1 Introduction and motivation

The search for methods distinguishing integrable systems (with regular dynamics) from non-integrable ones (with chaotic dynamics) has a long history. The majority of known tools have very serious limitations. They either restrict the admissible first integrals to a very narrow class of functions, or they require specific properties of a considered system. An important development took place in the 1980s when Ziglin [1] proposed an elegant method for studying the integrability of Hamiltonian systems in a wide class of meromorphic functions. Ziglin’s theory formulates necessary conditions for the Liouville integrability in terms of the monodromy group of variational equations along a particular solution of a considered system. This method has been effectively applied to various important dynamical systems, see for instance [2,3,4,5,6,7,8] and references therein.

At the end of the previous century, an algebraic version of Ziglin’s theory called the Morales–Ramis theory was created [9,10,11]. It connects integrability in the sense of Liouville with properties of the differential Galois group of variational equations. Its practical advantage over Ziglin’s theory is that it is easier to study the properties of the differential Galois group than the properties of the monodromy group. This is because, unlike the monodromy group, the differential Galois group is a linear algebraic group and has its own Lie algebra. Moreover, the differential Galois group contains the monodromy group, so its analysis gives stronger integrability conditions. The main theorem of the Morales–Ramis theory states the following.

Theorem 1

(Morales, 1999) If a Hamiltonian system is integrable with complex meromorphic first integrals in the Liouville sense in a neighborhood of a particular solution, then the identity component of the differential Galois group of the variational equations along this solution is Abelian.

During the last two decades, the Morales–Ramis theory has been successfully used in proving the non-integrability of many nonlinear dynamical systems. To exemplify this, we mention the non-integrability of the classical three-body problem proved by Boucher and Weil [12] and latter complemented in [13, 14]. Maciejewski and Przybylska showed that the Euler-Poisson equations are only integrable in three known cases: the Euler case, the Lagrange case and the Kovalevskaya case [15], while in [16] Acosta and co-workers analysed the Schrödinger equation with the help of the Morales–Ramis approach. It is worth mentioning that the Morales–Ramis theory has been already successfully applied also to non-Hamiltonian systems [17, 18]. Other examples of applications of the Morales–Ramis theory in recent years can be found in works [19,20,21,22,23,24,25,26,27,28,29,30,31,32,33].

However, the real advantage of this theory was shown in a case of study of natural Hamiltonian systems with homogeneous potentials

$$\begin{aligned} \mathcal {H}=\frac{1}{2}|\varvec{p} |^2+V(\varvec{q}), \end{aligned}$$
(1)

where \(\varvec{q}=(q_1,\ldots ,q_n)\in \mathbb {C}^n\) are generalized coordinates and \(\varvec{p}=(p_1,\ldots ,p_n)\in \mathbb {C}^n\) their corresponding canonical momenta. Potential \(V(\varvec{q})\) is a homogeneous function of a degree \(k\in \mathbb {Z}^*=\mathbb {Z}{\setminus }\{0\}\), which describes a field of acting forces. Integrability analysis of this model was possible thanks to the existence of a particular solution

$$\begin{aligned} \varvec{q}(t)=\varphi (t)\varvec{d},\quad \varvec{p}(t)=\dot{\varphi }(t)\varvec{d},\quad \ddot{\varphi }(t)=-\varphi (t)^{k-1}, \end{aligned}$$
(2)

where \(\varvec{d}\in \mathbb {C}^n\) is a non-zero solution of

$$\begin{aligned} V'(\varvec{d})=\gamma \varvec{d}, \quad \gamma \in \mathbb {C}\setminus \{ 0 \}, \end{aligned}$$
(3)

called the Darboux point. Integrability conditions determined by Morales–Ruiz [11] gave obstructions on eigenvalues \(\lambda _i\) of a Hessian matrix \(V''(\varvec{d})\). For our purpose, let us recall the main result formulated in the following theorem, see [9, 11].

Theorem 2

(Morales, 1999) Assume that the Hamiltonian system defined by Hamiltonian (1) with a homogeneous potential \(V(\varvec{q})\in \mathbb {C}(\varvec{q})\) of degree \(k\in \mathbb {Z}^{*}=\mathbb {Z}{\setminus }\{0\}\) integrable in the Liouville sense with meromorphic first integrals. Then, for each eigenvalue \(\lambda \) of \(\gamma ^{-1}V''(\varvec{d})\), pair \((k,\lambda _i)\) belongs to the following list:

  1. (i)

    \(k=\pm 2\) and \(\lambda _i\in \mathbb {C}\),

  2. (ii)

    \(k\in \mathbb {Z}\) and \(\lambda _i\in \{p+\frac{k}{2}p(p-1), \frac{1}{2}\big [\frac{k-1}{k}+p(p+1)k\big ]\}\),

  3. (iii)

    \(k=3\) and \(\lambda _i\in \{ -\frac{1}{24}+\frac{1}{6}(1+3p)^2, -\frac{1}{2}+\frac{3}{32}(1+4p)^2, -\frac{1}{24}+\frac{3}{50}(1+5p)^2,-\frac{1}{24}+\frac{3}{50}(2+5p)^2\}\),

  4. (iv)

    \(k=4\) and \(\lambda _i\in \{-\frac{1}{8}+\frac{2}{9}(1+3p)^2\}\),

  5. (v)

    \(k=5\) and \(\lambda _i \in \{-\frac{9}{40}+\frac{5}{18}(1+3p)^2, -\frac{9}{40}+\frac{1}{10}(2+5p)^2\}\),

  6. (vi)

    \(k=-3\) and \(\lambda _i \in \{\frac{25}{24}-\frac{1}{6}(1+3p)^2,-\frac{25}{24}-\frac{3}{32}(1+4p)^2,-\frac{25}{24}-\frac{3}{50}(1+5p)^2,\frac{25}{24}-\frac{3}{50}(2+5p)^2\}\),

  7. (vii)

    \(k=-4\) and \(\lambda _i \in \{\frac{9}{8}-\frac{2}{9}(1+3p)^2\}\),

  8. (viii)

    \(k=-5\) and \(\lambda _i \in \{\frac{49}{50}-\frac{5}{18}(1+3p)^2, \frac{49}{50}-\frac{1}{10}(2+5p)^2\}\),

where p is an integer.

It is really amazing result: proving the non-integrability of a classical homogeneous Hamiltonian system is reduced to purely algebraic calculations: search for non-zero solutions \(\varvec{d}\) of the algebraic system (3) and calculations of eigenvalues of \(\gamma ^{-1}V''(\varvec{d})\). Notice that except for cases when \(k=\pm 2\), if the system is integrable, then all eigenvalues \((\lambda _1, \ldots ,\lambda _n)\) are rational numbers. The above restriction to integrability has been already used in proving the non-integrability of various important physical systems. For instance, the Hénon-Heiles system [9, 34, 35]; the Hill lunar problem [36,37,38]; the colinear three-body problem [9]; the Armbruster-Guckenheimer-Kim galactic system [20]; the swinging Atwood machine [39,40,41], to mention just a few.

Having in mind the above-mentioned successes, it is natural to ask whether the approach used by Morales–Ruiz can be transferred to other classes of systems, especially those in the relativistic regime. Such systems are currently in a great activity of the scientific community. They appear in various fields of science and many real-life models have been already constructed. Let us mention, the Kapitza system [42,43,44,45] and Emden-Fowler type equations [46]; the relativistic hydrogen-like atom in a magnetic field [47, 48]; the relativistic two-dimensional harmonic and anharmonic oscillators in a uniform gravitational field [49,50,51]; the relativistic Lienard-type oscillators [52] or the relativistic time-dependent Ermakov–Milne–Pinney systems [53, 54]. These relativistic models find applications in physical systems, see e.g. the papers [55, 56] in which one can find the verified experimental realizations of the harmonic oscillator in the relativistic regime using the Bose-condensed lithium atoms in a two-dimensional optical lattice.

The above motivated us to make a more detailed analysis of relativistic systems in the context of their integrability. In the paper [57] we considered dynamics and the integrability of a relativistic particle moving in a flat space-time in an external potential \(V(\varvec{q})\) in the limit of a weak external field, that is when \(2V(\varvec{q})\ll mc^2\). The considered relativistic Hamiltonian has the form

(4)

where m is the particle rest mass and c is the speed of light. Such systems can be treated as a relativistic generalization of the classical Hamiltonian (1) with a homogeneous potential. Indeed, if \( |\varvec{p} |/m c\) is small, then for Hamiltonian we obtain the following expansion

Using the Morales–Ramis theory and the analysis of variational equations, we formulated very strong necessary integrability conditions for system (4), see Theorem 4 in [57]. Namely, if Hamilton’s equations are integrable, then the eigenvalues of the scaled Hessian matrix \(\gamma ^{-1}V''(\varvec{d})\) at any non-zero solution of the system \(V'(\varvec{d})=\gamma \varvec{d}\) must be integer numbers of appropriate form depending on k. We showed the true strength on the main theorem by its application to several examples of Hamiltonian systems of two degrees of freedom with polynomial homogeneous potentials. As its turns out, the obtained necessary integrability conditions for relativistic flat Hamiltonian (4) are very strong. Their direct applications to the famous Hénon–Heiles model and the anisotropic harmonic oscillator proved the non-integrability of these systems, definitely denying the existence of first integrals. It seems that the only integrable relativistic systems with homogeneous potentials are those depending only on one coordinate, or radial potentials. For more details see [57].

