INTRODUCTION

It is well known that research into the integrability of autonomous systems on a finite-dimensional configuration manifold Mn leads to the study of systems of order 2n on the tangent bundle TMn. A key point, along with the geometry of Mn, is the structure of the force field present in the system. For example, the problem of an n-dimensional pendulum moving on a generalized spherical hinge in a nonconservative force field leads to a dynamical system on the tangent bundle of the (n – 1)-dimensional sphere with a special metric on it induced by additional symmetry groups [1, 2]. The systems describing the motion of such a pendulum have dissipation of variable sign (referred to as alternating dissipation), and the complete list of first integrals consists of transcendental functions that can be expressed in terms of a finite combination of elementary functions [2, 3].

The problems of a point moving over multidimensional surfaces of revolution, in Lobachevsky spaces, etc., are also well known and complicated. Nevertheless, for dissipative systems, it is sometimes possible to find a complete list of first integrals consisting of transcendental functions (in the sense of complex analysis), since a complete list of continuous autonomous first integrals fails to be found. These results are important in the context of a nonconservative force field present in the system.

Generally, the problems under consideration have been addressed in numerous publications. Among them, we note only [4–6].

In this paper, we show the integrability of the classes of homogeneous geodesic, potential, and dissipative dynamical systems on tangent bundles of smooth four-dimensional manifolds. The force fields introduced into the systems give rise to alternating dissipation and generalize previously considered fields.

1 INTEGRATION OF GEODESIC EQUATIONS

It is well known that, in the case of an n-dimensional smooth Riemannian manifold Mn with coordinates (α, β), β = (β1, …, βn – 1), and affine connection \(\Gamma _{{jk}}^{i}\)(α, β), the equations of geodesic lines on the tangent bundle TMn\(\{ {{\alpha }^{ \bullet }},\beta _{1}^{ \bullet }, \ldots ,\beta _{{n - 1}}^{ \bullet };\alpha ,{{\beta }_{1}}, \ldots ,{{\beta }_{{n - 1}}}\} \), α = x1, \({{\beta }_{1}} = {{x}^{2}}, \ldots ,{{\beta }_{{n - 1}}}\) = xn, \(x = ({{x}^{1}}, \ldots ,{{x}^{n}})\), have the following form (the derivatives are taken with respect to a positive integer parameter):

$${{x}^{{i \bullet \bullet }}} + \sum\limits_{j,k = 1}^n {\Gamma _{{jk}}^{i}(x){{x}^{{j \bullet }}}{{x}^{{k \bullet }}}} = 0,\quad i = {\text{ }}1, \ldots ,n.$$
(1)

Let us study the structure of Eqs. (1) under a change of coordinates on TMn. For this purpose, we consider the following change of coordinates of the tangent space:

$${{x}^{{i \bullet }}} = \sum\limits_{j = 1}^n {{{R}^{{ij}}}{{z}_{j}}} ,$$
(2)

which can be inverted as \({{z}_{j}} = \sum\limits_{i = 1}^n {{{T}_{{ji}}}{{x}^{{i \bullet }}}} ;\) here, Rij and Tji, i, j = 1, …, n, are functions of x, and RT = E, where \(R = ({{R}^{{ij}}})\) and \(T = ({{T}_{{ji}}})\). Equations (2) will be referred to as new kinematic relations, i.e., linear relations on the tangent bundle TMn. It holds that

$$z_{i}^{ \bullet } = \sum\limits_{j,k = 1}^n {T_{{ij,k}}^{{}}{{x}^{{j \bullet }}}{{x}^{{k \bullet }}}} - \sum\limits_{j,p,q = 1}^n {T_{{ij}}^{{}}\Gamma _{{pq}}^{j}{{x}^{{p \bullet }}}{{x}^{{q \bullet }}}} ,$$
(3)

where \({{T}_{{ji,k}}} = \frac{{\partial {{T}_{{ji}}}}}{{\partial {{x}^{k}}}}\), j, i, k = 1, …, n; here, formulas (2) are substituted for \({{x}^{{i \bullet }}}\), i = 1, …, n, in (3), and the right-hand sides of compound system (2), (3) are homogeneous forms in the quasi-velocities z1, …, zn.

Proposition 1. System (1) in the domain where det\(R(x) \ne 0\) is equivalent to compound system (2), (3).

The result of passing from the geodesic equations (1) to equivalent system (2), (3) depends on both substitution (2) (i.e., on the introduced kinematic relations) and on the affine connection \(\Gamma _{{jk}}^{i}\)(α, β).

Consider a rather general case of kinematic relations specified as

$$\begin{gathered} {{\alpha }^{ \bullet }} = {{z}_{n}}{{f}_{n}}(\alpha ),\quad \beta _{1}^{ \bullet } = z_{{n - 1}}^{{}}{{f}_{1}}(\alpha ), \\ \beta _{2}^{ \bullet } = z_{{n - 2}}^{{}}{{f}_{2}}(\alpha ){{g}_{1}}({{\beta }_{1}}), \\ \beta _{3}^{ \bullet } = z_{{n - 3}}^{{}}{{f}_{3}}(\alpha ){{g}_{2}}({{\beta }_{1}}){{h}_{1}}({{\beta }_{2}}), \ldots , \\ \beta _{{n - 1}}^{ \bullet } = z_{1}^{{}}{{f}_{{n - 1}}}(\alpha ){{g}_{{n - 2}}}({{\beta }_{1}}){{h}_{{n - 3}}}({{\beta }_{2}}) \ldots {{i}_{1}}({{\beta }_{{n - 2}}}), \\ \end{gathered} $$
(4)

