# Branko. DragovichInstitute of Physics Belgrade | IPB · Institute of Physics

Branko. Dragovich

PhD

## About

147

Publications

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2,144

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Citations since 2017

## Publications

Publications (147)

Earlier constructed a simple nonlocal de Sitter gravity model has a cosmological solution in a very good agreement with astronomical observations. In this paper, we continue the investigation of the nonlocal de Sitter model of gravity, focusing on finding an appropriate solution for the Schwarzschild-de Sitter metric. We succeeded to solve the equa...

A bstract
This paper is devoted to a simple nonlocal de Sitter gravity model and its exact vacuum cosmological solutions. In the Einstein-Hilbert action with Λ term, we introduce nonlocality by the following way: $$ R-2\Lambda =\sqrt{R-2\Lambda}\kern1em \sqrt{R-2\Lambda}\to \sqrt{R-2\Lambda}\kern1em F\left(\square \right)\kern1em \sqrt{R-2\Lambda}...

This paper is devoted to a simple nonlocal de Sitter gravity model and its exact vacuum cosmological solutions. In the Einstein-Hilbert action with $\Lambda$ term, we introduce nonlocality by the following way: $R - 2 \Lambda = \sqrt{R-2\Lambda}\ \sqrt{R-2\Lambda} \to \sqrt{R-2\Lambda}\ F(\Box)\ \sqrt{R-2\Lambda} ,$ where ${F} (\Box) = 1 + \sum_{n=...

In this paper, we introduce a new type of matter that has origin in $p$-adic strings, i.e., strings with a $p$-adic worldsheet. We investigate some properties of this $p$-adic matter, in particular its cosmological aspects. We start with crossing symmetric scattering amplitudes for $p$-adic open strings and related effective nonlocal and nonlinear...

In this paper, we introduce a new type of matter that has origin in p-adic strings, i.e., strings with a p-adic worldsheet. We investigate some properties of this p-adic matter, in particular its cosmological aspects. We start with crossing symmetric scattering amplitudes for p-adic open strings and related effective nonlocal and nonlinear Lagrangi...

A nonlocal gravity model (2) was introduced and considered recently, and two exact cosmological solutions in flat space were presented. The first solution is related to some radiation effects generated by nonlocal dynamics on dark energy background, while the second one is a nonsingular time symmetric bounce. In the present paper, we investigate ot...

A nonlocal gravity model (2.1) was introduced and considered recently [49], and two exact cosmological solutions in flat space were presented. The first solution is related to some radiation effects generated by nonlocal dynamics on dark energy background, while the second one is a nonsingular time symmetric bounce. In the present paper we investig...

The principal objective of this article is a brief overview of the main parts of p-adic mathematics, which have already had valuable applications and may have a significant impact in the near future on the further development of some fields of theoretical and mathematical biology. In particular, we present the basics of ultrametrics, p-adic numbers...

This article is related to construction of zeta strings from $p$-adic ones. In addition to investigation of $p$-adic string for a particular prime number $p$, it is also interesting to study collective effects taking into account all primes $p$. An idea behind this approach is that a zeta string is a whole thing with infinitely many faces which we...

The genetic code (GC) plays a central role in all living organisms. From a mathematical point of view, the GC is a map from a set of 64 elements (which are codons) onto a set of 21 elements (which are 20 amino acids and 1 stop signal). The GC is the result of evolution, experimentally deciphered as early as the mid-1960s, but its satisfactory theor...

In this paper, we investigate a nonlocal modification of general relativity (GR) with action $S = \frac{1}{16\pi G} \int [ R- 2\Lambda + (R-4\Lambda) \, \mathcal{F}(\Box) \, (R-4\Lambda) ] \, \sqrt{-g}\; d^4x ,$ where $\mathcal{F} (\Box) = \sum_{n=1}^{+\infty} f_n \Box^n$ is an analytic function of the d'Alembertian $\Box$. We found a few exact cos...

In this paper, we investigate a nonlocal modification of general relativity (GR) with action S = 1 16 π G ∫ [ R − 2 Λ + ( R − 4 Λ ) F ( □ ) ( R − 4 Λ ) ] − g d 4 x , where F ( □ ) = ∑ n = 1 + ∞ f n □ n is an analytic function of the d’Alembertian □. We found a few exact cosmological solutions of the corresponding equations of motion. There are two...

