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Nonstandard Lagrangians and branched Hamiltonians: A brief review
Bijan Bagchi∗1, Aritra Ghosh†2, Miloslav Znojil‡3,4,5
1Brainware University, Barasat, Kolkata, West Bengal 700125, India
2School of Basic Sciences, Indian Institute of Technology Bhubaneswar, Jatni, Khurda, Odisha 752050, India
3The Czech Academy of Sciences, Nuclear Physics Institute, Hlavn´ı 130, 250 68 ˇ
Reˇz, Czech Republic
4Department of Physics, Faculty of Science, University of Hradec Kr´alov´e,
Rokitansk´eho 62, 50003 Hradec Kr´alov´e, Czech Republic
5Institute of System Science, Durban University of Technology, Durban, South Africa
(Dated: March 28, 2024)
Time and again, non-conventional forms of Lagrangians have found attention in the literature.
For one thing, such Lagrangians have deep connections with several aspects of nonlinear dynamics
including specifically the types of the Li´enard 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 Li´enard-type systems. We then take up other cases where the Lagrangians depend upon
the 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 upon the emergence of the notion of momentum-dependent mass in the theory of
branched Hamiltonians.
Keywords: Nonstandard Lagrangians; Branched Hamiltonians; Li´enard systems; Momentum-dependent mass
I. INTRODUCTION
During the past few years, the study of non-conventional types of dynamical systems, in particular those which
are controlled by Lagrangians that are not quadratic in the velocity has entered a new phase of intense development
[1–4]. Such Lagrangians lead to certain exotic Hamiltonians, commonly termed as branched Hamiltonians, that 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
(PT )-symmetric schemes [7], along with their relativistic counterparts [8].
A simple way to see how Lagrangians that are not quadratic in the velocity can lead to sensible dynamical systems
is to consider the following toy model [9, 10]:
L(x, ˙x) = (αx +β˙x)−1,(1)
where αand βare real numbers satisfying αβ > 0, and αx +β˙x= 0. Notice that the Lagrangian cannot be expressed
as the difference between the kinetic and potential energies; such Lagrangians shall be referred to as nonstandard. A
direct computation reveals that the Euler-Lagrange equation is
¨x+γ˙x+ω2
0x= 0,(2)
where γ=3α
2βand ω0=α
√2β. Eq. (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 Eq.
(2) upon invoking the Euler-Lagrange equation1. There exist various other families of nonstandard Lagrangians
(giving rise to different dynamical systems) which look quite different from Eq. (1); each family is endowed with
their own intriguing features. However, the common theme is the existence of Lagrangians that are not quadratic
in the velocity, thereby leading to a nonlinear relationship between the velocity and the momentum. This leads to
the notion of branching, typically yielding the so-called Riemann-surface phase-space structure and in consequence,
∗E-mail: bbagchi123@gmail.com
†E-mail: ag34@iitbbs.ac.in
‡E-mail: znojil@ujf.cas.cz
1One could recover the damped oscillator from a standard Lagrangian by using a Rayleigh dissipation function [11]. Alternatively, one
can consider the modified Euler-Lagrange equations from the Herglotz variational problem to describe the damped oscillator [12]. We
do not consider such situations here.
arXiv:2403.18801v1 [math-ph] 27 Mar 2024
2
certain interesting topological features [13].
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 [1–3]. This has triggered off a series of papers
by Curtright and Zachos [13–18] which were subsequently followed up by other works in a similar direction (see for
example, Refs. [5, 19, 20]). It bears mention 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 Painlev´e analysis, is in itself an interesting open
problem [13].
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 which 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 velocity. It also needs to be pointed out that the traditional approaches often do not always work as is the
case with certain PT -symmetric complex potentials possessing real spectra [21] or, on employing tractable non-local
generalizations [22].