where \({{f}_{k}}(\alpha )\) for k = 1, …, n – 1, \({{g}_{l}}({{\beta }_{1}})\) for l = 1, …, n – 2, and \({{h}_{m}}({{\beta }_{2}})\) for m = 1, …, n – 3, …, \({{i}_{1}}({{\beta }_{{n - 2}}})\) are smooth functions that do not vanish identically. Such coordinates z1, …, zn in the tangent space are introduced when we consider the following geodesic equations [7, 8] (in particular, on multidimensional surfaces of revolution, in Lobachevsky spaces, etc.) with n(n – 1) + 1 nonzero connection coefficients (here and below, a double index divided by a comma does not denote differentiation, in contrast to formulas (3)):

$$\begin{gathered} {{\alpha }^{{ \bullet \bullet }}} + \Gamma _{{\alpha \alpha }}^{\alpha }(\alpha ,\beta )\alpha _{{}}^{{ \bullet 2}} \\ \, + \Gamma _{{11}}^{\alpha }(\alpha ,\beta )\beta _{1}^{{ \bullet 2}} + \ldots + \Gamma _{{n - 1,n - 1}}^{\alpha }(\alpha ,\beta )\beta _{{n - 1}}^{{ \bullet 2}} = 0, \\ \beta _{1}^{{ \bullet \bullet }} + 2\Gamma _{{\alpha 1}}^{1}(\alpha ,\beta ){{\alpha }^{ \bullet }}\beta _{1}^{ \bullet } \\ \, + \Gamma _{{22}}^{1}(\alpha ,\beta )\beta _{2}^{{ \bullet 2}} + \ldots + \Gamma _{{n - 1,n - 1}}^{1}(\alpha ,\beta )\beta _{{n - 1}}^{{ \bullet 2}} = 0, \\ \beta _{2}^{{ \bullet \bullet }} + 2\Gamma _{{\alpha 2}}^{2}(\alpha ,\beta ){{\alpha }^{ \bullet }}\beta _{2}^{ \bullet } + 2\Gamma _{{12}}^{2}(\alpha ,\beta )\beta _{1}^{ \bullet }\beta _{2}^{ \bullet } \\ \, + \Gamma _{{33}}^{2}(\alpha ,\beta )\beta _{3}^{{ \bullet 2}} + \ldots + \Gamma _{{n - 1,n - 1}}^{2}(\alpha ,\beta )\beta _{{n - 1}}^{{ \bullet 2}} = 0, \\ \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots , \\ \beta _{{n - 2}}^{{ \bullet \bullet }} + 2\Gamma _{{\alpha ,n - 2}}^{{n - 2}}(\alpha ,\beta ){{\alpha }^{ \bullet }}\beta _{{n - 2}}^{ \bullet } + 2\Gamma _{{1,n - 2}}^{{n - 2}}(\alpha ,\beta )\beta _{1}^{ \bullet }\beta _{{n - 2}}^{ \bullet } \\ + \,\, \ldots + 2\Gamma _{{n - 3,n - 2}}^{{n - 2}}(\alpha ,\beta )\beta _{{n - 3}}^{ \bullet }\beta _{{n - 2}}^{ \bullet } + \Gamma _{{n - 1,n - 1}}^{{n - 2}}(\alpha ,\beta )\beta _{{n - 1}}^{{ \bullet 2}} = 0, \\ \beta _{{n - 1}}^{{ \bullet \bullet }} + 2\Gamma _{{\alpha ,n - 1}}^{{n - 1}}(\alpha ,\beta ){{\alpha }^{ \bullet }}\beta _{{n - 1}}^{ \bullet } \\ + \,2\Gamma _{{1,n - 1}}^{{n - 1}}(\alpha ,\beta )\beta _{1}^{ \bullet }\beta _{{n - 1}}^{ \bullet } + \ldots + 2\Gamma _{{n - 2,n - 1}}^{{n - 1}}(\alpha ,\beta )\beta _{{n - 2}}^{ \bullet }\beta _{{n - 1}}^{ \bullet } = 0. \\ \end{gathered} $$
(5)

