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.

Full text

Citation: Bagchi, B.; Ghosh, A.; Znojil,
M. A Reappraisal of Lagrangians with
Non-Quadratic Velocity Dependence
and Branched Hamiltonians.
Symmetry 2024,16, 860. https://
doi.org/10.3390/sym16070860
Academic Editor: Fernando Haas
Received: 1 June 2024
Revised: 26 June 2024
Accepted: 2 July 2024
Published: 7 July 2024
Copyright: © 2024 by the authors.
Licensee MDPI, Basel, Switzerland.
This article is an open access article
distributed under the terms and
conditions of the Creative Commons
Attribution (CC BY) license (https://
creativecommons.org/licenses/by/
4.0/).
symmetry
S
S
Article
A Reappraisal of Lagrangians with Non-Quadratic Velocity
Dependence and Branched Hamiltonians
Bijan Bagchi 1, Aritra Ghosh 2and Miloslav Znojil 3,4,5,*
1Department of Mathematics, Brainware University, Kolkata 700125, West Bengal, India;
bbagchi123@gmail.com
2School of Basic Sciences, Indian Institute of Technology Bhubaneswar, Jatni 752050, Odisha, India;
ag34@iitbbs.ac.in
3The Czech Academy of Sciences, Nuclear Physics Institute, Hlavní 130, 25068 ˇ
Rež, Czech Republic
4Department of Physics, Faculty of Science, University of Hradec Králové, Rokitanského 62,
50003 Hradec Králové, Czech Republic
5Institute of System Science, Durban University of Technology, Durban 4001, South Africa
*Correspondence: znojil@ujf.cas.cz
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.
Keywords: nonstandard Lagrangians; branched Hamiltonians; Liénard systems; momentum-
dependent mass
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
–
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 (
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,˙
x) = (αx+β˙
x)−1, (1)
where
α
and
β
are real numbers satisfying
αβ >
0. We may also require that
αx+β˙
x=
0,
i.e., that the velocity phase space accessible to the system is defined as a subset of R2.
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’
Symmetry 2024,16, 860. https://doi.org/10.3390/sym16070860 https://www.mdpi.com/journal/symmetry

Symmetry 2024,16, 860 2 of 16
would refer to a Lagrangian with a non-quadratic velocity dependence. (A linear depen-
dence on velocity makes the Hessian matrix singular, resulting in a singular Legendre
transform while passing from the Lagrangian to the Hamiltonian formalism (see, for ex-
ample, 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
¨
x+γ˙
x+ω2
0x=0, (2)
where
γ=3α
2β
and
ω0=α
√2β
.
(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 ‘stan-
dard’ 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 nonstan-
dard 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
,
˙
x)
defined for a system whose configuration space is a subset
of Ris called regular if its Hessian with respect to the velocity is non-vanishing, i.e.,
∂2L(x,˙
x)
∂˙
x2=0, (3)
and is of constant sign, allowing us to solve for the velocity
˙
x
in favor of the momentum
p(x
,
˙
x) = ∂L(x,˙
x)
∂˙
x
, i.e., we can write
˙
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
,
˙
x)
in favor of
˙
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
–
3
]. This has triggered a series of papers by Curtright and
Zachos [16–21],
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 simu-
lations 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
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

Symmetry 2024,16, 860 3 of 16
exchanged with the emergence of the notion of a momentum-dependent
mass [23,26–31].
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
Sections 4and 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 6and 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,˙
x) = 1
αµ(x) + β˙
x,α˙
x+βµ(x)=0, (4)
where
µ(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 differ-
ence between the kinetic and potential energies. However, the Euler–Lagrange equation gives
¨
x+f(x)˙
x+g(x) = 0,
with
f(x) = 3αµ′(x)
2β
and
g(x) = α2µ′(x)µ(x)
2β2
, where, for instance, picking
µ(x) = x
gives the linearly damped harmonic oscillator, while the choice
µ(x) = x2
implies
f(x)∝x and g(x)∝x3. Lagrangians of this type (4)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, (5)
where
δ
and
β
are real-valued and non-zero constant numbers. The Euler–Lagrange equation goes
as
¨
x+f(x)˙
x+g(x) =
0, with
f(x) = 2γµ′(x)
δ
and
g(x) = γ2µ(x)µ′(x)
δ2
. 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 R2.
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), (6)
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)] , (7)

