (3) fails for Lagrangians that are linear in velocity (see, for instance, Ref. [13]), but as mentioned earlier, they will not be our concern here. Instead, we shall be looking at Lagrangians for which solving the equation $p=(x,\dot{x})$ in favor of $\dot{x}$ leads to a non-unique solution, e.g., the appearance of a square root, which gives rise to what will be called branching. Such Lagrangians would not permit the construction of a Hamiltonian function in a unique way.

In the classical context, the problems associated with branched Hamiltonians and the ones that are inevitably posed after their quantization were addressed by Shapere and Wilczek [1,2,3]. This has triggered a series of papers by Curtright and Zachos [16,17,18,19,20,21], which were subsequently followed up by other works in a similar direction (see, for example, Refs. [5,22,23]). It bears mentioning that local branching is not so sufficient to ensure integrability. In particular, finding an integrable differential equation having solutions that are not locally finitely branched with a finitely sheeted Riemann surface, but not yet identified through Painlevé analysis, is in itself an interesting open problem [16].

Against this background, a new class of innovations on the description and simulations of quantum dynamics emerged in relation to the specific role played by certain models constructed appropriately. Not quite unexpectedly, Hamiltonians that are multi-valued functions of momenta confront us with some typical insurmountable ambiguities of quantization. In such cases, the underlying Lagrangian possesses time derivatives in excess of quadratic powers (or sometimes, inverse powers). The use of these models leads, on both classical and quantum grounds, to the necessity of a re-evaluation of the dynamical interpretation of the momentum, which, in principle, becomes a multi-valued function of the velocity. It also needs to be pointed out that the traditional approaches often do not always work as is the case with certain $\mathcal{PT}$-symmetric complex potentials possessing real spectra [24] or upon employing tractable non-local generalizations [25].

In the context of nonlinear models, certain Liénard-class systems present an intriguing feature of the Hamiltonian in which the roles of the position and momentum variables are exchanged with the emergence of the notion of a momentum-dependent mass [23,26,27,28,29,30,31]. Naturally, the presence of the damping as is the case for Liénard systems poses a problem whenever one tries to contemplate a quantization of the model. It is important to realize that the quantization is hard to tackle in the coordinate representation of the Schrödinger equation, but can be straightforwardly carried out in the momentum space [26,30] (see also Ref. [32]).

Although much has been said about the quantum mechanical formalisms, in this paper, we focus on the classical theory, (briefly) reviewing some aspects of nonstandard Lagrangians and the associated branched Hamiltonians. The theory is exemplified by focusing on various examples, which include some systems of the Liénard class, which are of great interest in the theory of dynamical systems. Apart from Liénard systems, we discuss some interesting toy Lagrangians, which contain time derivatives in excess of quadratic powers, leading to branched Hamiltonians. The basic features of the theory are discussed in light of these examples. However, we begin with a discussion on some simple nonstandard Lagrangians, which can be figured out via some guesswork, in Section 2. Following this, in Section 3, we discuss nonstandard Lagrangians and branched Hamiltonians in the context of Liénard systems, wherein we outline a systematic derivation of the Lagrangians, provided the system admits a certain integrability condition. In Section 4 and Section 5, we analyze various intriguing examples of Lagrangians in which time derivatives occur in excess of quadratic powers, while also discussing the associated Hamiltonians. We conclude with some remarks in Section 6 and in Appendix A, where a few further aspects of the problem of quantization are also discussed.

2. Some Illustrative Examples

Example 1.

Consider the following Lagrangian [9,10]:

$$L(x,\dot{x})=\frac{1}{\alpha \mu (x)+\beta \dot{x}},\alpha \dot{x}+\beta \mu (x)\ne 0,$$

(4)

where $\mu (x)$ is a well-behaved function (typically a polynomial), while α and β are real-valued and non-zero constant numbers. Obviously, it does not reveal the ‘standard’ form as the difference between the kinetic and potential energies. However, the Euler–Lagrange equation gives $\ddot{x}+f(x)\dot{x}+g(x)=0$, with $f(x)=\frac{3\alpha {\mu }^{\prime }(x)}{2\beta }$ and $g(x)=\frac{{\alpha }^2{\mu }^{\prime }(x)\mu (x)}{2{\beta }^2}$, where, for instance, picking $\mu (x)=x$ gives the linearly damped harmonic oscillator, while the choice $\mu (x)=x^2$ implies $f(x)\propto x$ and $g(x)\propto x^3$. Lagrangians of this type (4) are termed as reciprocal Lagrangians.

Example 2.