In the case of (4), Eqs. (3) become

$$\begin{gathered} z_{1}^{ \bullet } = - {{f}_{n}}(\alpha )[2\Gamma _{{\alpha ,n - 1}}^{{n - 1}}(\alpha ,\beta ) + D{{f}_{{n - 1}}}(\alpha )]{{z}_{1}}{{z}_{n}} \\ - \,{{f}_{1}}(\alpha )[2\Gamma _{{1,n - 1}}^{{n - 1}}(\alpha ,\beta ) + D{{g}_{{n - 2}}}({{\beta }_{1}})]{{z}_{1}}{{z}_{{n - 1}}} \\ - \,{{f}_{2}}(\alpha ){{g}_{1}}({{\beta }_{1}})[2\Gamma _{{2,n - 1}}^{{n - 1}}(\alpha ,\beta ) + D{{h}_{{n - 3}}}({{\beta }_{2}})]{{z}_{1}}{{z}_{{n - 2}}} \\ - \,\, \ldots - {{f}_{{n - 2}}}(\alpha ){{g}_{{n - 3}}}({{\beta }_{1}}){{h}_{{n - 4}}}({{\beta }_{2}}) \ldots {{r}_{1}}({{\beta }_{{n - 3}}})[2\Gamma _{{n - 2,n - 1}}^{{n - 1}}(\alpha ,\beta ) \\ \, + D{{i}_{1}}({{\beta }_{{n - 2}}})]{{z}_{1}}{{z}_{2}}, \\ z_{2}^{ \bullet } = - {{f}_{n}}(\alpha )[2\Gamma _{{\alpha ,n - 2}}^{{n - 2}}(\alpha ,\beta ) + D{{f}_{{n - 2}}}(\alpha )]{{z}_{2}}{{z}_{n}} \\ - \,{{f}_{1}}(\alpha )[2\Gamma _{{1,n - 2}}^{{n - 2}}(\alpha ,\beta ) + D{{g}_{{n - 3}}}({{\beta }_{1}})]{{z}_{2}}{{z}_{{n - 1}}} \\ - \,\, \ldots - {{f}_{{n - 3}}}(\alpha ){{g}_{{n - 4}}}({{\beta }_{1}}){{h}_{{n - 5}}}({{\beta }_{2}}) \ldots {{s}_{1}}({{\beta }_{{n - 4}}}) \\ \times \,[2\Gamma _{{n - 3,n - 2}}^{{n - 2}}(\alpha ,\beta ) + D{{r}_{1}}({{\beta }_{{n - 3}}})]{{z}_{2}}{{z}_{3}} \\ - \frac{{f_{{n - 1}}^{2}(\alpha )}}{{{{f}_{{n - 2}}}(\alpha )}}\frac{{g_{{n - 2}}^{2}({{\beta }_{1}})}}{{{{g}_{{n - 3}}}({{\beta }_{1}})}}\frac{{h_{{n - 3}}^{2}({{\beta }_{2}})}}{{{{h}_{{n - 4}}}({{\beta }_{2}})}} \ldots \frac{{r_{2}^{2}({{\beta }_{{n - 3}}})}}{{{{r}_{1}}({{\beta }_{{n - 3}}})}}i_{1}^{2}({{\beta }_{{n - 2}}}) \\ \times \,\Gamma _{{n - 1,n - 1}}^{{n - 2}}(\alpha ,\beta )z_{1}^{2}, \\ \end{gathered} $$
$$ \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots ,$$
(6)
$$\begin{gathered} z_{{n - 1}}^{ \bullet } = - {{f}_{n}}(\alpha )[2\Gamma _{{\alpha 1}}^{1}(\alpha ,\beta ) + D{{f}_{1}}(\alpha )]{{z}_{{n - 1}}}{{z}_{n}} \\ - \,\frac{{f_{2}^{2}(\alpha )}}{{{{f}_{1}}(\alpha )}}g_{1}^{2}({{\beta }_{1}})\Gamma _{{22}}^{1}(\alpha ,\beta )z_{{n - 2}}^{2} \\ - \,\, \ldots - \frac{{f_{{n - 1}}^{2}(\alpha )}}{{{{f}_{1}}(\alpha )}}g_{{n - 2}}^{2}({{\beta }_{1}})h_{{n - 3}}^{2}({{\beta }_{2}}) \ldots i_{1}^{2}({{\beta }_{{n - 2}}})\Gamma _{{n - 1,n - 1}}^{1}(\alpha ,\beta )z_{1}^{2}, \\ z_{n}^{ \bullet } = - {{f}_{n}}(\alpha )[\Gamma _{{\alpha \alpha }}^{\alpha }(\alpha ,\beta ) + D{{f}_{n}}(\alpha )]z_{n}^{2}\, - \,\frac{{f_{1}^{2}(\alpha )}}{{{{f}_{n}}(\alpha )}}\Gamma _{{11}}^{\alpha }(\alpha ,\beta )z_{{n - 1}}^{2} \\ - \,\frac{{f_{2}^{2}(\alpha )}}{{{{f}_{n}}(\alpha )}}g_{1}^{2}({{\beta }_{1}})\Gamma _{{22}}^{\alpha }(\alpha ,\beta )z_{{n - 2}}^{2} \\ - \,\, \ldots - \frac{{f_{{n - 1}}^{2}(\alpha )}}{{{{f}_{n}}(\alpha )}}g_{{n - 2}}^{2}({{\beta }_{1}})h_{{n - 3}}^{2}({{\beta }_{2}}) \ldots i_{1}^{2}({{\beta }_{{n - 2}}})\Gamma _{{n - 1,n - 1}}^{\alpha }(\alpha ,\beta )z_{1}^{2}, \\ \end{gathered} $$

\(Dj(\gamma ) = d{\text{ln}}\left| {j(\gamma )} \right|{\text{/}}d\gamma \), and the geodesic equations (5) are equivalent to compound system (4), (6) almost everywhere on the manifold TMn\(\{ {{z}_{n}}, \ldots ,{{z}_{1}};\alpha ,{{\beta }_{1}}\), ..., βn – 1}.

For complete integration of system (4), (6), it is necessary to know, generally speaking, 2n – 1 independent first integrals. Note that first integrals (in particular, for geodesic equations) can be sought in a more general form than that considered below.