Symmetry 2024,16, 860 4 of 16
B(x,˙
x) = αµ′(x)αµ(x) + βρ(˙
x)
2β2ρ′(˙
x)2−βρ′′(˙
x)[αµ(x) + βρ(˙
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
¨
x+f(x)˙
x+g(x) = 0 (9)
(often, it is sufficient to have
f(x)
,
g(x)∈C2(U
,
R)
, where
U⊂R
).
f(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) = 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) = x3
, 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)=
0, the system exhibits non-conservative dynamics because
(9)
does not stay
invariant under the transformation
t→ −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→ −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 La-
grangian has been the subject of much investigation [
37
–
42
] (see also Ref. [
43
]). In particular,
for Liénard systems satisfying a certain integrability condition, one can find nonstan-
dard 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:
¨
x=F(x,˙
x), (10)
one defines the last multiplier Mas that which satisfies
dln M
dt +∂F(x,˙
x)
∂˙
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=∂2L(x,˙
x)
∂˙
x2. (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Zf(x)dt. (13)
We may define a new nonlocal variable uas [41]
u=˙
x−g(x)
ℓf(x), (14)

Symmetry 2024,16, 860 5 of 16
with ℓ=−1, which is determined from
d
dx g(x)
f(x)+ℓ(ℓ+1)f(x) = 0. (15)
Then, using Equations
(14)
and
(15)
, we have
˙
u=ℓu f (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:
˙
u=ℓu f (x),˙
x=u+W(x), (16)
where
W(x) = ℓ−1g(x)/f(x)
. Since we have
˙
u=ℓu f (x)
, using
(13)
, we can write
M=
u1/ℓ
, which, upon using
(12)
, gives us (up to a gauge function, which can be ignored here)
L∼u2ℓ+1
ℓ. In terms of xand ˙
x, the Lagrangian reads as [23]
L(x,˙
x) = ℓ2
(ℓ+1)(2ℓ+1)˙
x−1
ℓ
g(x)
f(x)2ℓ+1
ℓ
, (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,˙
x) = ˙
x2
2+J(x)˙
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) = axα
, then (15)
dictates that g(x)must satisfy the following differential equation (see also Refs. [44,45]):
d
dx x−αg(x)+ℓ(ℓ+1)a2xα=0. (18)
A simple integration gives
g(x) = kxα−ℓ(ℓ+1)a2
α+1x2α+1, (19)
where
k∈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 Hamil-
tonian aspects. The conjugate momentum is found to be
p=∂L
∂˙
x=ℓ
ℓ+1˙
x−1
ℓ
g(x)
f(x)(ℓ+1)/ℓ
. (20)
Thus, the expression ˙
x=˙
x(p)may be multi-valued, depending on ℓ. It goes as
˙
x=K(l)pℓ/(ℓ+1)+1
ℓ
g(x)
f(x), (21)
where
K(ℓ)
is some function of
ℓ
and 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. (22)
Notice that, if (21) admits branching, then so does the Hamiltonian (22). Below, we discuss
a concrete example.

Symmetry 2024,16, 860 6 of 16
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
–
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−x3
[
23
]; (15) is satisfied for
ℓ=1, −2. For the sake of clarity, we will consider every such case separately.
3.3.1. Case with ℓ=1
In the case of ℓ=1, we have
p=p(x,˙
x) = 1
2˙
x+x2−12=⇒˙
x=˙
x(x,p) = 1−x2±p2p. (23)
This points towards branching. Notice that branching originates from the nonlinear depen-
dence between
p
and
˙
x
in the equation
p=∂L
∂˙
x
. The corresponding branched Hamiltonians
turn out to be
H±(x,p) = p1−x2±2
3p2p, (24)
exhibiting two distinct branches, where
p≥
0. We plot the function
˙
x=˙
x(p
,
x)
in
Figure 1, while Figure 2shows a plot of the branched pair of Hamiltonians,
H±=H±(x,p).
The branches coalesce at p=0.
Figure 1. Plot of
˙
x±=˙
x±(x
,
p)
for the case
ℓ=
1. We have
p≥
0 with the two branches meeting
at p=0.
Figure 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.