Consider another form of Lagrangians classified by [9]:

$$L(x,\dot{x})=ln[\gamma \mu (x)+\delta \dot{x}],\gamma \mu (x)+\delta \dot{x}>0,$$

(5)

where δ and β are real-valued and non-zero constant numbers. The Euler–Lagrange equation goes as $\ddot{x}+f(x)\dot{x}+g(x)=0$, with $f(x)=\frac{2\gamma {\mu }^{\prime }(x)}{\delta }$ and $g(x)=\frac{{\gamma }^2\mu (x){\mu }^{\prime }(x)}{{\delta }^2}$. Lagrangians that look like (5) are termed as logarithmic Lagrangians. The relation between logarithmic and reciprocal classes of Lagrangians has been explored in [11] (see also Ref. [12]). As with the system described by the Lagrangian (1), the systems given by (4) and (5) are defined only on appropriate regions of ${\mathbb{R}}^2$.

Example 3.

As another example, we point out that some equations that go as $\ddot{x}+A(x,\dot{x})\dot{x}+B(x,\dot{x})=0$, where $A(x,\dot{x})$ and $B(x,\dot{x})$ are suitable functions of $(x,\dot{x})$ can be derived from (reciprocal) Lagrangians that read

$$L(x,\dot{x})=\frac{1}{\alpha \mu (x)+\beta \rho (\dot{x})},$$

(6)

such that $\beta {\rho }^{\prime \prime }(\dot{x})[\alpha \mu (x)+\beta \rho (\dot{x})]\ne 2{\beta }^2{\rho }^{\prime }{(\dot{x})}^2$. Specifically, the functions $A(x,\dot{x})$ and $B(x,\dot{x})$ are

$$A(x,\dot{x})=\frac{2\alpha \beta {\rho }^{\prime }(\dot{x}){\mu }^{\prime }(x)}{2{\beta }^2{\rho }^{\prime }{(\dot{x})}^2-\beta {\rho }^{\prime \prime }(\dot{x})[\alpha \mu (x)+\beta \rho (\dot{x})]},$$

(7)

$$B(x,\dot{x})=\frac{\alpha {\mu }^{\prime }(x)\left[\alpha \mu (x)+\beta \rho (\dot{x})\right]}{2{\beta }^2{\rho }^{\prime }{(\dot{x})}^2-\beta {\rho }^{\prime \prime }(\dot{x})[\alpha \mu (x)+\beta \rho (\dot{x})]}.$$

(8)

However, there is a limited variety of differential equations that can be described by Lagrangians, which may be guessed; in general, it is often not possible to systematically derive a Lagrangian from which a given differential equation may emerge as the Euler–Lagrange equation. In what follows, we describe Liénard systems and demonstrate that, if a certain integrability condition is satisfied, then one may systematically find nonstandard Lagrangians describing such systems.

3. Liénard Systems

A Liénard system is a second-order ordinary differential equation that goes as

$$\ddot{x}+f(x)\dot{x}+g(x)=0$$

(9)

(often, it is sufficient to have $f(x),g(x)\in C^2(U,\mathbb{R})$, where $U\subset \mathbb{R}$). $f(x),g(x)\in C^{\infty }(\mathbb{R},\mathbb{R})$ can be suitably chosen. Interesting choices for $f(x)$ and $g(x)$ include $f(x)=1$ and $g(x)=x$, which is just the damped linear oscillator, while the choice $f(x)=(1-x^2)$ and $g(x)=x$ gives the van der Pol oscillator [33], known to admit limit-cycle behavior due to the particular choice of $f(x)$[34]. Another choice is $f(x)=1$ and $g(x)=x^3$, for which we have the linearly damped (nonlinear) Duffing oscillator (see, for example, Refs. [35,36]). It is noteworthy that, in any case with $f(x)\ne 0$, the system exhibits non-conservative dynamics because (9) does not stay invariant under the transformation $t\to -t$, namely time reversal. Furthermore, oscillatory dynamics can be obtained if $f(x)$ is an even function and if $g(x)$ is odd; this follows from the fact that the overall force (the second and third terms of (9)) should be odd under $x\to -x$ in order to support oscillations [35].

3.1. Chiellini Condition and Nonstandard Lagrangians

Given a second-order differential equation, the inverse problem of finding the Lagrangian has been the subject of much investigation [37,38,39,40,41,42] (see also Ref. [43]). In particular, for Liénard systems satisfying a certain integrability condition, one can find nonstandard Lagrangians from which they emerge as the Euler–Lagrange equation [41] (see also Refs. [23,26]). The idea is to make use of the so-called Jacobi last multiplier, which may be defined as in [37] (see Appendix A).