2 A system under consideration and the main result

The effective integrability analysis of the relativistic Hamiltonian in a flat space (4) raised a question about the integrability of mechanical systems in static curved spaces, where the source of space-time curvature is a scalar potential source. Our aim is to give an answer to this question

We consider the dynamics of a relativistic particle in a curved space-time in which a scalar potential \(V(\varvec{q})\) is the source of curvature. According to [58], the relativistic Hamiltonian takes the form

$$\begin{aligned} H=\sqrt{g_{00}}\sqrt{m^2c^4+c^2\varvec{p}^2},\qquad g_{00}=1+\frac{2V(\varvec{q})}{mc^2}, \nonumber \\ \end{aligned}$$
(5)

where m is a particle rest mass and c is the velocity of light. The canonical Hamiltonian equations of motion are as follows

$$\begin{aligned} \begin{aligned} \frac{\textrm{d}}{\textrm{d}\,t}\varvec{q}&= \frac{\partial H}{\partial \varvec{p}}=\sqrt{g_{00}}\frac{c\varvec{p}}{\sqrt{m^2c^2+\varvec{p}^2}}, \\ \frac{\textrm{d}}{\textrm{d}\,t}\varvec{p}&=- \frac{\partial H}{\partial \varvec{q}}=-\frac{V'(\varvec{q})}{\sqrt{g_{00}}}\sqrt{\frac{\varvec{p}^2}{m^2c^2}+1}, \end{aligned} \end{aligned}$$
(6)

where \(V'(\varvec{q})\) denotes the gradient of \(V(\varvec{q})\). These equations with prescribed forms of the metric element \(g_{00}\) appear e.g. in [58,59,60,61,62]. However, it seems that there is a lack of their detailed dynamics and integrability analysis.

Before we formulate our main result, let us make the following general remark. Assuming that \(2V(\varvec{q})/mc^2\), and \(|\varvec{p} |/ mc\), are small of the same order, we can expand Hamiltonian (5) into the power series

$$\begin{aligned} \begin{aligned} H&=mc^2\sqrt{1+\frac{2V(\varvec{q})}{mc^2}}\sqrt{1+\frac{\varvec{p}^2}{m^2c^2}}\\&=mc^2\left( 1 +\frac{V(\varvec{q})}{mc^2}+ \cdots \right) \left( 1 +\frac{\varvec{p}^2}{2m^2c^2}+ \cdots \right) \\&=mc^2\left( 1+\frac{\varvec{p}^2}{2m^2c^2}+\frac{V(\varvec{q})}{mc^2}+\cdots \right) \\&=mc^2+\mathcal {H}+\cdots . \end{aligned}\nonumber \\ \end{aligned}$$
(7)

Thus, Hamiltonian (5), after neglecting the constant term \(mc^2\), in the limit of a weak potential and small momentum takes the standard non-relativistic form (1). Moreover, if a system given by a Hamiltonian H has a first integral F, then F is the first integral of the system given by Hamiltonian \(H^2\). Thus, if a system generated by Hamiltonian H is integrable in the Liouville sense, then the same is true for the system generated by Hamiltonian \(H^2\). This simple observation is important when we apply the Morales–Ramis theorem. Notice that Hamiltonian (5) is not a meromorphic function, so we cannot apply directly Theorem 1 to study the integrability. However, it is an algebraic function, so we can extend the approach given in [63, 64]. But in the case of the system considered in this paper by its integrability, we will understand simply the integrability of Hamilton’s equations generated by \(H^2\).

From the above we can deduce, similarly as in [57], the following fact.

Proposition 1

If the relativistic Hamiltonian system governed by Hamiltonian (5) with a homogeneous potential \(V(\varvec{q})\) is integrable in the Liouville sense, then its non-relativistic counterpart defined by Hamiltonian \(\mathcal {H}\) given in (1) with the same \(V(\varvec{q})\) is also integrable in the Liouville sense.

Proof

By assumption, potential \(V(\varvec{q})\) is homogeneous of degree \(k\ne 0\). We introduce the weight-degree of \((\varvec{q},\varvec{p})\) in the following way. If \(k>0\), we set \(\deg _w(q_i) = 2 \) and \(\deg _w(p_i) = k \); if \(k<0\), we set \(\deg _w(q_i) = -2 \) and \(\deg _w(p_i) = -k \). This weight-homogeneity is consistent with the canonical structure. That is, a non-vanishing Poisson bracket \(\{F,G\}\) with a weight-homogeneous function of weight-degree \(l = \deg _w(F)\) and \(m = \deg _w(F)\) is weight-homogeneous of weight-degree \(\deg _w(\{F,G\})=l + m -s\), where \( s = (k+2){\text {sign}}(k)\).

A meromorphic function of \(f(\varvec{q},\varvec{p})\) has expansion

$$\begin{aligned} f(\varvec{q},\varvec{p}) = f^\circ (\varvec{q},\varvec{p})+\cdots , \end{aligned}$$
(8)

where \( f^\circ (\varvec{q},\varvec{p})\ne 0 \) is a weight-homogeneous function, and dots denote terms that have degrees of homogeneity higher than that of \(f^\circ (\varvec{q},\varvec{p})\).

As we noted, if a system with Hamiltonian H is integrable, then the system with Hamiltonian \(\widetilde{H} = H^2 -m^2c^4\) is also integrable. Notice that

$$\begin{aligned} \begin{aligned} \widetilde{H}&= 2 mc^2 \left( \frac{1}{2m}|\varvec{p} |^2 + V(\varvec{q})\right) + \frac{2}{m} |\varvec{p} |^2 V(\varvec{q})\\&= 2mc^2\mathcal {H}+ \frac{2}{m} |\varvec{p} |^2 V(\varvec{q}). \end{aligned} \end{aligned}$$
(9)

As it is easy to check \( \widetilde{H}^\circ = mc^2\mathcal {H}\). Now, let \(f(\varvec{q},\varvec{p})\) be a meromorphic first integral of the considered system. Then the expansion of the commutator gives

$$\begin{aligned} 0= \{\widetilde{H}, f\} = \{\widetilde{H}^\circ , f^\circ \} +\cdots . \end{aligned}$$
(10)

Hence, \( \{\widetilde{H}^\circ , f^\circ \} =0\), and so \(f^\circ \) is a first integral generated by classical Hamiltonian \(\mathcal {H}\). By the Ziglin Lemma, see [1], we can assume that \(f^\circ \) and \(\mathcal {H}\) are functionally independent. \(\square \)

The above proposition has the following important implications for the relativistic system with non-homogeneous potential.

Proposition 2

Assume the relativistic Hamiltonian system governed by Hamiltonian (5) is integrable in the Liouville sense, and potential \(V(\varvec{q})\) has expansion

$$\begin{aligned} V(\varvec{q}) = V_k(\varvec{q}) + \cdots , \end{aligned}$$
(11)

where \(V_k(\varvec{q})\) is homogeneous of degree k, and dots denote terms of homogeneous degree higher than k. Then the corresponding non-relativistic system defined by Hamiltonian \(\mathcal {H}\) given in (1) with \(V(\varvec{q})=V_k(\varvec{q})\) is also integrable in the Liouville sense.

Proof

We proceed as in the proof of Proposition 1. Inserting expansion (11) into \({\widetilde{H}}\) defined by (9) gives

$$\begin{aligned} \begin{aligned} \widetilde{H}&= 2 mc^2 \left( \frac{|\varvec{p} |^2}{2m} + (V_k(\varvec{q})+\cdots )\right) \\&\quad + \frac{2}{m} |\varvec{p} |^2 (V_k(\varvec{q})+\cdots )= \widetilde{H}^\circ + \cdots , \end{aligned} \end{aligned}$$

where

$$\begin{aligned} \widetilde{H}^\circ = 2mc^2\left( \frac{1}{2m}|\varvec{p} |^2 + V_k(\varvec{q})\right) . \end{aligned}$$
(12)

Then, the statement of the proposition follows. \(\square \)

Remark 1

It seems that there is no such a direct relation between the integrability of the mentioned two relativistic Hamiltonian equations generated by H and . In [58] the following expansion of H

(13)

was presented, where the first condition of weak potential \(2V(\varvec{q})\ll mc^2\), and later the condition of small momentum \(|\varvec{p} |\ll mc\) were applied. In this way the well-known relativistic Hamiltonian given in (4) was obtained. However, because the orders used in the above expansions are not indicated consistently a relation between the integrability of both Hamiltonian systems cannot be deduced.

Our main result is a theorem, which gives necessary integrability conditions for the relativistic curved system governed by Hamiltonian (5). It states as follows.

Theorem 3

(Main theorem) Assume that the Hamiltonian system defined by Hamiltonian (5) with a homogeneous meromorphic potential \(V(\varvec{q})\in \mathbb {C}(\varvec{q})\) of degree \(k\in \mathbb {Z}^{*}=\mathbb {Z}{\setminus }\{0\}\) satisfies the following conditions

  1. 1.

    there exists a non-zero \(\varvec{d}\in \mathbb {C}^n\) such that \(V'(\varvec{d})=\gamma \varvec{d}\), for a non-zero \(\gamma \in \mathbb {C}\), and

  2. 2.

    matrix \(\gamma ^{-1}V''(\varvec{d})\) is diagonalizable with eigenvalues \(\lambda _1, \ldots , \lambda _{n-1},\lambda _n=k-1\);

  3. 3.

    the system is integrable in the Liouville sense with complex meromorphic first integrals.

Then, at least one of the possibilities occurs:

  1. (i)

    \(k=\pm 2\), or

  2. (ii)

    \(k\in \mathbb {Z}^{*}\) and \(\lambda _i\in \{0,1\}\), for \(i=1,\ldots ,n-1\).

The above theorem tells us that if \(k = \pm 2\), then the Morales–Ramis theory does not give any obstruction for the integrability of the considered systems. It is somehow, the same result as the Morales–Ruiz obtained for the non-relativistic flat system (1). However, the true strength of our result is manifested in the second case of Theorem 3. Namely, checking whether a given potential V of degree \(k\in \mathbb {Z}{\setminus }\{0,\pm 2\}\) is integrable in the relativistic curved regime it comes down to check whether all eigenvalues of the rescaled Hessian \(\gamma ^{-1}V''(\varvec{d})\) at a Darboux point, belong to two-element set \(\{0,1\}\). Let us notice that these two values appear in item (ii) of Theorem 2 for the non-relativistic flat system (1) for \(p=0\) or \(p=1\), respectively.

Remark 2

If potential \(V(\varvec{q})\) in Hamiltonian (5) is not homogeneous, then still we can try to apply the Morales–Ramis theorem 1 provided that we know a particular solution. In general, a good method for finding a particular solution does not exist. However, in some cases we can find a straight-line solution analogously as for homogeneous potential. For example, assume that

$$\begin{aligned} V(\varvec{q}) = V_\textrm{r}(\varvec{q})+V_k(\varvec{q}), \end{aligned}$$
(14)

where \(V_k(\varvec{q})\) is a homogeneous potential of degree k, and \( V_\textrm{r} (\varvec{q})=W(r)\), with \(r^2 = \varvec{q}\cdot \varvec{q}\), is a radial potential. Now, if \( \varvec{d}\) is a Darboux point of \( V_k(\varvec{q})\), then it is also a Darboux point of \(V(\varvec{q})\).