Symmetry 2024,16, 860 7 of 16
3.3.2. Case with ℓ=−2
In the case where ℓ=−2, we have
p=p(x,˙
x) = 2˙
x+1−x2
21/2
=⇒˙
x=˙
x(x,p) = p2
4−(1−x2)
2, (25)
implying that there is no branching, because
˙
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) = p3
12 −p(1−x2)
2, (26)
wherein there is only one branch.
In Figures 3and 4, we plot
˙
x=˙
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) = x2
2p−1+U(p),U(p) = p3
12 −p
2. (27)
This resembles a standard Hamiltonian, only with the roles of the coordinate and momen-
tum being interchanged.
Figure 3. Plot of ˙
x=˙
x(x,p)for the case ℓ=−2. There are no branches.
Figure 4. Plot of the Hamiltonian H=H(x,p)arising for the case ℓ=−2, showing no branches.
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 con-
text 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 consis-
tent 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

Symmetry 2024,16, 860 8 of 16
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) = x2−V(v)
, rather than according to the usual recipe
L(x
,
v) = v2−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 v4Model
Shapere and Wilczek have discussed a concrete model depicting the non-convex nature
of the Lagrangian, which reads [1]
L(v) = 1
4v4−κ
2v2, (28)
where
v
is the velocity (now and in subsequent discussions, we will denote
˙
x=v
) and
κ>
0 is a coupling parameter. Corresponding to (28), the conjugate momentum is a cubic
function in vthat is given by
p(v) = v3−κv. (29)
Clearly,
p
is not monotonic in velocity, which may lead to branching. The corresponding
Hamiltonian is obtained as
H(p) = 3
4v4−κ
2v2,v=v(p), (30)
which, like L(v), is also a multi-valued function (with cusps) in the conjugate momentum
p, since each given pcorresponds 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)2k−1
2k+1−V(x),C=2k+1
2k−11
4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
(
2
k+
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)≈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 ≈Ct2−t1+1
3(x(t2)−x(t1)) + 1
9Zt2
t1
v2dt +Zt2
t1
O(v3)dt−Zt2
t1
V(x)dt. (32)

Symmetry 2024,16, 860 9 of 16
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
∂2L
∂v2
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) = 1
42
2k+11
(v−1)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±(p) = 1∓1
41
√p(2k+1)
. (34)
Corresponding to the two signs above, a pair of branches of the Hamiltonian, namely
H±(x
,
p)
, will appear. Specifically, for any positive-integer value of
k
, these may be identi-
fied to be
H±(x,p) = p±1
4k−21
√p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
∂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 [26,30].
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.
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

E
and

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
V(x
,
v)
in the manner as given by [22,29]:
L(x,v) = C(v−1)2k−1
2k+1−V(x,v),C=2k+1
2k−11
42
2k+1, (36)
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
v
and
x
, respectively. Using the standard
definition of the canonical momentum, we find its form to be
p=p(v) = 1
42
2k+1(v−1)−2
2k+1−U′(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. Neverthe-
less, the associated branches of the Hamiltonian can be straightforwardly written down
upon employing the Legendre transform as
H±(x,p) = p±1
4[p+U′(v)]−2k−1
22k+1
2k−1−p[p+U′(v)]−1+V(x,v),v=v(p). (38)

Symmetry 2024,16, 860 10 of 16
Unfortunately, since a Hamiltonian has to be a function of the coordinate and its corre-
sponding 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 (37) giving
v=v(p)
. We, therefore,
have to go for the specific cases of kand 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1
42
3(v−1)1
3−U(v)−V(x). (39)
A sample choice for U(v)could be [22]
U(v) = λv+3δ(v−1)1
3, (40)
in which
λ(≥
0
)
and
δ(<
4
−2
3)
are suitable real constants. The presence of the parameter
δ
scales the kinetic energy term in the Lagrangian. The canonical momentum
p
is now
given by
p=p(v) = µ(v−1)−2
3−λ, (41)
where the quantity
µ=
4
−2
3−δ>
0. We are, therefore, led to a pair of relations for the
velocity depending on p:
v=v±(p) = 1∓µ3
2(p+λ)−3
2. (42)
As a consequence, we find two branches of the Hamiltonian, which are expressible as
H±(x,p) = (p+λ)±2γ
pp+λ+V(x), (43)
where µ3/2 has been replaced by γ. As a final comment, the special case corresponding to
λ=0 and γ=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+σ(x)) 2m+1
2m−1−δ,Λ=1−2m
1+2m(δ)2
1−2m,δ>0, (44)
where we notice that the coefficient
Λ
is non-negative for 0
≤m<1
2
. The main difference
from (31) is in the choice of a general function
σ(x)
in place of
σ(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
δ
and 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, (45)
and a simple inversion yields
v=v±(x,p) = −σ(x) + δ±p−p2m−1. (46)
This means the Hamiltonian is obtained to be