In this manner, starting from an ordinary differential equation:

$$\ddot{x}=F(x,\dot{x}),$$

(10)

one defines the last multiplier M as that which satisfies

$$\frac{dlnM}{dt}+\frac{\partial F(x,\dot{x})}{\partial \dot{x}}=0.$$

(11)

As has been discussed in Whittaker’s classic textbook [37], if a second-order differential equation such as (10) follows from the Euler–Lagrange equations, then the Lagrangian is related to the latter multiplier as

$$M=\frac{{\partial }^{2}L(x,\dot{x})}{\partial {\dot{x}}^2}.$$

(12)

This allows one to determine the Lagrangian function for a given second-order differential equation, provided it admits a Lagrangian formalism.

For the Liénard system, a formal solution for the multiplier is found to be

$$M(t,x)=exp\left(\int f(x)dt\right).$$

(13)

We may define a new nonlocal variable u as [41]

$$u=\dot{x}-\frac{g(x)}{\ell f(x)},$$

(14)

with $\ell \ne -1$, which is determined from

$$\frac{d}{dx}\left(\frac{g(x)}{f(x)}\right)+\ell (\ell +1)f(x)=0.$$

(15)

Then, using Equations (14) and (15), we have $\dot{u}=\ell uf(x)$. In other words, if the condition (15) is true, the Liénard system (9) can be expressed as the following system of first-order equations:

$$\dot{u}=\ell uf(x),\dot{x}=u+W(x),$$

(16)

where $W(x)={\ell }^{-1}g(x)/f(x)$. Since we have $\dot{u}=\ell uf(x)$, using (13), we can write $M=u^{1/\ell }$, which, upon using (12), gives us (up to a gauge function, which can be ignored here) $L\sim u^{\frac{2\ell +1}{\ell }}$. In terms of x and $\dot{x}$, the Lagrangian reads as [23]

$$L(x,\dot{x})=\frac{{\ell }^2}{(\ell +1)(2\ell +1)}{\left(\dot{x}-\frac{1}{\ell }\frac{g(x)}{f(x)}\right)}^{\frac{2\ell +1}{\ell }},$$

(17)

which is of the nonstandard kind. (It may be noted that one cannot at this stage set $f(x)=0$ in (14)–(16) or (17) to recover the conservative case. However, one can set $f(x)=0$ in (13), which gives $M=1$, and consequently, from (12), one has $L(x,\dot{x})=\frac{{\dot{x}}^2}{2}+J(x)\dot{x}+K(x)$, where $K(x)$ is the (conservative) scalar potential, while $J(x)$ may be interpreted as a vector potential).

It is noteworthy that (15) represents what is known as the Chiellini condition, allowing one to recast the Liénard system in the form (16). Specifically, if $f(x)=a{x}^{\alpha }$, then (15) dictates that $g(x)$ must satisfy the following differential equation (see also Refs. [44,45]):

$$\frac{d}{dx}\left(x^{-\alpha }g(x)\right)+\ell (\ell +1)a^2{x}^{\alpha }=0.$$

(18)

A simple integration gives

$$g(x)=k{x}^{\alpha }-\frac{\ell (\ell +1)a^2}{\alpha +1}{x}^{2\alpha +1},$$

(19)

where $k\in \mathbb{R}$ is an integration constant. This gives the functional form of $g(x)$ so as to satisfy the Chiellini condition. In the cases where the Chiellini condition is satisfied, the Lagrangian is given by (17).

3.2. Hamiltonian Aspects

With the Lagrangian that describes the Liénard system, we can move on to its Hamiltonian aspects. The conjugate momentum is found to be

$$p=\frac{\partial L}{\partial \dot{x}}=\frac{\ell }{\ell +1}{\left(\dot{x}-\frac{1}{\ell }\frac{g(x)}{f(x)}\right)}^{(\ell +1)/\ell }.$$

(20)

Thus, the expression $\dot{x}=\dot{x}(p)$ may be multi-valued, depending on ℓ. It goes as

$$\dot{x}=K(l)p^{\ell /(\ell +1)}+\frac{1}{\ell }\frac{g(x)}{f(x)},$$

(21)

where $K(\ell )$ is some function of ℓ and is a constant. Using the above form, the Hamiltonian is found to be

$$H(x,p)=K(\ell )p^{\frac{2\ell +1}{\ell +1}}-\frac{g(x)}{\ell f(x)}p.$$

(22)

Notice that, if (21) admits branching, then so does the Hamiltonian (22). Below, we discuss a concrete example.