There have been efforts to construct simple systems with toy Lagrangians that lead to multi-valued Hamiltonian
systems. Such Hamiltonians would be defined in the momentum space revealing systems with momentum-dependent
masses [20, 23, 24]. In the context of nonlinear models, the Li´enard system presents an intriguing feature of
the Hamiltonian in which the roles of the position and momentum variables often get exchanged [23, 25]. Thus,
Li´enard systems are of potential importance in optics [26] as well as in non-Hermitian quantum mechanics [23, 27].
Naturally, the presence of the damping as is the case for Li´enard systems poses to be 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 Schr¨odinger equation but can be straightforwardly carried out in the momentum
space. Such a Hamiltonian, which has a branched character, may be interpreted to represent an effective-mass
quantum system [23, 28, 29].
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 Li´enard class, which are
of great interest in the theory of dynamical systems. Apart from Li´enard 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 the light of these examples. However, we begin with a discussion on
some simple nonstandard Lagrangians which can be figured out via some guesswork; we call them trial Lagrangians
and they are discussed in Section II. Following this, in Section III we discuss nonstandard Lagrangians and branched
Hamiltonians in the context of Li´enard systems, wherein we outline a systematic derivation of the Lagrangians,
provided the system admits a certain integrability condition. In Sections IV, V and VI 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 VII where, in particular, a few further aspects
of the problem of quantization are discussed. .
II. TRIAL LAGRANGIANS: RECIPROCAL AND LOGARITHMIC KINDS
Sometimes one can make educated guesses to construct plausible nonstandard Lagrangians. We cite a few examples.
Example 1 – Consider the following Lagrangian [9, 10]:
L(x, ˙x) = 1
αµ(x) + β˙x, α ˙x+βµ(x)= 0,(3)
where µ(x) is a well-behaved function (typically a polynomial), while αand βare 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 ¨x+f(x) ˙x+g(x) = 0, with f(x) = 3αµ′(x)
2βand g(x) = α2µ′(x)µ(x)
2β2, where
3
for instance, picking µ(x) = xgives the linearly-damped harmonic oscillator, while the choice µ(x) = x2implies
f(x)∝xand g(x)∝x3. Lagrangians of this type [Eq. (3)] are termed as reciprocal Lagrangians.
Example 2 – Consider another form of Lagrangians classified by [9]
L(x, ˙x) = ln[γµ(x) + δ˙x], γµ(x) + δ˙x= 0,(4)
where δand βare real-valued and non-zero constant numbers. The Euler-Lagrange equation goes as
¨x+f(x) ˙x+g(x) = 0, with f(x) = 2γ µ′(x)
δand g(x) = γ2µ(x)µ′(x)
δ2. Lagrangians given by Eq. (4) are termed
as logarithmic Lagrangians.
Example 3 – As another example, we point out that some equations that go as ¨x+A(x, ˙x) ˙x+B(x, ˙x) = 0, where
A(x, ˙x) and B(x, ˙x) are suitable functions of (x, ˙x) can be derived from (reciprocal) Lagrangians that read
L(x, ˙x) = 1
αµ(x) + βρ( ˙x),(5)
such that βρ′′( ˙x)[αµ(x) + βρ( ˙x)] = 2β2ρ′( ˙x)2. Specifically, the functions A(x, ˙x) and B(x, ˙x) are
A(x, ˙x) = 2αβρ′( ˙x)µ′(x)
2β2ρ′( ˙x)2−βρ′′( ˙x)[αµ(x) + βρ( ˙x)] ,(6)
B(x, ˙x) = αµ′(x)αµ(x) + βρ( ˙x)
2β2ρ′( ˙x)2−βρ′′( ˙x)[αµ(x) + βρ( ˙x)] .(7)
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 the equation may emerge as the
Euler-Lagrange equation. In what follows, we describe Li´enard systems and demonstrate that if a certain integrability
condition is satisfied, then one may systematically find nonstandard Lagrangians describing such systems.