3 Numerical analysis and applications of Theorem 3

In this section, we present the advantages of Theorem 3. We will show that the obtained integrability obstructions are effective and easy for applications. Indeed, to prove the non-integrability of the relativistic curved systems (5) and as well as to specify values of the parameters for which it is suspected to be integrable, we limit ourselves only to basic algebraic computations.

Moreover, to get a quick insight into the dynamics of classical systems and their corresponding relativistic curved counterparts, we perform a numerical analysis with the help of Poincaré cross-sections method. The purpose of it is to show how the addition of the relativistic correction to the integrable classical systems affects their dynamics and destroys integrability causing their chaotic behavior. That’s why we will compute Poincaré sections of classical and relativistic-curved systems for the same values of the parameters appearing in the potentials. We choose values of the parameters for which the classical systems are integrable. We show that the integrable cases of the Hénon–Heiles model, the Armbruster–Guckenheimer–Kim galactic system as well as the swinging Atwood machine are not integrable and they became chaotic in the relativistic-curved regime.

3.1 Example 1. Hénon–Heiles system

Consider the famous Hénon–Heiles system, which is defined by the following Hamiltonian

$$\begin{aligned} \mathcal {H}=\frac{1}{2}\left( p_1^2+p_2^2\right) +V,\qquad V=V_2+V_3, \end{aligned}$$
(15)

where

$$\begin{aligned} V_2=\frac{1}{2}\left( q_1^2+q_2^2\right) ,\quad V_3=\alpha q_1^2q_2+\frac{1}{3}\beta q_2^3. \end{aligned}$$
(16)

Here \(\alpha \) and \(\beta \) are real parameters. This model was investigated in various branches of physics and astronomy e.g. in celestial mechanics [34], in statistical and quantum mechanics [65] and recently in Hamiltonian neural networks [66] and in relativistic chaotic scattering [67, 68].

There are only three known integrable cases of the classical Hénon-Heiles system. Namely, when \(\alpha =0\), or \(\beta =\alpha \), or \(\beta =6\alpha \). For some time, it was unclear whether the Hénon-Heiles system could have other integrable cases. It seems that Ito [3] was the first who rigorously studied the integrability of (15) using Ziglin’s monodromy group integrability obstructions. Morales–Ruiz confirmed his result using the differential Galois approach, see [9, Secs. 5.1.3 and 6.4]. They showed that except for the above cases, and the case \(\beta =2\alpha \) the Hamiltonian system (15) is non-integrable with meromorphic first integrals. Although for \(\beta =2\alpha \) the numerical analysis can suggest the integrability of the system, the additional first integral has not been found. Only application of the integrability obstructions obtained from the analysis of the higher-order variational equations allowed Morales–Ruiz to prove the non-integrability in this case, see [9, Sec. 8.3.2] or [69].

Fig. 1
figure 1

Poincaré sections of the classical cases of Hénon–Heiles model (left) and their relativistic curved counterparts (right). The cross-section plane was specified as \(q_1=0\), with \( p_1>0\). Each color corresponds to a different initial condition. (Color figure online)

Full size image

Let us investigate the relativistic curved counterpart of the Hamiltonian (15). Hence, we study the system

$$\begin{aligned}{} & {} H=\sqrt{1+2V}\sqrt{1+p_1^2+p_2^2}, \nonumber \\{} & {} V=V_2+V_3, \end{aligned}$$
(17)

with the potentials (16). To get a quick insight into the dynamics of the Hamiltonian system (17), we make several Poincaré cross-sections. The main idea of the Poincaré cross-sections method is very simple. We select an energy level

Hence, is a three-dimensional manifold. Next, we choose a cross-section surface in . We require that in a certain domain on this surface, phase orbits intersect it transversally, return, and intersect it again. For the homogeneous potentials this property is not fulfilled, and this is why we consider the system with non-homogeneous potential \(V=V_2+V_3\). Each cross-section is depicted in a cross-section plane by a single point. As a result, we obtain a pattern of points on the plane, which is easy to visualize and interpret.

Fig. 2
figure 2

Poincaré sections of the classical Hénon-Heiles model (left) and its relativistic curved counterparts (right) constructed for \(\beta =2\alpha =-1\) at the levels \(E=E_\text {min}+0.183\). The cross-section plane was specified as \(q_1=0\), with \( p_1>0\). Each color corresponds to a different initial condition. (Color figure online)

Full size image

Figure 1 presents three pairs of Poincaré sections of the classical and the corresponding relativistic Hénon-Heiles model in curved space-time constructed for values of parameters corresponding to integrable classical cases

$$\begin{aligned} \begin{aligned}&(\textrm{i})\quad \alpha =0,\quad \beta =-\frac{1}{2},\quad (\textrm{ii}) \quad \alpha =\beta =-\frac{1}{2} \\&(\textrm{iii})\quad \alpha =-\frac{1}{12},\quad \beta =-\frac{1}{2}, \end{aligned} \end{aligned}$$
(18)

at constant energy levels. For convenience and to get a sufficiently large impact coming from the relativistic correction, we put \(m=1\) and \(c=1\). The surface of section is \(q_1 = 0\), and the direction was chosen by \(p_1 > 0\). Coordinates of the cross-section plane are specified as \((q_2,p_2)\). Here \(E_\textrm{min}\) denotes the energy minimum of the respective cases. For the non-relativistic system, we have \(E_\textrm{min}=0\), while for the relativistic case \(E_\textrm{min}=1\). As we can notice, the general shapes of the sections are similar, whereas the dynamics presented within them is completely different. In the non-relativistic cases, we obtain shapely elegant integrable curves with mostly quasi-periodic solutions. In the relativistic regime, however, we observe large regions responsible for the chaotic motions of the system with random-looking points corresponding to the fact that trajectories can freely wander over large regions of the phase space. The trajectories, unlike those for integrable systems, are no longer confined to the surfaces of a set of nested invariant tori, but they begin to move outside these tori. We say that the invariant tori are destroyed.

The numerical analysis suggests that the integrable cases of the classical Hénon–Heiles model are not integrable in the relativistic regime in curved space-time, compare left and right plots in Fig. 1. In the next paragraph we prove that indeed potential \(V=V_3\) is not integrable in the relativistic curved space-time framework.

Since our main Theorem 3 concerns only homogeneous potentials, we compare the integrability of the classical and the relativistic Hénon–Heiles systems with potential \(V=V_3\). It has two distinct non-trivial Darboux points

$$\begin{aligned} \begin{aligned} \varvec{d}_1&=\left( 0,\frac{1}{\beta }\right) , \quad \text {for}\quad \beta \ne 0, \\ \varvec{d}_2&=\left( \frac{1}{2}\sqrt{\frac{2\alpha -\beta }{\alpha ^3}}, \frac{1}{2\alpha }\right) ,\quad \text {for}\quad \alpha \ne 0. \end{aligned} \end{aligned}$$
(19)

The respective non-trivial eigenvalues of \(V_3''\) at points (19) are as follows

$$\begin{aligned} \begin{aligned} \varvec{d}_1:\quad \lambda =\frac{2\alpha }{\beta },\quad \text {and}\quad \varvec{d}_2: \quad \lambda =\frac{\beta }{\alpha }-1. \end{aligned} \end{aligned}$$
(20)

It is easy to verify that \(\lambda \in \{0,1\}\) only if \(\alpha =0\) or \(\beta =2\alpha \). Therefore, for these values of the parameters the necessary integrability conditions formulated in our main Theorem 3 are satisfied. However, Figs. 1a and 2 show that the relativistic non-homogeneous Hénon–Heiles model in curved space-time for \(\alpha =0\) or \(\beta =2\alpha \) is also not integrable. At the cross-section planes one can observe large regions responsible for the chaotic motion of the systems, ensuring their non-integrability. To prove analytically that the system in these cases is not integrable one can start with Remark 2 and then apply the Morales–Ramis Theorem 1.

3.2 Example 2. Armbruster–Guckenheimer–Kim galactic system

As the second system, we study Armbruster-Guckenheimer-Kim (AGK) galactic model [70]. It is defined by the following Hamiltonian

$$\begin{aligned} \mathcal {H}=\frac{1}{2}\left( p_1^2+p_2^2\right) +V,\qquad V=V_2+V_4, \end{aligned}$$
(21)

where

$$\begin{aligned}{} & {} V_2=\frac{1}{2}\left( q_1^2+q_2^2\right) ,\nonumber \\{} & {} V_4=-\frac{\alpha }{4}\left( q_1^2+q_2^2\right) ^2-\frac{\beta }{2}q_1^2q_2^2, \end{aligned}$$
(22)

and \(\alpha ,\beta \) are real parameters. Hamiltonian (21) appears in astrophysics, and it is related to a 2D-perturbed harmonic oscillator, which characterizes the local motion in the central area of a galaxy. This quartic potential is obtained by expanding the global galactic potentials in a Taylor series near a stable equilibrium point [70]. Many studies concerning the integrability and non-integrability of such systems have been done, see for instance [20, 71,72,73,74]. All these investigations show that the model is completely integrable in three cases: \(\beta =0\), or \(\beta =-\alpha \) or \(\beta =2\alpha \), with separable equations of motion.

Let us study numerically the dynamics and integrability of the relativistic AGK model in curved space-time governed by the Hamiltonian

$$\begin{aligned} H=\sqrt{1+2V}\sqrt{1+p_1^2+p_2^2},\qquad V=V_2+V_4,\nonumber \\ \end{aligned}$$
(23)

where \(V_2\) and \( V_4\) are given in (22). To get a quick insight into the dynamics of (23), we made Poincaré cross-sections. We choose values of the parameters for which the classical model (21) is integrable. Namely,

$$\begin{aligned} \begin{aligned}&(\textrm{i})\quad \alpha =\frac{1}{10},\quad \beta =0,\\&(\textrm{ii})\quad \alpha =\frac{1}{10},\quad \beta =-\frac{1}{10},\\&(\textrm{iii})\quad \alpha =\frac{1}{10},\quad \beta =\frac{1}{5}, \end{aligned} \end{aligned}$$
(24)

at constant energy levels.