Proposition 2. If the system of 1 + n(n – 1)/2 differential equalities

$$\begin{gathered} \Gamma _{{\alpha \alpha }}^{\alpha }(\alpha ,\beta ) + D{{f}_{n}}(\alpha ) \equiv 0, \\ f_{n}^{2}(\alpha )[2\Gamma _{{\alpha 1}}^{1}(\alpha ,\beta ) + D{{f}_{1}}(\alpha )] + f_{1}^{2}(\alpha )\Gamma _{{11}}^{\alpha }(\alpha ,\beta ) \equiv 0, \ldots , \\ f_{n}^{2}(\alpha )[2\Gamma _{{\alpha ,n - 1}}^{{n - 1}}(\alpha ,\beta ) + D{{f}_{{n - 1}}}(\alpha )] \\ + \,f_{{n - 1}}^{2}(\alpha )g_{{n - 2}}^{2}({{\beta }_{1}})h_{{n - 3}}^{2}({{\beta }_{2}}) \ldots i_{1}^{2}({{\beta }_{{n - 2}}})\Gamma _{{n - 1,n - 1}}^{\alpha }(\alpha ,\beta ) \equiv 0, \\ f_{1}^{2}(\alpha )[2\Gamma _{{12}}^{2}(\alpha ,\beta ) + D{{g}_{1}}({{\beta }_{1}})] \\ \, + f_{2}^{2}(\alpha )g_{1}^{2}({{\beta }_{1}})\Gamma _{{22}}^{1}(\alpha ,\beta ) \equiv 0, \ldots , \\ f_{1}^{2}(\alpha )[2\Gamma _{{1,n - 1}}^{{n - 1}}(\alpha ,\beta ) + D{{g}_{{n - 2}}}({{\beta }_{1}})] \\ + \,f_{{n - 1}}^{2}(\alpha )g_{{n - 2}}^{2}({{\beta }_{1}}) \ldots i_{1}^{2}({{\beta }_{{n - 2}}})\Gamma _{{n - 1,n - 1}}^{1}(\alpha ,\beta ) \equiv 0, \ldots , \\ f_{{n - 2}}^{2}(\alpha ) \ldots r_{1}^{2}({{\beta }_{{n - 3}}})[2\Gamma _{{n - 2,n - 1}}^{{n - 1}}(\alpha ,\beta ) + D{{i}_{1}}({{\beta }_{{n - 2}}})] \\ + \,f_{{n - 1}}^{2}(\alpha ) \ldots i_{1}^{2}({{\beta }_{{n - 2}}})\Gamma _{{n - 1,n - 1}}^{{n - 2}}(\alpha ,\beta ) \equiv 0 \\ \end{gathered} $$
(7)

holds everywhere, then system (4), (6) has an analytic first integral of the form

$$\Phi _{1}^{{}}({{z}_{n}}, \ldots ,{{z}_{1}}) = z_{1}^{2} + \ldots + z_{n}^{2} = C_{1}^{2} = {\text{const}}.$$
(8)

Examples. Equations (5) of geodesics in the multidimensional Lobachevsky space in the Klein model become

$$\begin{gathered} {{\alpha }^{{ \bullet \bullet }}} - \frac{1}{\alpha }({{\alpha }^{{ \bullet 2}}} - \beta _{1}^{{ \bullet 2}} - ... - \beta _{{n - 1}}^{{ \bullet 2}}) = 0, \\ ~\beta _{k}^{{ \bullet \bullet }} - \frac{2}{\alpha }{{\alpha }^{ \bullet }}\beta _{k}^{ \bullet } = 0,~\quad k = 1, \ldots ,n - 1. \\ \end{gathered} $$
(9)

It is possible to write a multiple-parameter system that is equivalent to Eqs. (9) and has a first integral of form (8). Equations of geodesics on multidimensional surfaces of revolution have similar properties.

System (7) can be treated as the possibility of transforming the quadratic form of the manifold’s metric into a canonical form with energy conservation law (8) (or see (21) below) depending on the problem under consideration. The history and the state of the art in this more general problem have been covered rather extensively (we note only [8, 9]). The search for both first integral (8) and others (see below) relies on additional symmetry groups present in the system.

It is possible to prove a separate theorem on the existence of a solution \({{f}_{k}}(\alpha )\), k = 1, …, n – 1, \({{g}_{l}}({{\beta }_{1}})\), l = 1, …, n – 2, \({{h}_{m}}({{\beta }_{2}})\), m = 1, …, n – 3, …, \({{i}_{1}}({{\beta }_{{n - 2}}})\) of system (7) for system (4), (6) to have analytical integral (8). However, some of the conditions in (7) are not required in the subsequent study of dynamical systems with dissipation. Nevertheless, we assume in what follows that the conditions

$${{f}_{1}}(\alpha ) \equiv \ldots \equiv {{f}_{{n - 1}}}(\alpha ) = f(\alpha )$$
(10)

holds for Eqs. (4) and the functions \({{g}_{l}}({{\beta }_{1}})\), l = 1, …, n – 2, \({{h}_{m}}({{\beta }_{2}})\), m = 1, …, n – 3, …, \({{i}_{1}}({{\beta }_{{n - 2}}})\), satisfy, generally speaking, (n – 1)(n – 2)/2 transformed equations in (7):

$$\begin{gathered} 2\Gamma _{{12}}^{2}(\alpha ,\beta ) + D{{g}_{1}}({{\beta }_{1}}) + g_{1}^{2}({{\beta }_{1}})\Gamma _{{22}}^{1}(\alpha ,\beta ) \equiv 0, \ldots , \\ 2\Gamma _{{1,n - 1}}^{{n - 1}}(\alpha ,\beta ) + D{{g}_{{n - 2}}}({{\beta }_{1}}) \\ + \,g_{{n - 2}}^{2}({{\beta }_{1}}) \ldots i_{1}^{2}({{\beta }_{{n - 2}}})\Gamma _{{n - 1,n - 1}}^{1}(\alpha ,\beta ) \equiv 0, \ldots , \\ g_{{n - 3}}^{2}({{\beta }_{1}}) \ldots r_{1}^{2}({{\beta }_{{n - 3}}})[2\Gamma _{{n - 2,n - 1}}^{{n - 1}}(\alpha ,\beta ) + D{{i}_{1}}({{\beta }_{{n - 2}}})] \\ + \,g_{{n - 2}}^{2}({{\beta }_{1}}) \ldots i_{1}^{2}({{\beta }_{{n - 2}}})\Gamma _{{n - 1,n - 1}}^{{n - 2}}(\alpha ,\beta ) \equiv 0. \\ \end{gathered} $$
(11)