We investigate probabilistic properties of random triangles in the space of finite sequences with the Hamming metrics. As a triangle is understood any triple of points with distances between them. Probability measure is given by the classical way. In particular, it is shown that randomly chosen triangle is approximately equilateral with high probab...

Хорошо известно, что Общая теория относительности (ОТО) имеет значительные феноменологические успехи и замечательные теоретические свойства. Однако, ОТО не является полной теорией гравитации. Следовательно, существуют многие попытки модифицировать ОТО. Одним из существенных подходов к более полной теории гравитации является нелокальная модификация...

Significant phenomenological success and nice theoretical properties of general relativity (GR) are well known. However, GR is not a complete theory of gravity. Hence, there are many attempts to modify GR. One of the current approaches to a more complete theory of gravity is a nonlocal modification of GR. The nonlocal gravity approach, which we con...

In this paper we consider modification of general relativity extending R−2Λ by nonlocal term of the form R−2ΛF(□)R−2Λ, where F(□) is an analytic function of the d'Alembert operator □. We have found some exact cosmological solutions of the corresponding equations of motion without matter and with Λ≠0. One of these solutions is a(t)=At23eΛ14t2, which...

In this paper we consider modification of general relativity extending $R - 2 \Lambda$ by nonlocal term of the form $\sqrt{R-2\Lambda}\, \mathcal{F}(\Box)\, \sqrt{R-2\Lambda} ,$ where $\mathcal{F}(\Box)$ is an analytic function of the d'Alembert operator $\Box$. We have found some exact cosmological solutions of the corresponding equations of motio...

We consider a modification of GR with a special type of a non-local f (R). The structure of the non-local operators is motivated by the string field theory and p-adic string theory. The spectrum is derived explicitly and the ghost-free condition for the model is formulated. We pay special attention to the classical stability of the de Sitter soluti...

Motivated by successful p-adic modeling of the genetic code, in this paper a p-adic (ultrametric) language is introduced. This language can serve as an effective and advanced constructive element of a future artificial intelligence. In this geometric approach to artificial language, we mainly use p-adic distance as the most powerful example of ultr...

We consider nonlocal modified Einstein gravity without matter, where nonlocal term has the form \(P(R) \mathcal {F}(\Box ) Q(R)\). For this model, in this paper we give the derivation of the equations of motion in detail. This is not an easy task and presented derivation should be useful to a researcher who wants to investigate nonlocal gravity. Al...

This paper contains a brief review of a very diverse and vast scientific work of Andrei Yurievich Khrennikov on the occasion of his 60th birthday.

The genetic code is a mapping from the set of 64 codons onto the set of 20 amino acids and one stop signal. The codons are ordered triplets composed of the nucleotides cytosine (C), adenine (A), uracil (U) (or thymine (T)), guanine (G) and they are contained in the genes. The amino acids are building blocks of the proteins. The vertebrate mitochond...

Summation of the p-adic functional series∑ εⁿ n! Pkε(n;x)xn, where Pkε(n; x) is a polynomial in x and n with rational coefficients, and ε = ±1, is considered. The series is convergent in the domain |x|p ≤ 1for all primes p. It is found the general form of polynomials Pkε (n; x) which provide rational sums when x ∈ Z. A class of generating polynomia...

A class of nonlocal gravity models, where nonlocal term contains an analytic function of the d'Alembert operator □, is considered. For simplicity, these models are considered without matter sector. Related equations of motion for gravitational field gμv(x) are presented and analyzed for a constant scalar curvature R. The corresponding solutions for...

p$-Adic mathematical physics is a branch of modern mathematical physics based on the application of $p$-adic mathematical methods in modeling physical and related phenomena. It emerged in 1987 as a result of efforts to find a non-Archimedean approach to the spacetime and string dynamics at the Planck scale, but then was extended to many other areas...

Ultrametric approach to the genetic code and the genome is considered and developed. $p$-Adic degeneracy of the genetic code is pointed out. Ultrametric tree of the codon space is presented. It is shown that codons and amino acids can be treated as $p$-adic ultrametric networks. Ultrametric modification of the Hamming distance is defined and noted...