Fig. 3
figure 3

Poincaré sections of the classical cases of AGK model (left) and their relativistic curved counterparts (right). The cross-section plane was specified as \(q_1=0\), with \( p_1>0\). Each color corresponds to a different initial condition. (Color figure online)

Full size image

Poincaré sections depicted in Fig. 3 illustrate the relation between the integrable cases of the classical AGK system and their relativistic counterparts. As we can notice, for cases \((\textrm{ii})\!-\!(\textrm{iii})\) given in (24) the relativistic model in curved space-time seems to be not integrable due to the appearance of regions at the Poincaré sections responsible for chaotic motion. However, for case \((\textrm{i})\) the Poincaré section shows shapely elegant quasi-periodic curves without any indications of the chaotic nature of the system. As it will be clear in a moment, the relativistic AGK galactic model in curved space-time for \(\alpha \ne 0\) and \(\beta =0\) is really integrable.

Since, as before, we have formulated the integrability obstructions only for homogeneous potentials, we will consider potential \(V_4\). It has only one Darboux point \(\varvec{d}=(0, 1/\sqrt{\alpha })\) with a non-trivial eigenvalue of the Hessian \(V_4''(\varvec{d})\) equals to \(\lambda =1+\beta /\alpha \). Hence, according to our main Theorem 3, the only possible cases for which the relativistic curved AGK system governed by Hamiltonian (23) satisfies the necessary integrability conditions are \(\beta =0\) or \(\beta =-\alpha \). For \(\beta =0\) the potential \(V=V_2+V_4\) is radial, and hence the relativistic curved Hamiltonian (23) is integrable with the angular momentum first integral \(I=q_1p_2-q_2p_1\). For \(\beta =-\alpha \), the Hamiltonian is not integrable. The Poincaré section visible in Fig. 3b shows the chaotic behaviour of the system.

3.3 Example 3. Swinging Atwood’s machine

Swinging Atwood’s machine (SAM) is a simple mechanical model, which consists of two masses M and m linked by a non-extensible string of a constant length. The mass m is allowed to swing in a plane, while the second mass M plays the role of a counterweight, and it is constrained to move only along the vertical direction. Usually SAM is defined in polar coordinates \((r,\varphi )\) and its motion is completely described by the following Hamiltonian

$$\begin{aligned} \mathcal {H}=\frac{1}{2}\left( \frac{p_r^2}{1+\mu }+\frac{p_\varphi ^2}{r^2}\right) +r(\mu -\cos \varphi ), \end{aligned}$$
(25)

where \(\mu =M/m\). Hamiltonian (25) was intensively studied by Tufillaro and co-workers [39, 40], and certain generalizations of SAM were recently studied in [75, 76]. Tufillaro showed that Hamiltonian (25) for \(\mu =3\) is integrable with a quadratic in momenta first integral. The application of the Morales–Ramis criterion for the integrability of classical homogeneous Hamiltonian systems formulated in Theorem 2 provides for classical SAM an infinite family of possible values of the parameter \(\mu = p(p+1)/(p(p+1)-4)\), where \(p\in \mathbb {Z}\), for which the system is suspected to be integrable [41]. This result was later completed in [77] by Martínez and Simó.

Fig. 4
figure 4

Poincaré sections of the classical (left) and the relativistic curved (right) model of the swinging Atwood’s machine. The plots were constructed for \(\mu =3\) at the energy level \(E=2\). The cross-section plane was specified as \(q_1=0\), with \( p_1>0\). Each color corresponds to a different initial condition. (Color figure online)

Full size image

To study the integrability of the relativistic curved SAM model, first we need to rewrite non-relativistic Hamiltonian (25) in the natural form (1) with the standard kinetic energy and potential. We do this by the following canonical transformation

$$\begin{aligned} \begin{aligned}&q_1=r \sin \left( \frac{\varphi }{\widetilde{\mu }}\right) , \quad p_1=\sin \left( \frac{\varphi }{\widetilde{\mu }}\right) p_r+\frac{\widetilde{\mu }}{r}\cos \left( \frac{\varphi }{\widetilde{\mu }}\right) p_\varphi ,\\&q_2=r \cos \left( \frac{\varphi }{\widetilde{\mu }}\right) , \quad p_2=\cos \left( \frac{\varphi }{\widetilde{\mu }}\right) p_r-\frac{\widetilde{\mu }}{r}\sin \left( \frac{\varphi }{\widetilde{\mu }}\right) p_\varphi , \end{aligned} \end{aligned}$$

where \(\widetilde{\mu }=\sqrt{1+\mu }\). Then, the dynamics of (25) is equivalent to the dynamics of the Hamiltonian

$$\begin{aligned} \mathcal {H}=\frac{1}{2}\left( p^{2}_{1}+p^{2}_{2}\right) +(1+\mu )V(q_1,q_2), \end{aligned}$$
(26)

where the potential \(V(q_1,q_2)\) under consideration is given by

$$\begin{aligned} V(q_1,q_2)=\sqrt{q_1^2+q_2^2}\left\{ \mu -\cos \hspace{-0.08cm}\left[ \widetilde{\mu }\arctan \left( \frac{q_1}{q_2}\right) \right] \right\} . \end{aligned}$$
(27)

Figure 4 shows the Poincaré cross sections of the integrable classical and the corresponding relativistic SAM model constructed for the potential (27). It is evident that the relativistic counterpart is not integrable. Indeed, large regions responsible for the chaotic motion of the system preclude its integrability. We prove this fact using Theorem 3.

Because potential (27) is a homogeneous function of degree \(k=1\), the integrability of relativistic SAM implies that the second item of Theorem 3 must hold. Hence, if the potential (27) is integrable, then at any Darboux point \(\varvec{d}\) a non-trivial eigenvalue \(\lambda \) of \(V''(\varvec{d})\) must belong to two-element set \(\{0,1\}\). Potential (27) has the Darboux point \(\varvec{d}=(0,\mu -1)\) with \(\lambda =2\mu /(\mu -1)\). Therefore, the only possible values of \(\mu \in \mathbb {R}\) for which \(\lambda \in \{0,1\}\) are \(\mu =0\) and \(\mu =-1\).

The above short analysis ensures the non-integrability of the swinging Atwood machine in the relativistic curved space-time regime for all \(\mu \in \mathbb {R}_{>0}\). Moreover, it is worth remarking that for the remaining values of \(\mu \) the system is really integrable. Namely, for \(\mu =0\) the potential reduces to \(V=q_2\) with the obvious cyclic variable \(q_1\) and the corresponding first integral \(I=p_1\). Whereas, for \(\mu =-1\) the potential is radial \(V=\sqrt{q_1^2+q_2^2}\) and the corresponding first integral is the angular momentum \(I=q_1p_2-q_2p_1\).

4 Integrable potentials and their first integrals

4.1 Quadratic integrable potentials of degree \(k=\pm 2\)

According to Theorem 3 for potentials of degree \(k=\pm 2\) there are no integrability obstructions. We show that these conditions are sometimes sufficient. To find an integrable Hamiltonian system with n degrees of freedom, we must give sets of n commuting first integrals including Hamiltonian. We obtain them using the direct method. For its applications see for instance [27].

Obviously, n-dimensional anisotropic harmonic oscillators

$$\begin{aligned} V=\sum _{i=1}^n A_iq_i^2, \end{aligned}$$
(28)

are integrable in the classical case. Really, the corresponding non-relativistic Hamiltonian (1) with potential (28) is separable in Cartesian coordinates and as functionally independent first integrals one can choose

$$\begin{aligned} H_i=\frac{1}{2}p_i^2+A_iq_i^2,\quad i=1,\ldots , n, \end{aligned}$$

which sum to the non-relativistic Hamiltonian and commute each other.

The integrability of the relativistic system (5) with anisotropic harmonic potential (28) in curved space-time is not obvious at all. Recall that in the limit of a weak external field the relativistic Hamiltonian system (4) in a flat space-time with harmonic oscillator potential is integrable if and only if it is isotropic harmonic i.e. with all equal frequencies, see [57]. Nevertheless, for the the relativistic system (5) in curved space-time we found \(n+1\) additional first integrals

$$\begin{aligned} \begin{aligned} F_0&=\frac{1}{2}\sum _{j=1}^n\frac{p_j^2}{A_j}+\left( \sum _{j=1}^np_jq_j\right) ^2+\sum _{j=1}^nq_j^2,\\ F_i&=\frac{1}{2}\prod _{{\mathop {j\ne i}\limits ^{j=1}}}^n (A_i-A_j)p_i^2\\&\quad +A_i\prod _{{\mathop {j\ne i}\limits ^{j=1}}}^n(A_i-A_j)q_i^2\left( 1+\sum _{i=1}^n p_i^2\right) \\&\quad +A_i\sum _{{\mathop {j\ne i}\limits ^{j=1}}}^n A_j\prod _{{\mathop {l\ne i,l\ne j}\limits ^{l=1}}}^n(A_i-A_l)(p_iq_j-p_jq_i)^2, \end{aligned} \end{aligned}$$

for \( i=1,\ldots ,n\). Functions \(F_1,\ldots ,F_n\) are functionally independent. Moreover, we have the following two relations between them and H

$$\begin{aligned}{} & {} \sum _{l=1}^n\prod _{{\mathop {i,j\ne l,i<j}\limits ^{i,j=1}}}^n (A_i-A_j)(-1)^{l-1}F_l\\{} & {} =\frac{1}{2}\prod _{{\mathop {i<j}\limits ^{i,j=1}}}^n(A_i-A_j)(H^2-1), \end{aligned}$$

and \(F_0\)

$$\begin{aligned}{} & {} \sum _{l=1}^n \prod _{{\mathop {i,j\ne l,i<j}\limits ^{i,j=1}}}^n A_j(A_i-A_j)(-1)^{l-1}F_l\\{} & {} =\frac{1}{2}\prod _{{\mathop {i<j}\limits ^{i,j=1}}}^nA_j(A_i-A_j)F_0, \end{aligned}$$

and all functions \(F_0,F_1,\ldots ,F_n,H\) pair-wise commute.