3.3. A Concrete Example

An illustrative example will help us understand the framework discussed above. In particular, this will also enable us to turn attention to the notion of momentum-dependent mass [26,27,28], a concept that has generated some interest in recent times, especially within the quantum mechanical physics-oriented model-building approaches.

We will consider the case where $f(x)=x$ and $g(x)=x-x^3$[23]; (15) is satisfied for $\ell =1,-2$. For the sake of clarity, we will consider every such case separately.

3.3.1. Case with $\ell =1$

In the case of $\ell =1$, we have

$$p=p(x,\dot{x})=\frac{1}{2}{\left(\dot{x}+x^2-1\right)}^2\Rightarrow \dot{x}=\dot{x}(x,p)=1-x^2\pm \sqrt{2p}.$$

(23)

This points towards branching. Notice that branching originates from the nonlinear dependence between p and $\dot{x}$ in the equation $p=\frac{\partial L}{\partial \dot{x}}$. The corresponding branched Hamiltonians turn out to be

$$H_{\pm }(x,p)=p\left(1-x^2\pm \frac{2}{3}\sqrt{2p}\right),$$

(24)

exhibiting two distinct branches, where $p\ge 0$. We plot the function $\dot{x}=\dot{x}(p,x)$ in Figure 1, while Figure 2 shows a plot of the branched pair of Hamiltonians, $H_{\pm }=H_{\pm }(x,p)$. The branches coalesce at $p=0$.

3.3.2. Case with $\ell =-2$

In the case where $\ell =-2$, we have

$$p=p(x,\dot{x})=2{\left(\dot{x}+\frac{1-x^2}{2}\right)}^{1/2}\Rightarrow \dot{x}=\dot{x}(x,p)=\frac{p^2}{4}-\frac{(1-x^2)}{2},$$

(25)

implying that there is no branching, because $\dot{x}$ can be extracted, uniquely, as a function of the momentum. A straightforward calculation reveals that the Hamiltonian turns out to be

$$H(x,p)=\frac{p^3}{12}-\frac{p(1-x^2)}{2},$$

(26)

wherein there is only one branch.

In Figure 3 and Figure 4, we plot $\dot{x}=\dot{x}(x,p)$ and $H=H(x,p)$. An intriguing aspect of the Hamiltonian (26) is that it may be expressed as

$$H(x,p)=\frac{x^2}{2p^{-1}}+U(p),U(p)=\frac{p^3}{12}-\frac{p}{2}.$$

(27)

This resembles a standard Hamiltonian, only with the roles of the coordinate and momentum being interchanged.

It is then certainly tempting to interpret $m(p)=p^{-1}$ as a momentum-dependent mass. Also, the quantization of such systems proceeds, in the momentum space, often in the context referring to the notion of momentum-dependent mass (see, for example, Ref. [26]). Still, in our particular model, such a mass becomes singular (infinite) in the zero-momentum limit. In the same limit, moreover, also the potential itself is vanishing, i.e., a consistent physical interpretation of the system would require a suitable regularization of the limiting process.

Marginally, let us add that one of the regularization recipes that appeared applicable to the quantum version of our very specific model (27) has been proposed and tested in an older paper [46]. Based on an ad hoc complexification of the momentum and on a certain rather sophisticated strong-coupling perturbation expansion technique, the recipe has been even found to provide the numerically fairly reliable spectra, in certain ranges of couplings at least.

4. A Generalized Class of Lagrangians Yielding Branched Hamiltonians

Let us note that, if one is given a single-valued Lagrangian $L(x,v)$ and defines it according to formula $L(x,v)=x^2-V(v)$, rather than according to the usual recipe $L(x,v)=v^2-V(x)$ and moreover, if the p or v dependence is non-convex, then, as a result of employing the Legendre transformation, the branched functions are always encountered despite our having started from a single-valued Lagrangian or Hamiltonian function.

4.1. The $v^4$ Model

Shapere and Wilczek have discussed a concrete model depicting the non-convex nature of the Lagrangian, which reads [1]

$$L(v)=\frac{1}{4}{v}^4-\frac{\kappa }{2}{v}^2,$$

(28)

where v is the velocity (now and in subsequent discussions, we will denote $\dot{x}=v$) and $\kappa >0$ is a coupling parameter. Corresponding to (28), the conjugate momentum is a cubic function in v that is given by

$$p(v)=v^3-\kappa v.$$

(29)

Clearly, p is not monotonic in velocity, which may lead to branching. The corresponding Hamiltonian is obtained as

$$H(p)=\frac{3}{4}{v}^4-\frac{\kappa }{2}{v}^2,v=v(p),$$

(30)

which, like $L(v)$, is also a multi-valued function (with cusps) in the conjugate momentum p, since each given p corresponds to one or three values of v, as shown in (29).