Summation of the $p$-adic functional series $\sum \varepsilon^n \, n! \, P_k^\varepsilon (n; x)\, x^n ,$ where $P_k^\varepsilon (n; x)$ is a polynomial in $x$ and $n$ with rational coefficients, and $\varepsilon = \pm 1$, is considered. The series is convergent in the domain $|x|_p \leq 1$ for all primes $p$. It is found the general form of polynom...

During hundred years of General Relativity (GR), many significant gravitational phenomena have been predicted and discovered. General Relativity is still the best theory of gravity. Nevertheless, some (quantum) theoretical and (astrophysical and cosmological) phenomenological difficulties of modern gravity have been motivation to search more genera...

We consider a modification of GR with a special type of a non-local f(R). The
structure of the non-local operators is motivated by the string field theory
and p-adic string theory. We pay special account to the stability of the de
Sitter solution in our model and formulate the conditions on the model
parameters to have a stable configuration. Relev...

We consider nonlocal modification of the Einstein theory of gravity in
framework of the pseudo-Riemannian geometry. The nonlocal term has the form
$\mathcal{H}(R) \mathcal{F}(\Box)\mathcal {G}(R)$, where $\mathcal{H}$ and
$\mathcal{G}$ are differentiable functions of the scalar curvature $R,$ and $
\mathcal{F}(\Box)= \displaystyle \sum_{n =0}^{\inf...

Summation of a large class of the functional series, which terms contain
factorials, is considered. We first investigated finite partial sums for
integer arguments. These sums have the same values in real and all p-adic
cases. The corresponding infinite functional series are divergent in the real
case, but they are convergent and have p-adic invari...

Despite many nice properties and numerous achievements, general relativity is
not a complete theory. One of actual approaches towards more complete theory of
gravity is its nonlocal modification. We present here a brief review of
nonlocal gravity with its cosmological solutions. In particular, we pay special
attention to two nonlocal models and the...

Modified gravity model with nonlocal term of the form R
−1
F(□)R is considered. Cosmological solutions for constant scalar curvature are found for all three values of curvature constant k = 0, ±1. Among these solutions there are two nonsingular bounce solutions and one singular cyclic cosmic solution.

We consider summation of some finite and infinite functional p-adic series
with factorials. In particular, we are interested in the infinite series which
are convergent for all primes p, and have the same integer value for an integer
argument. In this paper, we present rather large class of such p-adic
functional series with integer coefficients wh...

Besides great achievements and many nice properties, general relativity as
theory of gravity is not a complete theory. There are many attempts to its
modification. One of promising modern approaches towards more complete
theory of gravity is its nonlocal modification. We present here a brief
review of nonlocal gravity with some its cosmological sol...

We present a brief information on “The Workshop on p-Adic Methods for Modeling of Complex Systems”, which was held in the Center for Interdisciplinary Research (Zentrum für interdisziplinäre Forshung — ZiF), Bielefeld University, Bielefeld, Germany, April 15–19, 2013.

We consider some cosmological aspects of nonlocal modified gravity with
$\Lambda$ term, where nonlocality is of the type $R \mathcal{F}(\Box) R$. Using
ansatz of the form $\Box R = r R +s,$ we find a few a(t) nonsingular bounce
cosmological solutions for all three values of spatial curvature parameter k.
We also discuss this modified gravity model...

We present a brief biographical review of the scientific work and achievements of Vasiliy Sergeevich Vladimirov on the occasion of his death on November 3, 2012.

We consider a new modified gravity model with nonlocal term of the form R -1 ℱ(□)R. This kind of nonlocality is motivated by investigation of applicability of a few unusual ansätze to obtain some exact cosmological solutions. In particular, we find attractive and useful quadratic ansatz □R=qR 2 .

In this short paper I consider relation between measurements, numbers and
p-adic mathematical physics. p-Adic numbers are not result of measurements, but
nevertheless they play significant role in description of some systems and
phenomena. We illustrate their ability for applications referring to some
sectors of p-adic mathematical physics and rela...

Starting from p-adic string theory with tachyons, we introduce a new kind of
non-tachyonic matter which may play an important role in evolution of the
Universe. This matter retains nonlocal and nonlinear p-adic string dynamics,
but does not suffer of negative square mass. In space-time dimensions D = 2 +
4k, what includes D = 6, 10, ..., 26, the ki...