Relativistic n-dimensional system (5) with the inverse of anisotropic harmonic oscillator potential

$$\begin{aligned} V=\left( \sum _{i=1}^n A_iq_i^2\right) ^{-1}, \end{aligned}$$
(29)

has the following first integrals

$$\begin{aligned} \begin{aligned} F_i&=2\prod _{{\mathop {j\ne i}\limits ^{j=1}}}^n (A_i-A_j)p_i^2\\&\quad +A_i\sum _{{\mathop {j\ne i}\limits ^{j=1}}}^n A_j\prod _{{\mathop {l\ne i,l\ne j}\limits ^{l=1}}}^n(A_i-A_l)(p_iq_j -p_jq_i)^2\\&\quad -2A_i\prod _{{\mathop {j\ne i}\limits ^{j=1}}}^n(A_i-A_j)\frac{q_i^2\left( 1+\sum _{l=1}^np_l^2\right) }{\sum _{l=1}^n A_lq_l^2}, \end{aligned} \end{aligned}$$

for \(i=1,\ldots ,n\). These functions commute each other and satisfy the following relation

$$\begin{aligned} \sum _{l=1}^n\prod _{{\mathop {i,j\ne l,i<j}\limits ^{i,j=1}}}^n(A_i-A_j)(-1)^lF_l=2\prod _{{\mathop {i<j}\limits ^{i,j=1}}}^n(A_i-A_j). \end{aligned}$$

In the case \(n=2\), where there is no i, j different from l on the left-hand side, we take \(A_i-A_j=1\). Moreover, one can check that \(n-1\) functions from this set together with Hamiltonian give the set of functionally independent commuting first integrals. Let us notice that non-relativistic Hamiltonian (1) with potential (29) is also integrable with additional first integrals

$$\begin{aligned} \begin{aligned} F_i&= \sum _{{\mathop {j\ne i}\limits ^{j=1}}}^n A_j\prod _{{\mathop {l\ne i,l\ne j}\limits ^{l=1}}}^n(A_l-A_i)(p_jq_i-p_iq_j)^2\\&\quad +2\prod _{{\mathop {j\ne i}\limits ^{j=1}}}^n\frac{(A_j-A_i)q_i^2}{\sum _{l=1}^n A_lq_l^2}, \end{aligned} \end{aligned}$$

for \(i=1,\ldots ,n\), which all commute each other. There is a linear relation between them

$$\begin{aligned}{} & {} \sum _{l=1}^nA_l\prod _{{\mathop {i,j\ne l,i<j}\limits ^{i,j=1}}}^n(A_i-A_j)(-1)^{l-1}F_l\\{} & {} \quad =2(-1)^{n-1}\prod _{{\mathop {i<j}\limits ^{i,j=1}}}^n(A_i-A_j). \end{aligned}$$

Thus, among \(F_1,\ldots ,F_n\) one can choose the set of \(n-1\) functionally independent functions that together with Hamiltonian guarantee integrability in the Liouville sense.

4.2 Radial potentials

An example of potentials with all non-trivial eigenvalues of Hessian equal to 1 are radial potentials

$$\begin{aligned} V=r^k,\quad r=\sqrt{\sum _{i=1}^nq_i^2}. \end{aligned}$$
(30)

More precisely, these potentials have infinitely many Darboux points, but at each of them, all non-trivial eigenvalues are the same \(\lambda =1\). The potential (30) and relativistic Hamiltonian in curved space-time (5) are invariant under the group of rotations in n-dimensional space. This implies that \(\left( {\begin{array}{c}n\\ 2\end{array}}\right) =\tfrac{n(n-1)}{2}\) components of the angular momentum tensor

$$\begin{aligned} F_{ij}=q_ip_j-q_jp_i, \end{aligned}$$

where \(i,j=1,\ldots ,n\) and \(i<j\), are first integrals of the corresponding Hamilton equations (6). The same functions guarantee integrability of the corresponding non-relativistic Hamiltonian (1) with radial potentials (30). Functions \(F_{ij}\) are not all independent, among them only \(2n-3\) are independent, see e.g. Section 2.5 in [78]. They are generators of Lie algebra \(\textrm{so}(n,\mathbb {R})\). Using them together with Hamiltonian we can find the set of n commuting first integrals: Hamiltonian, Casimir functions and some functions \(F_{ij}\), for details see e.g. [79]. For \(n=3\) Casimir of Lie algebra \(\textrm{so}(3,\mathbb {R})\) is \(C=F_{12}^2+F_{23}^2+F_{31}^2\) and we can choose as commuting set H, C and \(F_{12}=q_1p_2-q_2p_1\).

5 Outline of the proof of Theorem 3

We assume that potential is a homogeneous function of integer degree k, that is \(V(\chi \varvec{q})=\chi ^kV(\varvec{q})\) for \(\chi >0\). Then, we look for straight-line particular solutions directed along a constant non-zero vector \(\varvec{d}\in \mathbb {C}^n\), that is \(\varvec{q}(t) = \varphi (t)\varvec{d}\), where \(\varphi =\varphi (t)\) is a scalar function that will be specified later. The velocity \(\varvec{v}=\dot{\varvec{q}}=\dot{\varphi }(t){\varvec{d}}\) and the corresponding momentum can be calculated from the first vector Hamilton equation (6). Hence, the particular solution is given by

$$\begin{aligned} \varvec{q}(t) = \varphi (t) \varvec{d}, \quad \varvec{p}(t)= \frac{m{\dot{\varphi }(t)} \varvec{d}}{\sqrt{1+2\frac{\varphi (t)^k V(\varvec{d})}{mc^2}-\frac{{\dot{\varphi }(t)^2}}{c^2}d^2}},\nonumber \\ \end{aligned}$$
(31)

where \(d^2=\varvec{d}\cdot \varvec{d}\). Substituting these expressions into the second vector Hamilton equation (6) gives the second order differential equation for the scalar function \(\varphi (t)\)

$$\begin{aligned} \ddot{\varphi }(t) =-\frac{1}{m}\left[ \gamma -\frac{\left( kV(\varvec{d})+\gamma d^2\right) \dot{\varphi }(t)^2}{c^2\left( 1+2\frac{\varphi (t)^k V(\varvec{d})}{mc^2}\right) }\right] \varphi (t)^{k-1}, \nonumber \\ \end{aligned}$$
(32)

provided

$$\begin{aligned} V'(\varvec{d}) = \gamma \varvec{d}, \quad \gamma \in \mathbb {C}\setminus \{ 0 \}. \end{aligned}$$
(33)

Non-zero solutions of this equation give the so-called Darboux points, compare with Eq. (3). As \(V(\varvec{q})\) is a homogeneous function of degree k the Euler’s homogeneous function theorem gives \(V'(\varvec{q})\cdot \varvec{q}=kV(\varvec{q})\), so

$$\begin{aligned} V'(\varvec{d})\cdot \varvec{d}=\gamma \varvec{d}^2=kV(\varvec{d}). \end{aligned}$$

Thus, we can rewrite (32) in the form

$$\begin{aligned} \begin{aligned}&\ddot{\varphi }(t) = -\frac{\gamma }{m}\left[ 1-\frac{2 d^2\dot{\varphi }(t)^2}{c^2S(t)}\right] \varphi (t)^{k-1},\\&S(t)=1+2\frac{\gamma d^2\varphi (t)^k}{kmc^2}. \end{aligned} \end{aligned}$$
(34)

The above equation has a first integral

$$\begin{aligned} \begin{aligned} h&=\frac{S(t)^2}{T(t)^2},\,\, T(t)=\sqrt{1+2\frac{\gamma d^2\varphi (t)^k }{kmc^2}-\frac{\dot{\varphi }(t)^2}{c^2}d^2}. \end{aligned} \nonumber \\ \end{aligned}$$
(35)

The variational equations along solution (31) have the form

$$\begin{aligned} \begin{aligned} \dot{\varvec{x}}&= \frac{\gamma \varphi (t)^{k-1}\dot{\varphi }(t)\varvec{d}\cdot \varvec{x})\varvec{d}}{mc^2S(t)}\\&\quad +\frac{T(t)}{m} \left[ \varvec{y}-\frac{\dot{\varphi }(t)^2(\varvec{d}\cdot \varvec{y})\varvec{d}}{c^2S(t)}\right] , \\ \dot{\varvec{y}}&= -\frac{\varphi (t)^{k-2}}{T(t)}\Big [V''(\varvec{d})\varvec{x}-\frac{\gamma ^2\varphi (t)^k(\varvec{d}\cdot \varvec{x})\varvec{d}}{mc^2S(t)} \Big ]\\&\quad -\frac{\gamma \varphi (t)^{k-1}\dot{\varphi }(t)(\varvec{d}\cdot \varvec{y})\varvec{d}}{mc^2S(t)}, \end{aligned} \end{aligned}$$
(36)

where we used the fact that \(V''(\varvec{q})\) is a homogeneous function of degree \(k-2\), and thus \(V''\left( \varphi (t) \varvec{d}\right) =\varphi ^{k-2}(t)V''(\varvec{d})\). If we will be able to show that the identity component of the differential Galois group of this system of variational equations or its closed subsystem is not Abelian, then, by Morales–Ramis theorem 1, we prove that the corresponding Hamiltonian system is not integrable. Our first step is a simplification of the above variational equations.