Thus, the functions \({{g}_{l}}({{\beta }_{1}})\), l = 1, …, n – 2, \({{h}_{m}}({{\beta }_{2}})\), m = 1, …, n – 3, …, \({{i}_{1}}({{\beta }_{{n - 2}}})\) depend on the connection coefficients. The constraints on the functions \(f(\alpha )\) and \({{f}_{n}}(\alpha )\) will be given below.

Proposition 3. If conditions (10) and (11) are satisfied and, additionally,

$$\Gamma _{{\alpha 1}}^{1}(\alpha ,\beta ) \equiv \ldots \equiv \Gamma _{{\alpha ,n - 1}}^{{n - 1}}(\alpha ,\beta ) = \Gamma _{1}^{{}}(\alpha ),$$
(12)

then system (4), (6) has a smooth first integral of the form

$$\begin{gathered} \Phi _{2}^{{}}({{z}_{{n - 1}}}, \ldots ,{{z}_{1}};\alpha ) \\ = \,\sqrt {z_{1}^{2} + \ldots + z_{{n - 1}}^{2}} \Phi _{0}^{{}}(\alpha ) = C_{2}^{{}} = {\text{const}}, \\ \Phi _{0}^{{}}(\alpha ) = f(\alpha )\exp \left\{ {2\int\limits_{{{\alpha }_{0}}}^\alpha {\Gamma _{1}^{{}}(b)db} } \right\}. \\ \end{gathered} $$
(13)

Proposition 4. If the conditions of Proposition 3 are satisfied and, additionally,

$${{g}_{1}}({{\beta }_{1}}) \equiv \ldots \equiv {{g}_{{n - 2}}}({{\beta }_{1}}) = g({{\beta }_{1}})$$
(14)

and

$$\Gamma _{{12}}^{2}(\alpha ,\beta ) \equiv \ldots \equiv \Gamma _{{1,n - 1}}^{{n - 1}}(\alpha ,\beta ) = \Gamma _{2}^{{}}({{\beta }_{1}}),$$
(15)

then system (4), (6) has a smooth first integral of the form

$$\begin{gathered} \Phi _{3}^{{}}({{z}_{{n - 2}}}, \ldots ,{{z}_{1}};\alpha ,{{\beta }_{1}}) \\ = \sqrt {z_{1}^{2} + \ldots + z_{{n - 2}}^{2}} \Phi _{0}^{{}}(\alpha ){{\Psi }_{1}}({{\beta }_{1}}) = C_{3}^{{}} = {\text{const}}, \\ \end{gathered} $$
(16)
$${{\Psi }_{1}}({{\beta }_{1}}) = g({{\beta }_{1}})\exp \left\{ {2\int\limits_{{{\beta }_{{10}}}}^{{{\beta }_{1}}} {\Gamma _{2}^{{}}(b)db} } \right\}.$$

Next, we prove by induction the necessary number of propositions (altogether n) and derive the following result (here and below, the ellipsis in a proposition means n propositions on n integrals).

Proposition 5. If the conditions of Propositions 3, 4, …, hold, and, additionally,

$$\Gamma _{{n - 2,n - 1}}^{{n - 1}}(\alpha ,\beta ) = \Gamma _{{n - 1}}^{{}}({{\beta }_{{n - 2}}}),$$
(17)

then system (4), (6) has a smooth first integral of the form

$$\begin{gathered} \Phi _{n}^{{}}({{z}_{1}};\alpha ,{{\beta }_{1}}, \ldots ,{{\beta }_{{n - 2}}}) \\ = {{z}_{1}}\Phi _{0}^{{}}(\alpha ){{\Psi }_{1}}({{\beta }_{1}}) \ldots {{\Psi }_{{n - 2}}}({{\beta }_{{n - 2}}}) = C_{n}^{{}} = {\text{const}}, \\ \end{gathered} $$
(18)
$${{\Psi }_{{n - 2}}}({{\beta }_{{n - 2}}}) = {{i}_{1}}({{\beta }_{{n - 2}}})\exp \left\{ {2\int\limits_{{{\beta }_{{n - 2,0}}}}^{{{\beta }_{{n - 2}}}} {\Gamma _{{n - 1}}^{{}}(b)db} } \right\}.$$

Proposition 6. If the conditions of Propositions 3, …, 5 hold, then system (4), (6) has a smooth first integral of the form

$$\begin{gathered} \Phi _{{n + 1}}^{{}}({{\beta }_{{n - 2}}},{{\beta }_{{n - 1}}}) \\ = {{\beta }_{{n - 1}}} \pm \int\limits_{{{\beta }_{{n - 2,0}}}}^{{{\beta }_{{n - 2}}}} {\frac{{{{C}_{n}}h(b)}}{{\sqrt {C_{{n - 1}}^{2}\Psi _{{n - 2}}^{2}(b) - C_{n}^{2}} }}db} = C_{{n + 1}}^{{}} = {\text{const}}, \\ \end{gathered} $$
(19)

where, after evaluating integral (19), the constants Cn – 1 and Cn can be formally replaced by the left-hand sides of the corresponding equalities.