III. LI ´
ENARD SYSTEMS
A Li´enard system is a second-order ordinary differential equation that goes as
¨x+f(x) ˙x+g(x) = 0,(8)
where2f(x), g(x)∈C∞(R,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 −x2) and g(x) = xgives the van
der Pol oscillator, known to admit limit-cycle behavior due to the particular choice of f(x) [30]. Another choice is
f(x) = 1 and g(x) = x3, for which we have the linearly-damped (nonlinear) Duffing oscillator. It is noteworthy that
in any case with f(x)= 0, the system exhibits non-conservative dynamics because Eq. (8) does not stay invariant
under the transformation t→ −t, namely, time reversal. Further, oscillatory dynamics can be obtained if f(x) is an
even function and if g(x) is odd; it follows from the fact that the overall force (the second and third terms of Eq. (8))
should be odd under x→ −xin order to support oscillations [31].
A. Cheillini condition and nonstandard Lagrangians
We now discuss a systematic derivation of nonstandard Lagrangians describing certain Li´enard systems [20, 23].
The idea is to make use of a function known as the Jacobi last multiplier which may be defined as follows [32]. Given
a second-order ordinary differential equation,
¨x=F(x, ˙x),(9)
2Often it is sufficient to have f(x),g (x)∈C2(R,R).
4
one defines the last multiplier Mas a solution of the following differential equation:
dln M
dt +∂F (x, ˙x)
∂˙x= 0.(10)
As has been discussed in Whittaker’s classic textbook [32], if a second-order differential equation such as Eq. (9)
follows from the Euler-Lagrange equations, then the Lagrangian is related to the last multiplier as
M=∂2L
∂˙x2.(11)
This allows one to determine the Lagrangian function for a given second-order differential equation, provided it admits
a Lagrangian formalism. For the Li´enard system, a formal solution for the last multiplier is found to be
M(t, x) = exp Zf(x)dt:= u1/ℓ,(12)
where uis some nonlocal variable and ℓis a suitable parameter whose significance appears in the following lemma:
Lemma 1 The Li´enard equation [Eq. (8)] can be written as the following system of first-order equations:
˙u=ℓuf (x),˙x=u+W(x),(13)
where W(x) = ℓ−1g(x)/f(x), with the parameter ℓbeing determined by the following condition:
d
dx g(x)
f(x)+ℓ(ℓ+ 1)f(x) = 0.(14)
Proof - From Eq. (12) we have
ln u=ℓZf(x)dt, (15)
which means ˙u=ℓuf(x). Simply putting ˙x=u+W(x), we find by differentiating with respect to t, the following
expression:
¨x= ˙u+W′(x) ˙x. (16)
Thereafter, substituting the expression for ˙ufrom the previous equation and after eliminating u, it follows that
¨x=ℓf(x)( ˙x−W(x)) + W′(x) ˙x. (17)
Comparison with Eq. (8) reveals that one must choose W′(x) = −(ℓ+ 1)f(x) and W(x) = ℓ−1g(x)/f (x). Consistency
between these two choices therefore demands that the condition appearing in Eq. (14) should be fulfilled.
It is noteworthy that Eq. (14) represents what is known as the Cheillini condition, allowing one to recast the
Li´enard system in the form suggested in Eq. (13). Specifically, if f(x) = axα, then Eq. (14) dictates that g(x) must
satisfy the following differential equation:
d
dx x−αg(x)+ℓ(ℓ+ 1)a2xα= 0.(18)
A simple integration gives
g(x) = kxα−ℓ(ℓ+ 1)a2
α+ 1 x2α+1 ,(19)
where k∈Ris an integration constant. This gives the functional form of g(x) so as to satisfy the Cheillini condition.