The Hessian \(V''(\varvec{d})\) of the potential V calculated at a Darboux point \(\varvec{d}\) is a symmetric matrix. Hence, in a generic case, there exists a complex orthogonal \(n\times n\) matrix A such that the canonical transformation

$$\begin{aligned} \varvec{x}= A \varvec{\eta }, \quad \varvec{y}= A \varvec{\xi }, \end{aligned}$$
(37)

transforms \(V''(\varvec{d})\) to its diagonal form with eigenvalues \((\widehat{\lambda }_1,\ldots , \widehat{\lambda }_n)\). A Darboux point \(\varvec{d}\) is an eigenvector of \(V''(\varvec{d})\) corresponding to the eigenvalue \(\widehat{\lambda }_n=\gamma (k-1)\). We can assume that \(\varvec{D}=A^T\varvec{d}=[0,\ldots ,0,d]^T\). Thus, the change of variables (37) transforms equations (36) into a direct product of two-dimensional systems

$$\begin{aligned} \begin{aligned} \dot{\eta }_i&= \frac{T(t)}{m}\,\xi _i, \quad \dot{\xi }_i = -\frac{\widehat{\lambda }_i}{T(t)}\varphi (t)^{k-2}\eta _i,\\ \dot{\eta }_n&= \frac{\gamma \varphi (t)^{k-1}\dot{\varphi }(t) d^2}{mc^2S(t)}\eta _n+\frac{1}{m}\frac{T(t)^3}{S(t)}\xi _n, \\ \dot{\xi }_n&=-\frac{\varphi (t)^{k-2}}{T(t)}\left[ \widehat{\lambda }_n-\frac{\gamma ^2\varphi (t)^kd^2}{mc^2S(t)} \right] \eta _n\\&\quad -\frac{\gamma \varphi (t)^{k-1}\dot{\varphi }(t) d^2}{mc^2S(t)}\xi _n, \end{aligned} \end{aligned}$$
(38)

where \(i=1,\ldots , n-1,\) or to the direct product of the scalar second-order equations

$$\begin{aligned} \begin{aligned}&\ddot{\eta }_i +a \dot{\eta }_i+b \eta _i=0,\qquad i = 1, \ldots , n-1,\\&a=-\frac{2\gamma \dot{\varphi }(t) \varphi (t)^{k-1}d^2}{mc^2S(t)},\quad b=\frac{\widehat{\lambda }_i}{m}\varphi (t)^{k-2}, \end{aligned} \end{aligned}$$
(39)

and for \(\eta _n\)

$$\begin{aligned} \begin{aligned}&\ddot{\eta }_n +a \dot{\eta }_n+b \eta _n=0,\\&a=-\frac{4 \gamma d^2 \varphi (t)^{k-1} \dot{\varphi }(t)}{mc^2 S(t)},\\&b=\frac{2 \gamma d^2 \varphi (t)^{k-2}\dot{\varphi }(t)^2}{ m^2c^4 S(t)^2}\left[ \frac{2 \gamma d^2 \varphi (t)^k}{k}- (k-1) mc^2\right] \\ {}&\qquad +\frac{\gamma }{m}(k-1) \varphi (t)^{k-2}. \end{aligned} \nonumber \\ \end{aligned}$$
(40)

We make the following change of the independent variable

$$\begin{aligned} t\longrightarrow z:= -\frac{2 d^2 \gamma }{k mc^2} \varphi (t)^k. \end{aligned}$$
(41)

To express \(\dot{\varphi }(t)^2\) as a function of z we use first integral (35). Setting \(h=e\) we get

$$\begin{aligned} \dot{\varphi }(t)^2=\frac{c^2}{ed^2}S(t)[e-S(t)] \end{aligned}$$

and then in S(t) we use substitution (41). The purpose of transformation (41) is to obtain equations with rational coefficients. As

$$\begin{aligned} \begin{aligned} \frac{\textrm{d}^2}{\textrm{d}t^2} x&= {\dot{z}}^2\, x'' + \ddot{z}\, x', \qquad '\equiv \frac{\textrm{d}}{\textrm{d}z}, \quad \text {where}\\&\dot{z}^2=\frac{2 \gamma k \varphi (t)^{k-2} z(z-1) (e+z-1)}{e m},\\&\ddot{z}=\frac{\gamma \varphi (t)^{k-2}}{e m}\Big [(e-2) (3 k-2) z\\&\quad -2 (e-1) (k-1) +2 (2 k-1) z^2\Big ], \end{aligned} \end{aligned}$$

Equations (39) transform to

$$\begin{aligned} \begin{aligned}&\eta _i''+p(z) \eta _i'+q(z)\eta _i=0, \quad i =1, \ldots n-1, \\&p(z)=\frac{k-1}{k z}-\frac{e}{2 (z-1) (e+z-1)},\\&q(z)= \frac{e \lambda _i }{2 k (z-1) z (e+z-1)}, \quad \lambda _i = {\widehat{\lambda }}_i/\gamma , \end{aligned} \end{aligned}$$
(42)

while Eq. (40) changes into

$$\begin{aligned} \begin{aligned}&\eta _n''+p (z) \eta _n'+q(z)\eta _n=0,\\&p(z)=\frac{k-1}{k z}+\frac{2-3 e-2 z}{2 (z-1) (e+z-1)},\\&q(z)= \frac{e (k z+k+z-1)+2 (z-1) (k+z-1)}{2 k (z-1)^2 z (e+z-1)}. \end{aligned} \nonumber \\ \end{aligned}$$
(43)

Equation (43) is solvable with the general solution

$$\begin{aligned} \begin{aligned} \eta _n(z)&= \widetilde{\eta }_n(z)\left[ C_1+C_2\int \frac{z^{\frac{1}{k}} \sqrt{z-1}}{z \left( z+e-1 \right) ^{\frac{3}{2}}}dz\right] ,\\ \widetilde{\eta }_n(z)&=\sqrt{z-1} \sqrt{e+z-1}, \end{aligned} \end{aligned}$$

where \(C_1\) and \(C_2\) are constants of integrations, so it does not give any obstruction to the integrability.

Now it is clear that if the Hamiltonian system (6) is integrable, then the identity component of the differential Galois group of each equation in (42) is Abelian. All of them are of the same form, so we drop the subscripts and we consider the single equation

$$\begin{aligned} \begin{aligned}&\eta ''+p(z) \eta '+q(z)\eta =0, \\&p(z)=\frac{k-1}{k z}-\frac{e}{2 (z-1) (e+z-1)},\\&q(z)= \frac{e \lambda }{2 k (z-1) z (e+z-1)}, \end{aligned} \end{aligned}$$
(44)

where \(\lambda =\lambda _i\), for a certain \(i\in \{1, \ldots , n-1\}\). For further analysis, it is convenient to transform equation (44) to its reduced form. To this end, we change the dependent variable

$$\begin{aligned} \eta = w \exp \left[ -\frac{1}{2} \int _{z_\star }^z p(s)\, ds \right] , \end{aligned}$$
(45)

transforming (44) to the form

$$\begin{aligned} w'' = r(z) w, \quad r(z) =\frac{1}{4}p(z)^2 + \frac{1}{2}p'(z) -q(z). \end{aligned}$$
(46)

The rational coefficient r(z) has the following decomposition

$$\begin{aligned} \begin{aligned} r&=\frac{1-k^2}{4 k^2 z^2}+\frac{5}{16 (z-1)^2}-\frac{3}{16 (e+z-1)^2}\\&\quad -\frac{2 e (k+2 \lambda -1)+k z}{8 k (z-1) z (e+z-1)}, \end{aligned} \end{aligned}$$
(47)

and its Laurent expansion at infinity is

$$\begin{aligned} r(z)=\frac{1-k^2}{4 k^2 z^2}+O\left( \dfrac{1}{z^3}\right) . \end{aligned}$$
(48)

The reduced equation (46) with coefficient (47) is Fuchsian, and it has four regular singular points over \(\mathbb {C}\mathbb {P}\), namely \(z_1=0\), \(z_{2}=1\), \(z_3=1 - e\) and \(z_4=\infty \) provided

$$\begin{aligned} e(e-1)k (k-1)(k+1)\ne 0. \end{aligned}$$
(49)

Differences of exponents at these singularities are as follows

$$\begin{aligned} \delta _1=\delta _4=\frac{1}{k},\quad \delta _2=\frac{3}{2},\quad \delta _3=\frac{1}{2}. \end{aligned}$$
(50)

The order of infinity is 2 provided \((k-1) (k+1)\ne 0\).

We have the freedom to fix the value of e arbitrarily. We will prove our results for all but a finite number of values of \(e\in \mathbb {R}\). Just from now, one can assume that \(e\ne 0\) and \(e\ne 1\).

Lemma 1

Assume that

$$\begin{aligned} |k |> 2, \quad \text {and} \quad \lambda \notin \{0,1\}. \end{aligned}$$
(51)

Then, for generic values of e the differential Galois group of Eq. (46) is \({\text {SL}}(2n,\mathbb {C})\).

Proof

Under our assumptions equations (46) with \(\lambda =\lambda _j\) for \(j=1,\ldots ,n-1\) have four regular singularities \(z_i\), \(i=1,\dots , 4\). In our further reasoning, we are going to apply the Kovacic algorithm [80]. Let us recall that in this algorithm we have to check successively three cases. Then, if our equation does not pass tests in all of these cases, then its differential Galois group is \({\text {SL}}(2n,\mathbb {C})\).

In each step we define set \(E_i\) associated with singularity \(z_i\), for \(i=1, \ldots , 4\). Then, we take elements \(\varvec{e}\) of the Cartesian product \(E=E_1\times \cdots \times E_4\), and check if there exists a polynomial solution P(z) of a certain linear differential equation (we call it a supplementary equation). The form of this equation as well as the degree of polynomial depends on \(\varvec{e}\) and on the considered case of the algorithm. All relevant details can be found in [80]. If a polynomial solution does not exist, then we pass to the next case. Otherwise, the equation is solvable, and we do not get obstructions for integrability.