A new approach to the wave function of the universe is suggested. The key idea is to take into account fluctuating number fields and present the wave function in the form of a Euler product. For this purpose we define a p-adic generalization of both classical and quantum gravitational theory. Elements of p-adic differential geometry are described....

We consider some aspects of nonlocal modified gravity, where nonlocality is
of the type $R \mathcal{F}(\Box) R$. In particular, using ansatz of the form
$\Box R = c R^\gamma,$ we find a few $R(t)$ solutions for the spatially flat
FLRW metric. There are singular and nonsingular bounce solutions. For late
cosmic time, scalar curvature R(t) is in low...

The genetic code is connection between 64 codons, which are building blocks
of the genes, and 20 amino acids, which are building blocks of the proteins. In
addition to coding amino acids, a few codons code stop signal, which is at the
end of genes, i.e. it terminates process of protein synthesis. This article is
a review of simple modelling of the...

We present a brief review of the scientific work and achievements of Igor V. Volovich on the occasion of his 65th birthday.

Change of signature by linear coordinate transformations in p-adic space-times is considered. It is shown that there exists arbitrary change of trivial signature in for all n≥1 if p≡1(mod 4). In other cases it is possible to change only even number of the signs of the signature. We suggest new concept of signature with respect to distinct quadratic...

We consider spectral problem for a free relativistic particle in p-adic and adelic quantum mechanics. In particular, we found p-adic and adelic eigenfunctions. Within adelic approach there exist quantum states that exhibit discrete structure of space–time at the Planck scale.

Feynman's path integral in adelic quantum mechanics is considered. The propagator for one-dimensional adelic systems with quadratic Lagrangians is analytically evaluated. Obtained exact general formula has the form which is invariant under interchange of the number fields ℝ and ℚp.

We present a short review of adelic quantum mechanics pointing out its non-Archimedean and noncommutative aspects. In particular,
p-adic path integral and adelic quantum cosmology are considered. Some similarities between p-adic analysis and q-analysis are noted. The p]-adic Moyal product is introduced

We consider nonlocal field theory aspects of some p-adic strings. In particular, Lagrangians of p-adic open scalar strings, for single p as well as for collective primes p, are reviewed. They contain space-time nonlocality through the d'Alembertian 2 in the argument of exponential and the Riemann zeta function.

Feynman’s path integrals in ordinary, p-adic and adelic quantum mechanics are considered. The corresponding probability amplitudes K(x″, t″; x′, t′) for two-dimensional systems with quadratic Lagrangians are evaluated analytically and obtained expressions are generalized
to any finite-dimensional spaces. These general formulas are presented in the...

We consider the construction of Lagrangians that might be suitable for describing the entire p-adic sector of an adelic open
scalar string. These Lagrangians are constructed using the Lagrangian for p-adic strings with an arbitrary prime number p.
They contain space-time nonlocality because of the d’Alembertian in the argument of the Riemann zeta f...

We consider construction of Lagrangians which may be suitable for description of p‐adic sector of an open scalar string. Such Lagrangians have their origin in Lagrangian for a single p‐adic string and they contain the Riemann zeta function with the d’Alembertian in its argument. However, investigation of the field theory with Riemann zeta function...

Living organisms are the most complex, interesting and significant objects regarding all substructures of the universe. Life science is regarded as a science of the 21st century and one can expect great new discoveries in the near futures. This article contains an introductory brief review of genetic information, its coding and translation of genes...

We study the construction of Lagrangians that can be considered the Lagrangians of the p-adic sector of an adelic open scalar
string. Such Lagrangians are closely related to the Lagrangian for a single p-adic string and contain the Riemann zeta function
with the d’Alembertian in its argument. In particular, we present a new Lagrangian obtained by a...

We consider possible adelic structure of the Universe. Theoretically this is mainly motivated by developments in p-adic mathematical physics, and especially in p-adic and adelic string theory. Phenomenological motivation is related to accelerated expansion of the Universe, dark matter and dark energy. The main direction of this research is investig...

Classical and quantum mechanics based on an extended Heisenberg algebra with additional canonical commutation relations for
position and momentum coordinates are considered. In this approach additional noncommutativity is removed from the algebra
by a linear transformation of coordinates and transferred to the Hamiltonian (Lagrangian). This linear...

p-Adic strings are important objects of string theory, as well as of p-adic mathematical physics and nonlocal cosmology. By a concept of adelic string one can unify and simultaneously study various aspects of ordinary and p-adic strings. By this way, one can consider adelic strings as a very useful instrument in the further investigation of modern...