For systems with a non-convex Lagrangian as sampled by (28), the routine construction of the corresponding Hamiltonian in the conjugate momentum variable is not unique. An analogous incertitude is encountered in cosmology models [47,48], in generalized schemes of Einstein gravity, which involve topological invariants, and in theories of higher curvature gravity [49].

4.2. Velocity-Independent Potentials

Curtright and Zachos [20] extended the analysis of [1] by considering a generalized class of non-quadratic Lagrangians that go as

$$L(x,v)=C{(v-1)}^{\frac{2k-1}{2k+1}}-V(x),C=\frac{2k+1}{2k-1}{\left(\frac{1}{4}\right)}^{\frac{2}{2k+1}},$$

(31)

where the traditional kinetic energy term is replaced by a fractional function of the velocity variable v and $V(x)$ represents a convenient local interaction potential. The fractional powers facilitate the derivation of supersymmetric partner forms of the potential á la Witten [50]. We remark that the $(2k+1)$st root of the first term in $L(x,v)$ is required to be real, and $>0$ or $<0$ for $v>1$ or $v<1$, respectively.

Let us focus on the case with $k=1$. Performing a Taylor expansion for v near zero, we can write $L(x,v)\approx C(-1+\frac{v}{3}+\frac{v^2}{9}+O(v^3))-V(x)$. While the first term is merely a constant and the second term contributes to the boundary of the action and, therefore, does not influence the equations of motion, the third term yields the kinetic structure:

$$\begin{array}{c}\hfill A={\int }_{t_1}^{t_2}L(x,v)dt\approx C\left(t_2-t_1+\frac{1}{3}(x(t_2)-x(t_1))+\frac{1}{9}{\int }_{t_1}^{t_2}{v}^{2}dt+{\int }_{t_1}^{t_2}O(v^3)dt\right)-{\int }_{t_1}^{t_2}V(x)dt.\end{array}$$

(32)

Thus, for small velocities, the action results in the usual Newtonian form of the equations of motion.

For the large velocities, on the other hand, we have a less trivial scenario, which leads (for finite, positive-integer values of k) to a non-convex function of v. The curvature term corresponding to the quantity $\frac{{\partial }^{2}L}{\partial v^2}$ changes sign at the point $v=1$. Thus, $L(x,v)$ may be interpreted as a single pair of convex functions that have been judiciously pieced together. Now, from the Lagrangian (31), the canonical momentum can be calculated:

$$p=p(v)={\left(\frac{1}{4}\right)}^{\frac{2}{2k+1}}\frac{1}{{(v-1)}^{\frac{2}{2k+1}}}.$$

(33)

Inverting the relation, we observe that the velocity variable $v(p)$ emerges as a double-valued function of p:

$$v=v_{\pm }(p)=1\mp \frac{1}{4}{\left(\frac{1}{\sqrt{p}}\right)}^{(2k+1)}.$$

(34)

Corresponding to the two signs above, a pair of branches of the Hamiltonian, namely $H_{\pm }(x,p)$, will appear. Specifically, for any positive-integer value of k, these may be identified to be

$$H_{\pm }(x,p)=p\pm \frac{1}{4k-2}{\left(\frac{1}{\sqrt{p}}\right)}^{2k-1}+V(x).$$

(35)

From a classical perspective, in order to avoid an imaginary $v(p)$, one needs to address a non-negative p. This in turn implies that the slope $\frac{\partial L}{\partial v}$ is always positive. It is interesting to note that, for the $k=1$ case, we are led to the quantum mechanical supersymmetric structure for the difference $H_{\pm }(x,p)-V(x)$, which reads $p\pm \frac{1}{2\sqrt{p}}$, in the momentum space. The associated spectral properties have been analyzed in the literature [26,30].

We end our discussion on this example by noting that, in the special case where $V(x)=x^2$, the branched Hamiltonian is $H_{\pm }(x,p)=x^2+U_{\pm }(p)$, wherein it appears as if the roles of the coordinate and the momentum have been interchanged, with $U_{\pm }(p)$ being a momentum-dependent potential that exhibits two branches.