In the first case of the Kovacic algorithm sets \(E_i\) are

$$\begin{aligned} E_i = \left\{ \frac{1}{2}(1\pm \delta _i)\right\} , \qquad i=1, \ldots , 4. \end{aligned}$$
(52)

For a given \(\varvec{e}=(e_1, \ldots ,e_4)\in E\), the degree of polynomial P(z) is \({\textsf{d}}(\varvec{e})=e_4 -e_1-e_2-e_3\). Thus, we consider only such \(\varvec{e}\) for which \({\textsf{d}}(\varvec{e})\in \mathbb {N}_0\). For our differential equation (46) with coefficient (47) taking into account assumption \(|k|>2\) sets \(E_i\) are following

$$\begin{aligned} \begin{aligned}&E_1=E_4=\left\{ \frac{1}{2}-\frac{1}{2k},\frac{1}{2}+\frac{1}{2k} \right\} ,\quad E_2=\left\{ -\frac{1}{4},\frac{5}{4}\right\} ,\\&E_3=\left\{ \frac{1}{4},\frac{3}{4}\right\} , \end{aligned} \end{aligned}$$

and we have two choices of \(\varvec{e}\)

$$\begin{aligned} \begin{aligned} \varvec{e}_1&=\left( \frac{1}{2}+\frac{1}{2k}, -\frac{1}{4}, \frac{1}{4},\frac{1}{2}+\frac{1}{2k} \right) ,\quad {\textsf{d}}\left( \varvec{e}_1\right) =0,\\ \varvec{e}_2&=\left( \frac{1}{2}-\frac{1}{2k}, -\frac{1}{4}, \frac{1}{4},\frac{1}{2}-\frac{1}{2k} \right) ,\quad {\textsf{d}}\left( \varvec{e}_2\right) =0. \end{aligned} \end{aligned}$$

For \(\varvec{e}_1\), if the supplementary equation has a polynomial solution of degree 0, then \(e(\lambda -1)=0\), but our assumptions exclude this possibility. Similarly, for \(\varvec{e}_2\), if the supplementary equation has a polynomial solution of degree 0, then \(e\lambda =0\), and again it is impossible by our assumptions.

In the second case of the Kovacic algorithm sets \(E_i\) are defined as

$$\begin{aligned} E_i = \left\{ 2+p\delta _i\, |\, p=0,\pm 2\right\} \cap \mathbb {Z}. \end{aligned}$$

Thus for our differential equation (46) with coefficient (47) these sets become

$$\begin{aligned} E_1=E_4=\left\{ 2 \right\} ,\quad E_2=\left\{ -1,2,5\right\} ,\quad E_3=\left\{ 1,2,3\right\} , \end{aligned}$$

and we have one choice for \(\varvec{e}\in \) such that \({\textsf{d}}(\varvec{e}):=\tfrac{1}{2}(e_4 -e_1-e_2-e_3)\) is a non-negative integer, namely

$$\begin{aligned} \varvec{e}=(2,-1,1,2),\quad {\textsf{d}}(\varvec{e})=0, \end{aligned}$$

but the condition of the existence of a polynomial of degree zero gives the system

$$\begin{aligned} e (-2 \lambda k+k-1)=0,\quad e (-2 \lambda k+k+1)=0, \end{aligned}$$

which has only solution \(e=0\), but it is excluded by assumptions.

In case 3 of the Kovacic algorithm sets \(E_i\) are defined as

$$\begin{aligned} E_i = \left\{ 6+p\delta _i\, |\, p=0,\pm 1,\pm 2,\pm 3,\pm 4,\pm 5,\pm 6\right\} \cap \mathbb {Z}, \end{aligned}$$

and we look for \(\varvec{e}=(e_1, \ldots ,e_4)\in E\), such that \({\textsf{d}}(\varvec{e}):=e_4 -e_1-e_2-e_3\in \mathbb {N}_0\). For our differential equation (46) with coefficient (47) sets \(E_i\) are the following

$$\begin{aligned} \begin{aligned}&E_1=E_4=\Big \{6\pm \frac{6}{k},6\pm \frac{5}{k},6\pm \frac{4}{k},6\pm \frac{3}{k},6\pm \frac{2}{k},\\&6\pm \frac{1}{k},6\Big \}\cap \mathbb {Z},\\&E_2=\{-3, 0, 3, 6, 9, 12, 15\},\\&E_3=\{3, 4, 5, 6, 7, 8, 9\}. \end{aligned} \end{aligned}$$

For \(k\in \mathbb {Z}\) such that \(|k |>6\) we have \(E_1=E_4=\{6\}\), and the only \(\varvec{e}\) yielding non-negative integer \({\textsf{d}}(\varvec{e})\) is

$$\begin{aligned} \varvec{e}=\{6,-3,3,6\},\quad {\textsf{d}}(\varvec{e})=0. \end{aligned}$$

However, the corresponding supplementary equation has a polynomial solution of degree 0 only if \(e=0\), but this value of e is excluded by assumption.

Till this point the application of the Kovacic algorithm was easy. In the remaining cases when \( |k| = 3,4,5,6\) calculations are more complicated, and they cannot be performed without the help of a computer algebra system. Namely, we have to check whether polynomial P(z) of degree \(l={\textsf{d}}(\varvec{e})\) really exists. We postulate the form of this polynomial with undetermined coefficients \(P(z)=\sum _{i=0}^{l}p_iz^{l-i}\) and substitute to the recursive differential formula for this polynomial, see page 23 in [80], that gives the supplementary equation. Comparing it to zero gives a polynomial of two variables z and e that all coefficients must vanish that leads to a linear homogeneous system

$$\begin{aligned} \begin{bmatrix} a_{11}&{}a_{12}&{}\cdots &{}a_{1l+1}\\ a_{21}&{}a_{22}&{}\cdots &{}a_{2l+1}\\ \vdots &{}\vdots &{}\cdots &{}\vdots \\ a_{s1}&{}a_{s2}&{}\cdots &{}a_{sl+1}\\ \end{bmatrix} \begin{bmatrix} p_0\\ p_1\\ \vdots \\ p_l \end{bmatrix}= \begin{bmatrix} 0\\ 0\\ \vdots \\ 0 \end{bmatrix} \end{aligned}$$
(53)

for undetermined coefficients \(p_0,p_1,\ldots ,p_l\). The \(s\times (l+1)\) matrix of this system \(\varvec{A}=[a_{ij}]\) has polynomial entries \(a_{ij}=a_{ij}(\lambda )\). Because we look for a non-zero solution of system (53) with \(p_0\ne 0\) all \((l+1)\times (l+1)\) minors of this matrix must vanish. As these minors are polynomials in \(\lambda \) we have to check their greatest common divisor (GCD) has roots only at \(\lambda =0\) or at \(\lambda =1\). In the considered problem, the maximal value of l is 4, and in this case \(s=237\). Hence, there \(\left( {\begin{matrix} s \\ l+1 \end{matrix}} \right) = 5971985487\) minors, and it seems that the problem is not tractable. However, we can proceed in the following way. We select at random two non-zero minors and check their GCD. If it vanishes only at \(\lambda =0\) or at \(\lambda =1\), we have done. Otherwise, we select the third non-zero minor and check the GCD of all three minors.

For \(k=\pm 6,\pm 5,\pm 4\) sets \(E_1\) and \(E_4\) are bigger \(E_1=E_4=\{5, 6, 7\}\), and we have 10 elements \(\varvec{e}\in E\) with \({\textsf{d}}\left( \varvec{e}_i\right) \in \mathbb {N}_0\):

$$\begin{aligned} \begin{aligned} \varvec{e}_1&=\{5,-3,3,5\},\qquad {\textsf{d}}\left( \varvec{e}_1 \right) =0,\\ \varvec{e}_2&=\{5,-3,3,6\},\qquad {\textsf{d}}\left( \varvec{e}_2 \right) =1,\\ \varvec{e}_3&=\{5,-3,3,7\},\qquad {\textsf{d}}\left( \varvec{e}_3 \right) =2,\\ \varvec{e}_4&=\{5,-3,4,6\},\qquad {\textsf{d}}\left( \varvec{e}_4 \right) =0,\\ \varvec{e}_5&=\{5,-3,4,7\},\qquad {\textsf{d}}\left( \varvec{e}_5 \right) =1,\\ \varvec{e}_6&=\{5,-3,5,7\},\qquad {\textsf{d}}\left( \varvec{e}_6 \right) =0,\\ \varvec{e}_7&=\{6,-3,3,6\},\qquad {\textsf{d}}\left( \varvec{e}_7 \right) =0,\\ \varvec{e}_8&=\{6,-3,3,7\},\qquad {\textsf{d}}\left( \varvec{e}_8 \right) =1,\\ \varvec{e}_9&=\{6,-3,4,7\},\qquad {\textsf{d}}\left( \varvec{e}_9 \right) =0,\\ \varvec{e}_{10}&=\{7,-3,3,7\},\qquad {\textsf{d}}\left( \varvec{e}_{10} \right) =0. \end{aligned} \end{aligned}$$

If \(k=\pm 6\), then the conditions for the existence a polynomial solution of the a supplementary equation can be fulfilled for a generic value of e only for \(\lambda =0\) or \(\lambda =1\). More precisely, for \(\varvec{e}_1\) condition of existence of a polynomial of degree 0 gives a polynomial of two variables e and z which factorizes by \(\lambda \) and the remaining polynomial part contains coefficients which are non-vanishing constants.

For \(\varvec{e}_2\) the searched polynomial is \(P(z)=p_0z+p_1\), the dimension of the system (53) in this case is \(s=80\). Hence, the number of all minors is 3160. All non-zero minors have factor \(\lambda \). Taking two such minors, e.g., \(M_1=\lambda (5 \lambda -2)\) and \(M_2=\lambda (312378585131 \lambda -126796514318)\), we get \(\textrm{GCD}(M_1,M_2) = \lambda \). So, if \(\lambda \in \mathbb {R}{\setminus }\{0,1\}\) the rank of matrix \(\varvec{A}\) is 2.

For \(\varvec{e}_3\), the searched polynomial is \(P(z)=p_0z^2+p_1z+p_2\), the dimension of the system (53) is \(s=84\) and the number of \(3\times 3\) minors is 94742. All these minors have a common factor \(\lambda (\lambda -1)\). In fact, according to our procedure it is enough to pick up two of them: \(M_1=\lambda (\lambda -1) (36 \lambda -17)\) and \(M_2=\lambda ^2(\lambda -1) (3 \lambda +5) (3 \lambda +32) (5 \lambda -2) (12 \lambda -7) (75 \lambda -64)\). Clearly, \(\textrm{GCD}(M_1,M_2) = \lambda (\lambda -1)\), so if \(\lambda \in \mathbb {R}{\setminus }\{0,1\}\) the rank of matrix \(\varvec{A}\) is 3. Similarly, one can analyze all cases with \(k\in \{=\pm 6,\pm 5,\pm 4\}\) and all \(\varvec{e}_i\) for \(i=1,\ldots ,10\). The conclusion is that if \(\lambda \in \mathbb {R}{\setminus }\{0,1\}\) the third case of the Kovacic algorithm does not occur for them.