A brief review of some selected topics in p-adic mathematical physics is presented. Comment: 36 pages

Using basic properties of p-adic numbers, we consider a simple new approach to describe main aspects of DNA sequence and the genetic code. In our investigation
central role plays an ultrametric p-adic information space whose basic elements are nucleotides, codons and genes. We show that a 5-adicmodel is appropriate
for DNA sequence. This 5-adicmode...

We consider some nonlocal and nonpolynomial scalar field models originating from p-adic string theory. An infinite number
of space-time derivatives is determined by the operator-valued Riemann zeta function through the d’Alembertian □ in its argument.
The construction of the corresponding Lagrangians L starts with the exact Lagrangian $
\mathcal{L}...

We consider construction of some Lagrangians which contain the Riemann zeta function. The starting point in their construction is p-adic string theory. These Lagrangians describe some nonlocal and nonpolynomial scalar field models, where nonlocality is controlled by the operator valued Riemann zeta function. The main motivation for this research is...

Some nonlocal and nonpolynomial scalar field models originated from p-adic string theory are considered. Infinite number of spacetime derivatives is governed by the Riemann zeta function through d'Alembertian $\Box$ in its argument. Construction of the corresponding Lagrangians begins with the exact Lagrangian for effective field of p-adic tachyon...

We present a very brief review of p-adic, adelic and zeta strings. Details can be found in cited literature.

Linear fractional (M̈obius) transformations: f(x) = a + bx c + dx , ad- bc = 0, are related to many parts of pure mathematics and to some its applications. Parameters a, b, c, d may be some complex or p-adic numbers, or adeles. It is of a particular interest to consider linear fractional recurrences (LFR) xn+1 = a + bxn c + dxn , n= 0, 1, 2... as d...

We consider the noncommutative minisuperspace classical and quantum cosmologies.

Application of adeles in modern mathematical physics is briefly reviewed. In particular, some adelic products are presented.

This paper presents the foundations of p-adic modelling in genomics. Considering nucleotides, codons, DNA and RNA sequences, amino acids and proteins as information
systems, we have formulated the corresponding p-adic formalisms for their investigations. Each of these systems has its characteristic prime number used for construction
of the related...

In the framework of adelic approach we consider real and p-adic properties of dynamical system given by linear fractional map f (x) = (a x + b)/(c x + d), where a, b, c and d are rational numbers. In particular, we investigate behavior of this adelic dynamical system when fixed points are rational. It is shown that any of rational fixed points is p...

Degeneracy of the genetic code is a biological way to minimize effects of the undesirable mutation changes. Degeneration has a natural description on the 5-adic space of 64 codons $\mathcal{C}_5 (64) = \{n_0 + n_1 5 + n_2 5^2 : n_i = 1, 2, 3, 4 \} ,$ where $n_i$ are digits related to nucleotides as follows: C = 1, A = 2, T = U = 3, G = 4. The small...

Using an adelic approach we simultaneously consider real and p-adic aspects of dynamical systems whose states are mapped by linear fractional transformations isomorphic to some subgroups of GL(2, ℚ), SL(2, ℚ) and SL(2, ℤ) groups. In particular, we investigate behaviour of these adelic systems when fixed points are rational. It is shown that any of...

We introduce nonlinear scalar field models for open and open-closed strings with spacetime derivatives encoded in the operator valued Riemann zeta function. The corresponding two Lagrangians are derived in an adelic approach starting from the exact Lagrangians for effective fields of $p$-adic tachyon strings. As a result tachyons are absent in thes...

Adelic quantum mechanics is form invariant under an interchange of real and p-adic number fields as well as rings of p-adic integers. We also show that in adelic quantum mechanics Feynman's path integrals for quadratic actions with rational coefficients are invariant under changes of their entries within nonzero rational numbers.

Classical and quantum mechanics for an extended Heisenberg algebra with canonical commutation relations for position and momentum coordinates are considered. In this approach additional noncommutativity is removed from the algebra by linear transformation of phase space coordinates and transmitted to the Hamiltonian (Lagrangian). This transformatio...