4.3. Velocity-Dependent Potentials

Lines of force can be ascertained with the help of velocity-dependent potentials, which ensure that particles take certain specified paths [14,51]. In electrodynamics, the field vectors $\overrightarrow{E}$ and $\overrightarrow{B}$ can be determined given such a potential function when the trajectories of a charged particle’s motion are specified. In the present context, we proceed to set up an extended scheme where the Lagrangian depends on a velocity-dependent potential $\mathcal{V}(x,v)$ in the manner as given by [22,29]:

$$L(x,v)=C{(v-1)}^{\frac{2k-1}{2k+1}}-\mathcal{V}(x,v),C=\frac{2k+1}{2k-1}{\left(\frac{1}{4}\right)}^{\frac{2}{2k+1}},$$

(36)

where $\mathcal{V}(x,v)$ is assumed to be given in a separable form, i.e., $\mathcal{V}(x,v)=U(v)+V(x)$; here, $U(v)$ and $V(x)$ are well-behaved functions of v and x, respectively. Using the standard definition of the canonical momentum, we find its form to be

$$p=p(v)={\left(\frac{1}{4}\right)}^{\frac{2}{2k+1}}{(v-1)}^{-\frac{2}{2k+1}}-U^{\prime }(v).$$

(37)

The complexity of the right side does not facilitate an easy inversion of the above relation that would reveal the multi-valued nature of velocity in a closed, tractable form. Nevertheless, the associated branches of the Hamiltonian can be straightforwardly written down upon employing the Legendre transform as

$$H_{\pm }(x,p)=p\pm \frac{1}{4}{[p+U^{\prime }(v)]}^{-\frac{2k-1}{2}}\left(\frac{2k+1}{2k-1}-p{[p+U^{\prime }(v)]}^{-1}\right)+\mathcal{V}(x,v),v=v(p).$$

(38)

Unfortunately, since a Hamiltonian has to be a function of the coordinate and its corresponding canonical momentum, the generality of the form of $H_{\pm }(x,p)$ as derived above is of little use unless we have an explicit inversion of (37) giving $v=v(p)$. We, therefore, have to go for the specific cases of k and $U(v)$.

A Special Case

Indeed, the case $k=1$ proves to be particularly worthwhile to understand the spectral properties of the Hamiltonian. It corresponds to the Lagrangian as given by

$$L(x,v)=3{\left(\frac{1}{4}\right)}^{\frac{2}{3}}{(v-1)}^{\frac{1}{3}}-U(v)-V(x).$$

(39)

A sample choice for $U(v)$ could be [22]

$$U(v)=\lambda v+3\delta {(v-1)}^{\frac{1}{3}},$$

(40)

in which $\lambda (\ge 0)$ and $\delta (<4^{-\frac{2}{3}})$ are suitable real constants. The presence of the parameter $\delta$ scales the kinetic energy term in the Lagrangian. The canonical momentum p is now given by

$$p=p(v)=\mu {(v-1)}^{-\frac{2}{3}}-\lambda ,$$

(41)

where the quantity $\mu =4^{-\frac{2}{3}}-\delta >0$. We are, therefore, led to a pair of relations for the velocity depending on p:

$$v=v_{\pm }(p)=1\mp {\mu }^{\frac{3}{2}}{(p+\lambda )}^{-\frac{3}{2}}.$$

(42)

As a consequence, we find two branches of the Hamiltonian, which are expressible as

$$H_{\pm }(x,p)=(p+\lambda )\pm \frac{2\gamma }{\sqrt{p+\lambda }}+V(x),$$

(43)

where ${\mu }^{3/2}$ has been replaced by $\gamma$. As a final comment, the special case corresponding to $\lambda =0$ and $\gamma =1/4$ conforms to the Hamiltonian (35) advanced in [20].

5. Three More Forms of Hamiltonians

5.1. Higher Power Lagrangians

As an extension of (31), the following higher power Lagrangian was proposed in [5]:

$$L(x,v)=C{(v+\sigma (x))}^{\frac{2m+1}{2m-1}}-\delta ,\mathsf{\Lambda }=\left(\frac{1-2m}{1+2m}\right){(\delta )}^{\frac{2}{1-2m}},\delta >0,$$

(44)

where we notice that the coefficient $\mathsf{\Lambda }$ is non-negative for $0\le m<\frac{1}{2}$. The main difference from (31) is in the choice of a general function $\sigma (x)$ in place of $\sigma (x)=-1$ as in (31). The other point is that the inverse exponent with respect to the model of Curtright and Zachos [20] has been taken for the convenience of calculus. We have omitted the explicit potential function assuming that the interaction re-appears in a more natural manner via a suitable choice of an auxiliary free parameter $\delta$ and that of a nontrivial function $\sigma (x)$. As long as our Lagrangian $L(x,v)$ is of a nonstandard type, we will not feel disturbed by the absence of the explicit potential $V(x)$.