Theorem 1. If the conditions of Propositions 2, …, 5 hold, then system (4), (6) has n + 1 independent first integrals of form (8), (13), (16), …, (18), (19).

The fact that the complete set consists of n + 1, rather than 2n – 1, first integrals will be shown below.

2 INTEGRATION OF THE EQUATIONS OF MOTION IN A POTENTIAL FORCE FIELD

Modifying (4), (6) yields a conservative system.  Namely, we introduce a smooth (external) force   field in the projections onto the \(z_{k}^{ \bullet }\) axes, k = 1, …, n, respectively: \({{F}_{1}}({{\beta }_{{n - 1}}}){{f}_{{n - 1}}}(\alpha ){{g}_{{n - 2}}}({{\beta }_{1}}) \ldots {{i}_{1}}({{\beta }_{{n - 2}}})\), F2(βn – 2)fn – 2(α)\(g{}_{{n - 3}}({{\beta }_{1}}) \ldots {{r}_{1}}({{\beta }_{{n - 3}}})\), …, \({{F}_{{n - 1}}}({{\beta }_{1}}){{f}_{1}}(\alpha )\), \({{F}_{n}}(\alpha ){{f}_{n}}(\alpha )\). The considered system on the tangent bundle TMn\(\{ {{z}_{n}}, \ldots ,{{z}_{1}};\alpha ,{{\beta }_{1}}, \ldots ,{{\beta }_{{n - 1}}}\} \) becomes

$$\begin{gathered} {{\alpha }^{ \bullet }} = {{z}_{n}}{{f}_{n}}(\alpha ), \\ z_{n}^{ \bullet } = {{F}_{n}}(\alpha ){{f}_{n}}(\alpha ) - {{f}_{n}}(\alpha )[\Gamma _{{\alpha \alpha }}^{\alpha }(\alpha ,\beta ) + D{{f}_{n}}(\alpha )]z_{n}^{2} \\ \, - \frac{{f_{1}^{2}(\alpha )}}{{{{f}_{n}}(\alpha )}}\Gamma _{{11}}^{\alpha }(\alpha ,\beta )z_{{n - 1}}^{2} \\ - \frac{{f_{2}^{2}(\alpha )}}{{{{f}_{n}}(\alpha )}}g_{1}^{2}({{\beta }_{1}})\Gamma _{{22}}^{\alpha }(\alpha ,\beta )z_{{n - 2}}^{2} \\ - ... - \frac{{f_{{n - 1}}^{2}(\alpha )}}{{{{f}_{n}}(\alpha )}}g_{{n - 2}}^{2}({{\beta }_{1}})h_{{n - 3}}^{2}({{\beta }_{2}})...i_{1}^{2}({{\beta }_{{n - 2}}})\Gamma _{{n - 1,n - 1}}^{\alpha }(\alpha ,\beta )z_{1}^{2}, \\ \end{gathered} $$
$$\begin{gathered} z_{{n - 1}}^{ \bullet } = {{F}_{{n - 1}}}({{\beta }_{1}}){{f}_{1}}(\alpha ) - {{f}_{n}}(\alpha )[2\Gamma _{{\alpha 1}}^{1}(\alpha ,\beta ) + D{{f}_{1}}(\alpha )]{{z}_{{n - 1}}}{{z}_{n}} \\ - \frac{{f_{2}^{2}(\alpha )}}{{{{f}_{1}}(\alpha )}}g_{1}^{2}({{\beta }_{1}})\Gamma _{{22}}^{1}(\alpha ,\beta )z_{{n - 2}}^{2} - ... \\ - \frac{{f_{{n - 1}}^{2}(\alpha )}}{{{{f}_{1}}(\alpha )}}g_{{n - 2}}^{2}({{\beta }_{1}})h_{{n - 3}}^{2}({{\beta }_{2}})...i_{1}^{2}({{\beta }_{{n - 2}}})\Gamma _{{n - 1,n - 1}}^{1}(\alpha ,\beta )z_{1}^{2}, \\ ..........................................................................., \\ \end{gathered} $$
$$\begin{gathered} z_{2}^{ \bullet } = {{F}_{2}}({{\beta }_{{n - 2}}}){{f}_{{n - 2}}}(\alpha ){{g}_{{n - 3}}}({{\beta }_{1}})...{{r}_{1}}({{\beta }_{{n - 3}}}) \\ - \,{{f}_{n}}(\alpha )[2\Gamma _{{\alpha ,n - 2}}^{{n - 2}}(\alpha ,\beta ) + D{{f}_{{n - 2}}}(\alpha )]{{z}_{2}}{{z}_{n}} \\ - {{f}_{1}}(\alpha )[2\Gamma _{{1,n - 2}}^{{n - 2}}(\alpha ,\beta ) + D{{g}_{{n - 3}}}({{\beta }_{1}})]{{z}_{2}}{{z}_{{n - 1}}} - ... \\ - {{f}_{{n - 3}}}(\alpha ){{g}_{{n - 4}}}({{\beta }_{1}}){{h}_{{n - 5}}}({{\beta }_{2}})...{{s}_{1}}({{\beta }_{{n - 4}}}) \\ \times \,[2\Gamma _{{n - 3,n - 2}}^{{n - 2}}(\alpha ,\beta ) + D{{r}_{1}}({{\beta }_{{n - 3}}})]{{z}_{2}}{{z}_{3}} \\ - \frac{{f_{{n - 1}}^{2}(\alpha )}}{{{{f}_{{n - 2}}}(\alpha )}}\frac{{g_{{n - 2}}^{2}({{\beta }_{1}})}}{{{{g}_{{n - 3}}}({{\beta }_{1}})}}\frac{{h_{{n - 3}}^{2}({{\beta }_{2}})}}{{{{h}_{{n - 4}}}({{\beta }_{2}})}}...\frac{{r_{2}^{2}({{\beta }_{{n - 3}}})}}{{{{r}_{1}}({{\beta }_{{n - 3}}})}} \\ \times \,i_{1}^{2}({{\beta }_{{n - 2}}})\Gamma _{{n - 1,n - 1}}^{{n - 2}}(\alpha ,\beta )z_{1}^{2}, \\ \end{gathered} $$
(20)
$$\begin{gathered} z_{1}^{ \bullet } = {{F}_{1}}({{\beta }_{{n - 1}}}){{f}_{{n - 1}}}(\alpha ){{g}_{{n - 2}}}({{\beta }_{1}})...{{i}_{1}}({{\beta }_{{n - 2}}}) \\ - \,{{f}_{n}}(\alpha )[2\Gamma _{{\alpha ,n - 1}}^{{n - 1}}(\alpha ,\beta ) + D{{f}_{{n - 1}}}(\alpha )]{{z}_{1}}{{z}_{n}} \\ - \,{{f}_{1}}(\alpha )[2\Gamma _{{1,n - 1}}^{{n - 1}}(\alpha ,\beta ) + D{{g}_{{n - 2}}}({{\beta }_{1}})]{{z}_{1}}{{z}_{{n - 1}}} \\ - \,{{f}_{2}}(\alpha ){{g}_{1}}({{\beta }_{1}})[2\Gamma _{{2,n - 1}}^{{n - 1}}(\alpha ,\beta ) + D{{h}_{{n - 3}}}({{\beta }_{2}})]{{z}_{1}}{{z}_{{n - 2}}} \\ - ... - {{f}_{{n - 2}}}(\alpha ){{g}_{{n - 3}}}({{\beta }_{1}}){{h}_{{n - 4}}}({{\beta }_{2}})...{{r}_{1}}({{\beta }_{{n - 3}}}) \\ \times \,[2\Gamma _{{n - 2,n - 1}}^{{n - 1}}(\alpha ,\beta ) + D{{i}_{1}}({{\beta }_{{n - 2}}})]{{z}_{1}}{{z}_{2}}, \\ \end{gathered} $$
$$\begin{gathered} \beta _{1}^{ \bullet } = {{z}_{{n - 1}}}{{f}_{1}}(\alpha ),\quad \beta _{2}^{ \bullet } = {{z}_{{n - 2}}}{{f}_{2}}(\alpha ){{g}_{1}}({{\beta }_{1}}),..., \\ \beta _{{n - 1}}^{ \bullet } = {{z}_{1}}{{f}_{{n - 1}}}(\alpha ){{g}_{{n - 2}}}({{\beta }_{1}}){{h}_{{n - 3}}}({{\beta }_{2}})...{{i}_{1}}({{\beta }_{{n - 2}}}), \\ \end{gathered} $$