In the cases where the Cheillini condition is satisfied, one can find the Lagrangian as follows:
Theorem 1 Consider a Li´enard system with choices of f(x)and g(x)that satisfy the Cheillini condition. Then, the
second-order differential equation can be obtained as the Euler-Lagrange equation of the Lagrangian that reads
L(x, ˙x) = ℓ2
(ℓ+ 1)(2ℓ+ 1) ˙x−1
ℓ
g(x)
f(x)(2ℓ+1)/ℓ
.(20)
5
Proof - If the functions f(x) and g(x) satisfy Eq. (14), then we have from Eqs. (11) and (12), the following
condition:
∂2L
∂˙x2=˙x−1
ℓ
g(x)
f(x)1/ℓ
.(21)
Then, it follows straightforwardly by simple integration that the Lagrangian is given by Eq. (20). Clearly, a Lagrangian
such as the one appearing in Eq. (20) is of the nonstandard form because there is no identification of kinetic and
potential energies. It may be emphasized that not all choices of f(x) and g(x) would satisfy the Cheillini condition,
meaning that such equations cannot be cast into the form of Eq. (13); an example is the van der Pol oscillator for
which f(x) = (1−x2) and g(x) = x. Therefore, not all Li´enard systems are known to admit a Lagrangian description.
B. Hamiltonian aspects
Now that we are equipped with the Lagrangian that describes the Li´enard system, we can move on to its Hamiltonian
aspects. The conjugate momentum is found to be
p=∂L
∂˙x=ℓ
ℓ+ 1 ˙x−1
ℓ
g(x)
f(x)(ℓ+1)/ℓ
.(22)
Thus, the expression ˙x= ˙x(p) may be multi-valued, depending upon ℓ. It goes as
˙x=K(l)pℓ/(ℓ+1) +1
ℓ
g(x)
f(x),(23)
where K(ℓ) is some function of ℓand is a constant. Using the above form, the Hamiltonian is found to be
H(x, p) = K(ℓ)p2ℓ+1
ℓ+1 −g(x)
ℓf(x)p. (24)
Notice that if Eq. (23) admits branching, then so does the Hamiltonian [Eq. (24)]. Below, we discuss a concrete
example.
C. A concrete example
We discuss an example now, which will help us understand the framework discussed above. In particular, we
will also encounter the notion of momentum-dependent mass [23–25], a concept that has generated some interest in
the recent times, especially within the quantum-mechanical framework. We consider the case where f(x) = xand
g(x) = x−x3[20]; Eq. (14) is satisfied for ℓ= 1,−2. We consider each case separately now.
1. Case with ℓ= 1
In the case where ℓ= 1, we have
p=p(x, ˙x) = 1
2˙x+x2−12=⇒˙x= ˙x(x, p) = 1 −x2±p2p, (25)
which points towards branching. Notice that branching originates from the nonlinear dependence between pand ˙xin
the equation p=∂L
∂˙x. The corresponding branched Hamiltonians turn out to be
H±(x, p) = p1−x2±2
3p2p,(26)
exhibiting two distinct branches, where p≥0. We have plotted the function ˙x= ˙x(p, x) in Fig. (1), while Fig. (2)
shows a plot of the branched pair of Hamiltonians, H±=H±(x, p). The branches coalesce at p= 0.
6
FIG. 1: Plot of ˙x±= ˙x±(x, p) for the case ℓ= 1. We have p≥0 with the two branches meeting at p= 0.
FIG. 2: Plot of the branched Hamiltonian H±=H±(x, p) arising for the case ℓ= 1, with p≥0 and the two branches coalesce
at p= 0.