For this particular model, the canonical momentum reads as

$$p=p(x,v)=-{(\delta )}^{\frac{2}{1-2m}}{(v+\sigma (x))}^{\frac{2}{2m-1}},$$

(45)

and a simple inversion yields

$$v=v_{\pm }(x,p)=-\sigma (x)+\delta {\left(\pm \sqrt{-p}\right)}^{2m-1}.$$

(46)

This means the Hamiltonian is obtained to be

$$H_{\pm }(x,p)=(-p)\sigma (x)-\frac{2\delta }{2m+1}{\left(\pm \sqrt{-p}\right)}^{2m+1}+\delta .$$

(47)

Special Case

The specific case with $m=0$ is of interest as it allows us to easily derive the (double-valued) velocity profile, which reads as

$$v=v_{\pm }(x,p)=-\sigma (x)\pm \frac{\delta }{\sqrt{-p}},$$

(48)

implying that the Hamiltonian branches out into components:

$$H_{\pm }(x,p)=(-p)\sigma (x)\mp 2\delta \sqrt{-p}+\delta .$$

(49)

The nature of the two Hamiltonians depends on the sign of p. Once we specify the following choice of $\sigma (x)$, namely

$$\sigma (x)=\frac{\lambda }{2}{x}^2+\frac{9{\lambda }^2}{2k^2},\lambda >0,$$

(50)

together with the choice $\delta =\frac{9{\lambda }^2}{2k^2}$, then upon imposing a simple translation $p\to \frac{2k}{3\lambda }p-1$, the Hamiltonians $H_{\pm }$ acquire the forms that go as

$$H_{\pm }(x,p)=\frac{9{\lambda }^2}{2k^2}\left[2\mp 2{\left(1-\frac{2kp}{3\lambda }\right)}^{\frac{1}{2}}+\frac{k^2{x}^2}{9\lambda }-\frac{2kp}{3\lambda }-\frac{2k^3{x}^{2}p}{27{\lambda }^2}\right].$$

(51)

These are readily identifiable as a set of plausible Hamiltonians representing a nonlinear Liénard system [23,26,28]. The appearance of the coordinate–momentum coupling is noteworthy and leads us to the notion of a momentum-dependent mass as

$$H_{\pm }(x,p)=\frac{x^2}{2m(p)}+U_{\pm }(p),m(p)={\left[\lambda -\frac{2kp}{3}\right]}^{-1},U_{\pm }(p)=\frac{9{\lambda }^2}{2k^2}\left[2\mp 2{\left(1-\frac{2kp}{3\lambda }\right)}^{\frac{1}{2}}-\frac{2kp}{3\lambda }\right].$$

(52)

From a classical perspective, the momentum p needs to be restricted to the range $-\infty <p\le \frac{3\lambda }{2k}$ to account for the physical properties of the system in the real space; this also ensures that the momentum-dependent mass is positive and finite. However, because of a branch-point singularity at $p=\frac{3\lambda }{2k}$, a thorough analytical study of $H_{\pm }(x,p)$ becomes greatly involved. Observe that, when $p=\frac{3\lambda }{2k}$, we find the coincidence of the two Hamiltonians $H_{\pm }(x,p)$.

5.2. Rational Function Lagrangians

In another characteristic example, let us pick up an illustration where $L(x,v)$ is of the reciprocal kind and is defined to be [5]

$$L(x,v)=\frac{1}{s}{\left(\frac{1}{3}s{x}^2+\frac{3}{s}\lambda -v\right)}^{-1},$$

(53)

where s is a real parameter. The canonical momentum comes out as

$$p=p(x,v)=\frac{1}{s}{\left(\frac{1}{3}s{x}^2+\frac{3}{s}\lambda -v\right)}^{-2},$$

(54)

which, when inverted, yields

$$v=v_{\pm }(x,p)=\frac{1}{3}s{x}^2+\frac{3}{s}\lambda \pm \frac{1}{\sqrt{sp}}.$$

(55)

The accompanying Hamiltonian corresponding to the above Lagrangian has two branches:

$$H_{\pm }(x,p)=\frac{s}{3}{x}^{2}p+\frac{3}{s}\lambda p\pm 2\sqrt{\frac{p}{s}}.$$

(56)

It should be remarked that, as $\lambda \to 0$, (53) is just the trial Lagrangian (6), for the choice $\mu (x)=a{x}^2$ and under a suitable identification of the constant parameters. We end by noting that (56) can be re-expressed as a model:

$$H_{\pm }(x,p)=\frac{x^2}{2m(p)}+U_{\pm }(p),m(p)=\frac{3}{2sp},U_{\pm }(p)=\frac{3}{s}\lambda p\pm 2\sqrt{\frac{p}{s}}.$$

(57)

with a momentum-dependent mass.