Symmetry 2024,16, 860 11 of 16
H±(x,p) = (−p)σ(x)−2δ
2m+1±p−p2m+1+δ. (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±(x,p) = −σ(x)±δ
√−p, (48)
implying that the Hamiltonian branches out into components:
H±(x,p) = (−p)σ(x)∓2δp−p+δ. (49)
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, (50)
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∓21−2kp
3λ1
2+k2x2
9λ−2kp
3λ−2k3x2p
27λ2#. (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±(x,p) = x2
2m(p)+U±(p),m(p) = λ−2kp
3−1
,U±(p) = 9λ2
2k2"2∓21−2kp
3λ1
2−2kp
3λ#. (52)
From a classical perspective, the momentum
p
needs to be restricted to the range
−∞<
p≤3λ
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=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).
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) = 1
s1
3sx2+3
sλ−v−1
, (53)
where sis a real parameter. The canonical momentum comes out as
p=p(x,v) = 1
s1
3sx2+3
sλ−v−2
, (54)
which, when inverted, yields
v=v±(x,p) = 1
3sx2+3
sλ±1
√sp . (55)
The accompanying Hamiltonian corresponding to the above Lagrangian has two branches:
H±(x,p) = s
3x2p+3
sλp±2rp
s. (56)

Symmetry 2024,16, 860 12 of 16
It should be remarked that, as
λ→
0, (53) is just the trial Lagrangian (6), for the choice
µ(x) = ax2
and under a suitable identification of the constant parameters. We end by
noting that (56) can be re-expressed as a model:
H±(x,p) = x2
2m(p)+U±(p),m(p) = 3
2s p ,U±(p) = 3
sλp±2rp
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) = −mcpc2−v2,v<c. (58)
The conjugate momentum is obtained as
p=p(v) = ∂L(v)
∂v=mcv
√c2−v2. (59)
This implies that p2c2−p2v2−m2c2v2=0. Solving for the velocity gives
v=v±(p) = ±pc
pp2+m2c2. (60)
Thus, the Hamiltonian reads
H±(p) = c(±p2−m2c2)
pp2+m2c2. (61)
In particular, one may consider the ultrarelativistic limit (
v≈c
) in which
H±(p) = ±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 empha-
sized, 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 (PT -symmetry; see, for example, Refs. [56,57] for a pedagogic and

Symmetry 2024,16, 860 13 of 16
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 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 [61–63].
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.
Author Contributions: Conceptualization, B.B. and M.Z.; methodology, B.B. and A.G. and M.Z.;
software, A.G.; validation, M.Z.; formal analysis, B.B.; investigation, A.G. and M.Z.; resources,
B.B. and M.Z.; data curation, A.G.; writing and original draft preparation, B.B. and A.G. and M.Z.;
review and editing, B.B. and A.G. and M.Z.; visualization, A.G.; supervision, B.B. and M.Z.; project
administration, B.B.; funding acquisition, A.G. All authors have read and agreed to the published
version of the manuscript.
Funding: This research was funded by the Ministry of Education (MoE), Government of India, by the
Prime Minister’s Research Fellowship grant number 1200454, and from the budget of the Brainware
University and of the University of Hradec Kralove.
Data Availability Statement: Data are contained within the article.
Acknowledgments: We thank 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), the 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.
Conflicts of Interest: The authors declare no conflict of interest.
Appendix A. Jacobi Last Multiplier
The Jacobi last multiplier [
37
] is a very useful tool in the mathematics of dynamical
systems. On the one hand, given a dynamical system on an
m
-dimensional phase space
with
(m−
2
)
linearly independent and known first integrals, it allows one to determine
the ‘last’, i.e.,
(m−
1
)
th independent first integral; on the other hand, given a second-order
(ordinary) differential equation, it facilitates the computation of a suitable Lagrangian
function that describes the dynamics via the Euler–Lagrange equation [
37
–
40
]. Thus,
the last multiplier is intimately related to the integrability properties of a dynamical system.
Naively, one can define it as follows. Given a vector field
X
on the phase space, the Jacobi
last multiplier
M
is a factor such that
MX
has zero divergence. For Hamiltonian systems
for which Liouville’s theorem holds, i.e.,
div ·X=
0, the last multiplier is just a constant
number as
X
is already divergence-free. On the other hand,
M
assumes a more non-trivial