2. Case with ℓ=−2
In the case where ℓ=−2, we have
p=p(x, ˙x)=2˙x+1−x2
21/2
=⇒˙x= ˙x(x, p) = p2
4−(1 −x2)
2,(27)
implying that there is no branching, because ˙xcan be solved uniquely as a function of the momentum. A straightfor-
ward calculation reveals that the Hamiltonian turns out to be
H(x, p) = p3
12 −p(1 −x2)
2,(28)
wherein, there is only one branch. In Figs. (3) and (4), we have plotted ˙x= ˙x(x, p) and H=H(x, p). An intriguing
aspect of the Hamiltonian given in Eq. (28) is that it may be expressed as
H(x, p) = x2
2p−1+U(p), U(p) = p3
12 −p
2.(29)
This resembles a standard Hamiltonian with the roles of coordinate and momentum being interchanged; its then
tempting to interpret m(p) = p−1as a momentum-dependent mass. The quantization of such systems proceeds in
7
FIG. 3: Plot of ˙x= ˙x(x, p) for the case ℓ=−2. There are no branches.
FIG. 4: Plot of the Hamiltonian H=H(x, p) arising for the case ℓ=−2, showing no branches.
the momentum space, often with the notion of momentum-dependent mass (see for example, Ref. [23]).
IV. THE v4MODEL
Shapere and Wilczek have discussed a concrete model depicting a non-convex nature of the Lagrangian which reads
[1]
L(v) = 1
4v4−κ
2v2,(30)
where vis the velocity3and κ > 0 is a coupling parameter. Corresponding to Eq. (30), the conjugate momentum is
a cubic function in vthat is given by
p(v) = v3−κv . (31)
Clearly, pis not monotonic in velocity, which may lead to branching. The corresponding Hamiltonian is obtained as
H(p) = 3
4v4−κ
2v2, v =v(p),(32)
3Now and in subsequent discussions, we will denote ˙x=v.
8
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 vas shown in Eq. (31).
Hence, for systems with a non-convex Lagrangian such as Eq. (30), the construction of the corresponding Hamil-
tonian in conjugate momentum variable is not unique. A similar incertitude is also encountered in cosmology models
[33, 34], in generalized schemes of Einstein gravity which involve topological invariants, and in theories of higher-
curvature gravity [35]. We note in passing that if one is given a single-valued Lagrangian L(x, v), and define it
according to L(x, v) = x2−V(v) , rather than the usual form L(x, v) = v2−V(x) , and that if the por vdependence
is non-convex, then branched functions are always encountered, as a result of employing the Legendre transformation,
despite having started with a single-valued Lagrangian or Hamiltonian function.
V. A GENERALIZED CLASS OF LAGRANGIANS AND BRANCHED HAMILTONIANS
A. Velocity-independent potentials
Curtright and Zachos [17] extended the analysis of [1] by considering a generalized class of non-quadratic Lagrangians
that go as
L(x, v) = C(v−1) 2k−1
2k+1 −V(x), C =2k+ 1
2k−11
42
2k+1
,(33)
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 ´a la Witten [36]. 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 vnear zero, we can write L(x, v)≈
C(−1 + v
3+v2
9+O(v3)) −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:
A=Zt2
t1
L(x, v)dt ≈Ct2−t1+1
3(x(t2)−x(t1)) + 1
9Zt2
t1
v2dt +Zt2
t1
O(v3)dt−Zt2
t1
V(x)dt. (34)
Thus, for small velocities, the action results in the usual Newtonian form of the equations of motion.
On the other hand, for large velocities, we have a rather nontrivial scenario which leads (for finite, positive-integer
values of k) to a non-convex function of v. The curvature term corresponding to the quantity ∂2L
∂v2changes sign at
just 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 given in Eq. (33), the canonical momentum can be easily calculated to
be
p=p(v) = 1
42
2k+1 1
(v−1) 2
2k+1
,(35)
Inverting the relation we observe at once that the velocity variable v(p) emerges as a double-valued function of p:
v=v±(p) = 1∓1
41
√p(2k+1)
.(36)
Corresponding to the two signs above, a pair branches of the Hamiltonian, namely, H±(x, p) will appear. Specifically,
for any positive-integer value of k, these may be identified to be
H±(x, p) = p±1
4k−21
√p2k−1
+V(x).(37)
9
From a classical perspective, in order to avoid an imaginary v(p), one needs to address a nonnegative p. This in turn
implies that the slope ∂ L
∂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±(x, p)−V(x), which reads p±1
2√p, in the momentum space.