5.3. Relativistic Free Particle

As a final example of the dynamics due to non-quadratic Lagrangians, let us re-examine the much-studied problem of a relativistic (free) particle, which is described by the Lagrangian:

$$L(v)=-mc\sqrt{c^2-v^2},v<c.$$

(58)

The conjugate momentum is obtained as

$$p=p(v)=\frac{\partial L(v)}{\partial v}=\frac{mcv}{\sqrt{c^2-v^2}}.$$

(59)

This implies that $p^2{c}^2-p^2{v}^2-m^2{c}^2{v}^2=0$. Solving for the velocity gives

$$v=v_{\pm }(p)=\pm \frac{pc}{\sqrt{p^2+m^2{c}^2}}.$$

(60)

Thus, the Hamiltonian reads

$$H_{\pm }(p)=\frac{c(\pm p^2-m^2{c}^2)}{\sqrt{p^2+m^2{c}^2}}.$$

(61)

In particular, one may consider the ultrarelativistic limit ($v\approx c$) in which $H_{\pm }(p)=\pm pc$. Related to the above example, the reader is referred to Ref. [52] for the example of a spinning particle with the Lagrangian being non-quadratic both in the position and spin variables. Another physically interesting example is that of an axially symmetric charged body in an electromagnetic field, which is governed by Euler–Poisson equations [53].

6. Concluding Remarks

In the present treatment on the existence of nonstandard Lagrangians, we emphasized, first of all, the existence of certain unusual aspects of their relationship with the associated branched Hamiltonians. Various different examples were discussed; in all of them, the velocity dependence of the Lagrangian was not of (homogenous) degree two, but contained either powers larger than two or negative powers. This resulted in a nonlinear relationship between the generalized velocity and the conjugate momentum, leading to a multi-valued behavior of the velocity when solved as a function of the momentum (and, perhaps, the coordinate).

We observed that, in the description of Hamiltonians emerging from nonstandard Lagrangians, the notion of momentum-dependent mass is often encountered. It is then as if the coordinate of the particle played the role of momentum and vice versa, with a function of the momentum variable appearing as an ‘effective mass’ describing the system. Such systems can be quantized straightforwardly in the momentum space [26,30,32]. Naturally, this reopens a few mathematically deeper questions concerning their quantization. Indeed, the technicalities of canonical quantization can be perceived as widely assessed in the literature (see, for example, Ref. [54]), wherein it is not infrequent to encounter certain fundamental difficulties. For example, in certain ‘anomalous’ quantum systems with non-Hermitian Hamiltonians supporting real eigenvalues, it has been shown that the quantum wave functions themselves could still, in finite time, diverge [55]. Moreover, after one admits the unusual forms of the Hamiltonians characterized, typically, by the popular parity-time-symmetry ($\mathcal{PT}$-symmetry; see, for example, Refs. [56,57] for a pedagogic and introductory discussion on such specific variants of non-self-adjoint models), the anomalies may occur even when the $\mathcal{PT}$-symmetry itself remains unbroken.

Several unusual forms of the latter anomalies may appear in both the spectra and eigenfunctions, materialized as Kato’s exceptional points [58,59] or the so-called spectral singularities [60]. In particular, exceptional points can be regarded as a typical feature of non-Hermitian systems related to a branch-point singularity where two or more discrete eigenvalues, real or complex, and corresponding to two different quantum states, along with their accompanying eigenfunctions, coalesce [61,62,63].

Naturally, the possible relevance of the latter anomalies in the quantum systems controlled by the branched Hamiltonians is more than obvious. One only has to emphasize the difference between the systems characterized by the unitary and non-unitary evolution. In the former case, indeed, one is mainly interested in the description of the systems of stable bound states. In the latter setting, the scope of the theory is broader; the states are resonant and unstable in general. In the related models, one deals with Hamiltonians that are manifestly non-Hermitian and that undergo non-unitary quantum evolution; they generally represent open systems with balanced gain and loss [64,65]. Exceptional points occur there as experimentally measurable phenomena. In this connection, it is also relevant to point out the occurrence of certain theoretical anomalies like the possible breakdown of the adiabatic theorem [66] or the feature of stability-loss delay [67], etc. In all of these contexts, one encounters the possibility of interpreting branched Hamiltonians as an innovative theoretical tool admitting a coalescence of the branched pairs of operators at an exceptional point. Thus, preliminarily, let us conclude that the (related) possible innovative paths towards quantization look truly promising.