If \(k=\pm 3\), then \(E_1=E_4=\{4,5, 6, 7,8\}\), and there are 39 elements \(\varvec{e}\) of E for which \({\textsf{d}}(\varvec{e})\in \mathbb {N}_0\). More precisely, \(d(e)\in \{0,1,2,3,4\}\). Using method described above we can check that if \(\lambda \in \mathbb {R}{\setminus }\{0,1\}\), then for all 39 admissible values of \(\varvec{e}\) the third case of the Kovacic algorithm does not occur. \(\square \)

Lemma 2

Assume that

$$\begin{aligned} k=\pm 1, \quad \text {and} \quad \lambda \notin \{0,1\}. \end{aligned}$$
(54)

Then, for generic values of e the differential Galois group of equation (46) is \({\text {SL}}(2n,\mathbb {C})\).

Proof

Cases with \(k=\pm 1\) need separate analysis because then the coefficient of the reduced equation takes the following form for \(k=1\)

$$\begin{aligned} \begin{aligned} r&=\frac{5}{16 (z-1)^2}-\frac{3}{16 (e+z-1)^2}+\frac{e \lambda }{2 (e-1)z}\\&\quad +\frac{-4 e \lambda -1}{8 e(z-1)} +\frac{-4 e \lambda +e-1}{8 (e-1) e (e+z-1)} \end{aligned} \nonumber \\ \end{aligned}$$
(55)

and for \(k=-1\)

$$\begin{aligned} r= & {} \frac{5}{16 (z-1)^2}-\frac{3}{16 (e+z-1)^2}+\frac{e-e \lambda }{2 (e-1)z}\nonumber \\{} & {} +\frac{4 e \lambda -4 e-1}{8e (z-1)}+\frac{4 e \lambda -3 e-1}{8 (e-1) e (e+z-1)}, \end{aligned}$$
(56)

respectively that are different from the previous cases with \(|k|>2\). Thus, in both cases \(k=\pm 1\) singularity \(z_1=0\) becomes a pole of the first order and the degree of the infinity is 3. More precisely, equation (46) with coefficient (55) or (56) has three singularities: \(z_1=0\), which is a pole of the first order, \(z_2=1\) and \(z_3=1-e\), which are poles of the second order. The order of infinity is 3 provided \(e(e-1)\lambda (\lambda -1)\ne 0\).

Now, the proof consists in a direct application of the Kovacic algorithm. These easy calculations are left to the reader. \(\square \)

Now, proof of our Theorem 3 follows directly from Lemma 1 and Lemma  2. Indeed, let us assume that the relativistic Hamiltonian system (5) with a homogenous potential \(V(\varvec{q})\) is integrable in the Liouville sense and it admits a Darboux point \(\varvec{d}\) satisfying \( V'(\varvec{d})=\gamma \varvec{d}\) with diagonalisable rescaled Hessian \(\gamma ^{-1}V''(\varvec{d})\). Then Hamilton equations (6) admit a non-equilibrium particular solution and variational equations are the direct product of the reduced second-order linear equations (46) for eigenvalues of \(\gamma ^{-1}V''(\varvec{d})\). If assumptions formulated in Lemma 1 and in Lemma  2 are satisfied, then the identity component of the differential Galois group of the variational equations is not Abelian. Thus, by Theorem 1 the Hamiltonian system is not integrable. This contradiction finishes the proof.

The following two remarks show that in cases listed in Theorem 3 variational equations are solvable and necessary integrability conditions are satisfied.

Remark 3

Solutions for \(\lambda =0\) and \(\lambda =1\)

General solution of (46) for \(\lambda =0\) is

$$\begin{aligned} w(z)=\widetilde{w}(z) z^{-\frac{1}{2 k}}\Big [C_1+C_2\int \frac{\sqrt{\frac{z-1}{z-1+e}}\, z^{\frac{1}{k}}}{z} dz \Big ], \end{aligned}$$

and for \(\lambda =1\) equals

$$\begin{aligned} w(z)= \widetilde{w}(z)z^{\frac{1}{2 k}}\left[ C_1+C_2\int \frac{\sqrt{\frac{z-1}{z-1+e}}}{z \,z^{\frac{1}{k}}}dz\right] , \end{aligned}$$

respectively, where

$$\begin{aligned} \widetilde{w}(z)=\sqrt{z}\, \left( \frac{z-1+e}{z-1}\right) ^{\frac{1}{4}}, \end{aligned}$$

and \(C_1\) and \(C_2\) are constants of integration.

Remark 4

Solutions for \(k=2\) and \(k=-2\)

General solution of (46) for \(k=2\) is

$$\begin{aligned} \begin{aligned} w(z)&=\root 4 \of {\frac{e z}{z-1}+z} \sqrt{\lambda (z-1)+1}\\&\qquad \cdot \left[ C_1e^{\widetilde{w}(z)} +C_2e^{- \widetilde{w}(z)}\right] ,\\ \widetilde{w}(z)&=\int \frac{\lambda (z-1) \sqrt{e (1-\lambda )-\frac{1}{\lambda }+1}}{2 \sqrt{(z-1) z (e+z-1)} (\lambda (z-1)+1)}dz, \end{aligned} \end{aligned}$$

and for \(k=-2\) equals

$$\begin{aligned} \begin{aligned} w(z)&=\root 4 \of {\frac{e z}{z-1}+z} \sqrt{(\lambda -1) z-\lambda }\\&\qquad \cdot \left[ C_1e^{\widetilde{w}(z)} +C_2e^{- \widetilde{w}(z)}\right] ,\\ \widetilde{w}(z)&=\int \frac{(\lambda -1) (z-1) \sqrt{\lambda \left( e+\frac{1}{\lambda -1}\right) }}{2 \sqrt{(z-1) z (e+z-1)} ((\lambda -1) z-\lambda )} dz, \end{aligned} \end{aligned}$$

respectively, where \(C_1\) and \(C_2\) are constants of integration.

Remark 5

On the energy level \(e=1\) coalescence of singularities \(z_1\) and \(z_3\) appears and coefficients p(z) and q(z) of normal equations (42) simplify to

$$\begin{aligned}{} & {} p=\frac{3 k-2}{2 k z}-\frac{1}{2 (z-1)},\nonumber \\ {}{} & {} q=-\frac{\lambda }{2 k z^2}+\frac{\lambda }{2 k (z-1) z}. \end{aligned}$$
(57)

So equations (42) take the form of the Riemann P equation

$$\begin{aligned} \begin{aligned}&\dfrac{\textrm{d}^2 \eta }{\textrm{d}z^2}+\left( \dfrac{1-\rho _1-\rho _2}{z}+ \dfrac{1-\gamma _1-\gamma _2}{z-1}\right) \dfrac{\textrm{d}\eta }{\textrm{d}z}\\&\quad +\left( \dfrac{\rho _1\rho _2}{z^2}+\dfrac{\gamma _1\gamma _2}{(z-1)^2}+ \dfrac{\tau _1\tau _2-\rho _1\rho _2-\gamma _1\gamma _2}{z(z-1)}\right) \eta =0, \end{aligned}\nonumber \\ \end{aligned}$$
(58)

where \(\eta \equiv \eta _i\), \((\rho _1,\rho _2)\), \((\gamma _1,\gamma _2)\) and \((\tau _1,\tau _2)\) are the exponents at the respective singular points: \(0,1,\infty \). For our equation (42) exponents are the following

$$\begin{aligned} \begin{aligned}&\rho _1=\frac{2-k+\sqrt{8 k \lambda +(k-2)^2}}{4 k},\\&\rho _2=\frac{2-k-\sqrt{8 k \lambda +(k-2)^2}}{4 k},\\&\sigma _1=\frac{3}{2},\qquad \sigma _2=0,\qquad \tau _1=0,\qquad \tau _2=-\frac{1}{k} \end{aligned} \end{aligned}$$

and its differences equal to

$$\begin{aligned} \begin{aligned}&\rho = \rho _1-\rho _2=\frac{\sqrt{(k-2)^2+8 k \lambda }}{2 k}, \\&\sigma =\sigma _1-\sigma _2=\frac{3}{2}, \\&\tau =\tau _1 -\tau _2=\frac{1}{k}. \end{aligned} \end{aligned}$$

Hence, for the energy \(e=1\), the integrability obstructions for the relativistic Hamiltonian system (5) in curved space-time are the same as the ones for the corresponding classical non-relativistic Hamiltonian system (1) with the same potential formulated in Theorem 2.

6 Conclusions

In this paper, we study the dynamics and integrability of a relativistic point mass moving in curved space-time that curvature is caused by scalar potential \(V(\varvec{q})\). Such systems are currently in a great activity of the scientific community. This is due to the fact of their potential experimental realisations. In the case when potential \(V(\varvec{q})\) is a homogeneous potential of integer non-zero degree k, we obtained very strong necessary integrability conditions for relativistic Hamiltonian equations that either \(k=\pm 2\), or all non-trivial eigenvalues of the re-scaled Hessian \(\gamma ^{-1}V''(\varvec{d})\) are either 0, or 1.

Moreover, the integrability of the corresponding non-relativistic flat Hamiltonian with the same \(V(\varvec{q})\) is a necessary condition for the integrability of its relativistic counterpart in curved space-time. In fact, well-known necessary integrability conditions for a non-relativistic flat Hamiltonian with homogeneous potential \(V(\varvec{q})\) formulated in Theorem 2 can be obtained from analysis of variational equations for the relativistic Hamiltonian in curved space-time on a special energy level corresponding to the coalescence of two singularities. Quite unexpectedly, however, it seems that there is no relation between the integrability of a relativistic Hamiltonian system H defined in (5) in curved space-time and the integrability of the corresponding relativistic Hamiltonian system in flat space-time (4) obtained in the weak potential limit when \(2V(\varvec{q})\ll mc^2\). This is due to orders used in expansions (13) are not indicated consistently. Therefore, a relation between the integrability of both Hamiltonian systems cannot be deduced.