and it is almost everywhere equivalent to the system

$$\begin{gathered} {{\alpha }^{{ \bullet \bullet }}} - {{F}_{n}}(\alpha )f_{n}^{2}(\alpha ) + \Gamma _{{\alpha \alpha }}^{\alpha }(\alpha ,\beta )\alpha _{{}}^{{ \bullet 2}} \\ + \,\Gamma _{{11}}^{\alpha }(\alpha ,\beta )\beta _{1}^{{ \bullet 2}} + \ldots + \Gamma _{{n - 1,n - 1}}^{\alpha }(\alpha ,\beta )\beta _{{n - 1}}^{{ \bullet 2}} = 0, \\ \beta _{1}^{{ \bullet \bullet }} - {{F}_{{n - 1}}}({{\beta }_{1}})f_{1}^{2}(\alpha ) + 2\Gamma _{{\alpha 1}}^{1}(\alpha ,\beta ){{\alpha }^{ \bullet }}\beta _{1}^{ \bullet } \\ + \,\Gamma _{{22}}^{1}(\alpha ,\beta )\beta _{2}^{{ \bullet 2}} + \ldots + \Gamma _{{n - 1,n - 1}}^{1}(\alpha ,\beta )\beta _{{n - 1}}^{{ \bullet 2}} = 0, \\ \end{gathered} $$
$$ \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots ,$$
$$\begin{gathered} \beta _{{n - 2}}^{{ \bullet \bullet }} - {{F}_{2}}({{\beta }_{{n - 2}}})f_{{n - 2}}^{2}(\alpha )g_{{n - 3}}^{2}({{\beta }_{1}}) \ldots r_{1}^{2}({{\beta }_{{n - 3}}}) \\ + \,2\Gamma _{{\alpha ,n - 2}}^{{n - 2}}(\alpha ,\beta ){{\alpha }^{ \bullet }}\beta _{{n - 2}}^{ \bullet } + 2\Gamma _{{1,n - 2}}^{{n - 2}}(\alpha ,\beta )\beta _{1}^{ \bullet }\beta _{{n - 2}}^{ \bullet } \\ + \,\, \ldots + 2\Gamma _{{n - 3,n - 2}}^{{n - 2}}(\alpha ,\beta )\beta _{{n - 3}}^{ \bullet }\beta _{{n - 2}}^{ \bullet } + \Gamma _{{n - 1,n - 1}}^{{n - 2}}(\alpha ,\beta )\beta _{{n - 1}}^{{ \bullet 2}} = 0, \\ \beta _{{n - 1}}^{{ \bullet \bullet }} - {{F}_{1}}({{\beta }_{{n - 1}}})f_{{n - 1}}^{2}(\alpha )g_{{n - 2}}^{2}({{\beta }_{1}}) \ldots i_{1}^{2}({{\beta }_{{n - 2}}}) \\ \, + 2\Gamma _{{\alpha ,n - 1}}^{{n - 1}}(\alpha ,\beta ){{\alpha }^{ \bullet }}\beta _{{n - 1}}^{ \bullet } \\ + 2\Gamma _{{1,n - 1}}^{{n - 1}}(\alpha ,\beta )\beta _{1}^{ \bullet }\beta _{{n - 1}}^{ \bullet } + \ldots + 2\Gamma _{{n - 2,n - 1}}^{{n - 1}}(\alpha ,\beta )\beta _{{n - 2}}^{ \bullet }\beta _{{n - 1}}^{ \bullet } = 0. \\ \end{gathered} $$