The associated spectral properties have been analyzed in the literature [23, 29]. We end our discussion on this example
by noting that in the special case where V(x) = x2, the branched Hamiltonian is H±(x, p) = x2+U±(p), wherein it
appears as if the roles of the coordinate and the momentum have been interchanged with U±(p) being a momentum-
dependent potential that exhibits two branches.
B. Velocity-dependent potentials
Lines of force can be ascertained with the help of velocity-dependent potentials which ensure that particles take to
certain specified paths [11, 37]. In electrodynamics, the field vectors
Eand
Bcan 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 upon a velocity-dependent potential V(x, v) in the manner as
given by [19, 27]
L(x, v) = C(v−1) 2k−1
2k+1 − V(x, v), C =2k+ 1
2k−11
42
2k+1
,(38)
where V(x, v) is assumed to be given in a separable form, i.e., V(x, v) = U(v) + V(x); here U(v) and V(x) are
well-behaved functions of vand x, respectively. Using the standard definition of the canonical momentum, we find
its form to be
p=p(v) = 1
42
2k+1
(v−1)−2
2k+1 −U′(v).(39)
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 on employing the Legendre transform as
H±(x, p) = p±1
4[p+U′(v)]−2k−1
22k+ 1
2k−1−p[p+U′(v)]−1+V(x, v), v =v(p).(40)
Unfortunately, since a Hamiltonian has to be a function of the coordinate and its corresponding canonical momentum,
the generality of the form of H±(x, p) as derived above is of little use unless we have an explicit inversion of Eq. (39)
giving v=v(p). We therefore have to go for the specific cases of kand U(v).
1. 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)=31
42
3
(v−1)1
3−U(v)−V(x).(41)
A sample choice for U(v) could be [19]
U(v) = λv + 3δ(v−1)1
3,(42)
in which λ(≥0) and δ(<4−2
3) are suitable real constants. The presence of the parameter δscales the kinetic-energy
term in the Lagrangian. The canonical momentum pis now given by
p=p(v) = µ(v−1)−2
3−λ , (43)
where the quantity µ= 4−2
3−δ > 0. We are therefore led to a pair of relations for the velocity depending on p:
10
v=v±(p)=1∓µ3
2(p+λ)−3
2.(44)
In consequence, we find two branches of the Hamiltonian which are expressible as
H±(x, p)=(p+λ)±2γ
√p+λ+V(x),(45)
where µ3/2has been replaced by γ. As a final comment, the special case corresponding to λ= 0 and γ= 1/4 conforms
to the Hamiltonian advanced in [17] [Eq. (37)].
VI. TWO MORE CLASSES OF BRANCHED HAMILTONIANS
A. Example 1
As an extension of Eq. (33), the following higher-power Lagrangian was proposed in [5]:
L(x, v) = C(v+σ(x)) 2m+1
2m−1−δ, Λ = 1−2m
1+2m(δ)2
1−2m, δ > 0,(46)
where we notice that the coefficient Λ is non-negative for 0 ≤m < 1
2. The main difference from Eq. (33) is in
the choice of a general function σ(x) in place of σ(x) = −1 as in Eq. (33). The other point is that the inverse
exponent with respect to the model of Curtright and Zachos [17] has been taken for 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 δand that of a nontrivial function σ(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) = −(δ)2
1−2m(v+σ(x)) 2
2m−1,(47)
and a simple inversion yields
v=v±(x, p) = −σ(x) + δ±√−p2m−1.(48)
This means, the Hamiltonian is obtained to be
H±(x, p) = (−p)σ(x)−2δ
2m+ 1±√−p2m+1 +δ . (49)
1. A 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±(x, p) = −σ(x)±δ
√−p,(50)
implying that the Hamiltonian branches out into components:
H±(x, p) = (−p)σ(x)∓2δ√−p+δ . (51)
The nature of the two Hamiltonians depends on the sign of p. Once we specify the following choice of σ(x), namely,
σ(x) = λ
2x2+9λ2
2k2, λ > 0,(52)
11
together with the choice δ=9λ2
2k2, then upon imposing a simple translation p→2k
3λp−1, the Hamiltonians H±acquire
the forms that go as
H±(x, p) = 9λ2
2k2"2∓21−2kp
3λ1
2
+k2x2
9λ−2kp
3λ−2k3x2p
27λ2#.(53)
These are readily identifiable as a set of plausible Hamiltonians representing a nonlinear Li´enard system [20, 23, 25].