Symmetry 2024,16, 860 14 of 16
form if
X
has non-vanishing divergence, i.e., if
div ·X=
0, say, for the Liénard equation,
which is dissipative and, hence, is not volume-conserving (see, for example, Refs. [44,45]).
Consider a vector field
X
; in local coordinates
{xj}
, where
j=
1, 2,
. . .
,
m
, one can write
its components as
˙
xj=Xj({xj})
, which define the dynamical system as a system of first-
order equations. Further, let there be a certain number of first integrals
(F1
,
F2
,
. . .
,
Fk)
, where
k<m
. For any open subset
Ω⊂Rm
, we define the Jacobi last multiplier to be a function
M:Rm→R, which is non-negative and defines an invariant measure RΩMdmx, i.e.,
ZΩMdx1∧dx2∧. . . ∧dxm=Zϕt(Ω)M∂(x1,x2, . . . , xk,xk+1, . . . , xm)
∂(F1,F2, . . . , Fk,xk+1, . . . , xm)dF1∧dF2∧. . . ∧dFk∧dxk+1∧. . . dxm, (A1)
where
ϕt(Ω)
is the transformation of the region
Ω
under the flow of
X
. Notice that it
is necessary that
(F1
,
F2
,
. . .
,
Fk)
be independent, ensuring that
dF1∧dF2∧. . . ∧dFk=
0.
The above-mentioned invariance condition leads to the equation:
d
dt M∂(x1,x2, . . . , xk,xk+1, . . . , xm)
∂(F1,F2, . . . , Fk,xk+1, . . . , xm)!=0, (A2)
which, upon employing the chain rule, gives
dM
dt
∂(x1,x2, . . . , xk,xk+1, . . . , xm)
∂(F1,F2, . . . , Fk,xk+1, . . . , xm)+M
m
∑
j=1
∂Xj
∂xj
∂(x1,x2, . . . , xk,xk+1, . . . , xm)
∂(F1,F2, . . . , Fk,xk+1, . . . , xm)=0. (A3)
This is just equivalent to
d
dt ln M+
m
∑
j=1
∂Xj
∂xj
=0, (A4)
which coincides with (11) as appropriate for the system (10). Notice that
∑m
j=1
∂Xj
∂xj
is just the
divergence of
X
, and therefore,
M=
1 or some constant if the vector field has zero divergence.
Now, having defined the Jacobi last multiplier, let us demonstrate its use in deriving
Lagrangians for a second-order ordinary differential equation with a two-dimensional
phase space. Consider the system (10). If it is derivable from the Euler–Lagrange equation:
d
dt ∂L(x,˙
x)
∂˙
x=∂L(x,˙
x)
∂x, (A5)
or ∂2L(x,˙
x)
∂˙
x2¨
x+∂2L(x,˙
x)
∂˙
x∂x˙
x−∂L(x,˙
x)
∂x=0, (A6)
then, from (10), one should have
∂2L(x,˙
x)
∂˙
x2F(x,˙
x) + ∂2L(x,˙
x)
∂˙
x∂x˙
x−∂L(x,˙
x)
∂x=0. (A7)
Differentiating both sides with respect to ˙
xgives
∂
∂˙
x∂2L(x,˙
x)
∂˙
x2F(x,˙
x)+∂3L(x,˙
x)
∂˙
x2∂x˙
x=0. (A8)
Defining Σ(x,˙
x) = ∂2L(x,˙
x)
∂˙
x2, (A8) gives
∂
∂˙
x(Σ(x,˙
x)F(x,˙
x)) + ∂
∂x(Σ(x,˙
x)˙
x) = 0. (A9)
But, this is just Equation (A4) of the last multiplier (if
M
has no explicit time dependence),
i.e., we should identify
Σ(x
,
˙
x) = M(x
,
˙
x)
, and this readily gives (12). See Refs. [
37
–
40
] for
more details.

Symmetry 2024,16, 860 15 of 16
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