A brief review of p‐adic and adelic cosmology is presented. In particular, p‐adic and adelic aspects of gravity, classical cosmology,
quantum mechanics,
quantum cosmology and the wave function of the universe are considered.
p‐Adic worlds made of p‐adic matters, which are different from real world of ordinary matter, are introduced. Real world a...

The next natural step in the development of the p-adic physics is the consideration of p-adic fermions and supersymmetric theories. Some results in this direction were proposed by V. Vladimirov and I. Volovich in [125]. They considered the superspace over an arbitrary locally compact ultrametric number field and proposed a number of results about a...

In the framework of adelic approach we consider real and p-adic properties of dynamical system given by linear fractional map f(x) = (a x + b)/(cx + d), where a, b, c, and d are rational numbers. In particular, we investigate behavior of this adelic dynamical system when fixed points are rational. It is shown that any of rational fixed points is p-...

In the last years noncommutative quantum mechanics has been investigated
intensively. We consider the influence of magnetic field on decoherence of a
system in the noncommutative quantum plane. Particularly, we point out a model
in which the magnetic field allows {\it in situ} dynamical control of
decoherence as well as, in principle, observation o...

We consider classical and quantum mechanics related to an additional noncommutativity, symmetric in position and momentum coordinates. We show that such mechanical system can be transformed to the corresponding one which allows employment of the usual formalism. In particular, we found explicit connections between quadratic Hamiltonians and Lagrang...

We consider an extension of the Feynman path integral to the quantum mechanics of noncommuting spatial coordinates and formulate the corresponding formalism for noncommutative classical dynamics related to quadratic Lagrangians (Hamiltonians). The basis of our approach is that a quantum mechanical system with a noncommutative configuration space ca...

A brief review of the previous research on the Heisenberg uncertainty relations at the Planck scale is given. In this work, investigation of the uncertainty principle extends to p-adic and adelic quantum mechanics. In particular, p-adic analogs of the Heisenberg algebra and uncertainty relation are introduced. Unlike ordinary quantum theory, adelic...

The summation formula $$ \sum^{n-1}_{i=0}\epsilon^i i! (i^k+u_k) = v_k+\epsilon^{n-1} n! A_{k-1}(n) $$ $(\epsilon=\pm 1; k=1,2,...; u_k, v_k\in \msbm\hbox{Z}; A_{k-1}$ is a polynomial) is derived and its various aspects are considered. In particular, divisibility with respect to $n$ is investigated. Infinitely many equivalents to Kurepa's hypothesi...

Using the Weyl quantization we formulate one-dimensional adelic quantum
mechanics, which unifies and treats ordinary and $p$-adic quantum mechanics on
an equal footing. As an illustration the corresponding harmonic oscillator is
considered. It is a simple, exact and instructive adelic model. Eigenstates are
Schwartz-Bruhat functions. The Mellin tra...

Adelic quantum mechanics is formulated. The corresponding model of the harmonic oscillator is considered. The adelic harmonic oscillator exhibits many interesting features. One of them is a softening of the uncertainty relation.

Some p-adic series with factorials are considered.

We obtained the region of convergence and the summation formula for some modified generalized hypergeometric series (1.2). We also investigated rationality of the sums of the power series (1.3). As a result the series (1.4) cannot be the same rational number in all Z_p.

We consider Feynman's path integral approach to quantum mechanics with a noncommutativity in position and momentum sectors of the phase space. We show that a quantum-mechanical system with this kind of noncommutativity is equivalent to the another one with usual commutative coordinates and momenta. We found connection between quadratic classical Ha...

A brief review of a superanalysis over real and $p$-adic superspaces is presented. Adelic superspace is introduced and an adelic superanalysis, which contains real and $p$-adic superanalysis, is initiated.

p-Adic mathematical physics emerged as a result of efforts to find a non-Archimedean approach to the spacetime and string dynamics at the Planck scale. One of its main achievements is a successful formulation and development of p-adic and adelic quantum mechanics, which have complex-valued wave functions of p-adic and adelic arguments, respectively...

This is a brief review article of various applications of non-Archimedean geometry, p-adic numbers and adeles in modern mathematical physics.

In order to evaluate the Feynman path integral in noncommutative quantum mechanics, we consider properties of a Lagrangian related to a quadratic Hamiltonian with noncommutative spatial coordinates. A quantum-mechanical system with noncommutative coordinates is equivalent to another one with commutative coordinates. We found connection between quad...