The appearance of the coordinate-momentum coupling is noteworthy, and leads us to the notion of a momentum-
dependent mass as
H±(x, p) = x2
2m(p)+U±(p), m(p) = λ−2kp
3−1
, U±(p) = 9λ2
2k2"2∓21−2kp
3λ1
2
−2kp
3λ#.(54)
From a classical perspective, the momentum pis needed to be restricted to the range −∞ < p ≤3λ
2kto 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=3λ
2k, a thorough analytical study of H±(x, p) becomes
greatly involved. Observe that when p=3λ
2k, we find the coincidence of the two Hamiltonians H±(x, p).
B. Example 2
As a final example, let us turn to an illustration where L(x, v) is of the reciprocal kind and is defined to be [5]
L(x, v) = 1
s1
3sx2+3
sλ−v−1
,(55)
where sis a real parameter. The canonical momentum comes out as
p=p(x, v) = 1
s1
3sx2+3
sλ−v−2
,(56)
which, when inverted yields
v=v±(x, p) = 1
3sx2+3
sλ±1
√sp .(57)
The accompanying Hamiltonian corresponding to the above Lagrangian has two branches:
H±(x, p) = s
3x2p+3
sλp ±2rp
s.(58)
It should be remarked that as λ→0, Eq. (55) is just the trial Lagrangian given in Eq. (5), for the choice µ(x) = ax2
and under a suitable identification of the constant parameters. We end by noting that Eq. (58) can be expressed with
a momentum-dependent mass as
H±(x, p) = x2
2m(p)+U±(p), m(p) = 3
2sp, U±(p) = 3
sλp ±2rp
s.(59)
VII. CONCLUDING REMARKS
In our present short review on the existence of nonstandard Lagrangians, we emphasized upon the associated
branched Hamiltonians. Various different examples were discussed; in all of them, the velocity dependence of
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
12
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 [23, 28, 29]. 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. [38]) 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 [39]. Moreover, after one admits the unusual forms of the Hamiltonians
characterized, typically, by the popular parity-time symmetry (PT -symmetry – see for example, Refs. [40, 41] for a
pedagogic and introductory discussion on such specific variants of non-self-adjoint models), the anomalies may occur
even when the PT -symmetry itself remains unbroken.
Several unusual forms of the latter anomalies may appear in both the spectra and eigenfunctions, materialized as
the Kato’s exceptional points [44, 45] or the so-called spectral singularities [46]. 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 [47–49].
Naturally, the possible relevance of the latter anomalies in the quantum systems controlled by the branched Hamil-
tonians 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 which
undergo non-unitary quantum evolution; they generally represent open systems with balanced gain and loss [42, 43].
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 [50],
or the feature of stability-loss delay [51], 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.
Acknowledgments
We thank Prof. Anindya Ghose Choudhury for discussions and for his interest in this work. B.B. thanks Brainware
University for infrastructural support. A.G. thanks the Ministry of Education (MoE), Government of India, for
financial support in the form of a Prime Minister’s Research Fellowship (ID: 1200454). M.Z. is financially supported
by the Faculty of Science of UHK.
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