Proposition 7. If the conditions of Proposition 2 are satisfied, then system (20) has a smooth first integral of the form

$$\begin{gathered} \Phi _{1}^{{}}({{z}_{n}}, \ldots ,{{z}_{1}};\alpha ,\beta ) \\ = z_{1}^{2} + \ldots + z_{n}^{2} + V(\alpha ,\beta ) = C_{1}^{{}} = {\text{const}}, \\ \end{gathered} $$
(21)
$$\begin{gathered} V(\alpha ,\beta ) = {{V}_{n}}(\alpha ) + \sum\limits_{k = 1}^{n - 1} {{{V}_{{n - k}}}({{\beta }_{k}})} \\ = - 2\int\limits_{{{\alpha }_{0}}}^\alpha {{{F}_{n}}(a)da} - 2\sum\limits_{k = 1}^{n - 1} {\int\limits_{{{\beta }_{{k0}}}}^{{{\beta }_{k}}} {{{F}_{{n - k}}}(b)db} } . \\ \end{gathered} $$

The following assertions hold in a more general form, but we state then as follows.

Proposition 8. Suppose that \({{F}_{{n - k}}}({{\beta }_{k}}) \equiv 0\), k = 1, …, n – 1. If the conditions of Propositions 3, …, 5 hold, then system (20) has n smooth first integrals of form (13), (16), …, (18), (19).

Theorem 2. Suppose that \({{F}_{{n - k}}}({{\beta }_{k}}) \equiv 0\), k = 1, …, n – 1. If the conditions of Propositions 2, …, 5 hold, then system (20) has n + 1 independent first integrals of form (21), (13), (16), …, (18), (19).

The fact that, under certain conditions, the complete set consists of n + 1, rather than 2n – 1, first integrals will be shown below.

3 INTEGRATION OF EQUATIONS OF MOTION IN A FORCE FIELD WITH DISSIPATION

Now slightly modifying (20) with conditions (10)–(12), (14), (15), …, (17) and \({{F}_{{n - k}}}({{\beta }_{k}}) \equiv 0\), k = 1, …, n – 1, we obtain a system with alternating dissipation. Its presence is characterized not only by the coefficient \(b\delta (\alpha )\), \(b > 0\), in the first equation in (22), but also by the following dependence of the (external) force field in the projections onto the \(z_{k}^{ \bullet }\) axes, k = 1, …, n, respectively: \({{z}_{1}}F_{{}}^{1}(\alpha )\), …, \({{z}_{{n - 1}}}F_{{}}^{1}(\alpha )\), \({{F}_{n}}(\alpha ){{f}_{n}}(\alpha ) + {{z}_{n}}F_{n}^{1}(\alpha )\). The considered system on the tangent bundle TMn\(\{ {{z}_{n}}, \ldots ,{{z}_{1}};\alpha ,{{\beta }_{1}}, \ldots ,{{\beta }_{{n - 1}}}\} \) becomes

$$\begin{gathered} {{\alpha }^{ \bullet }} = {{z}_{n}}{{f}_{n}}(\alpha ) + b\delta (\alpha ), \\ z_{n}^{ \bullet } = {{F}_{n}}(\alpha ){{f}_{n}}(\alpha ) - {{f}_{n}}(\alpha )[\Gamma _{{\alpha \alpha }}^{\alpha }(\alpha ,\beta ) \\ \, + D{{f}_{n}}(\alpha )]z_{n}^{2} - \frac{{{{f}^{2}}(\alpha )}}{{{{f}_{n}}(\alpha )}}\Gamma _{{11}}^{\alpha }(\alpha ,\beta )z_{{n - 1}}^{2} \\ - \,\,\frac{{f_{{}}^{2}(\alpha )}}{{{{f}_{n}}(\alpha )}}[g_{{}}^{2}({{\beta }_{1}})\Gamma _{{22}}^{\alpha }(\alpha ,\beta )z_{{n - 2}}^{2} \\ + \,\, \ldots + \,g_{{}}^{2}({{\beta }_{1}})h_{{}}^{2}({{\beta }_{2}}) \ldots i_{{}}^{2}({{\beta }_{{n - 2}}})\Gamma _{{n - 1,n - 1}}^{\alpha }(\alpha ,\beta )z_{1}^{2}] + {{z}_{n}}F_{n}^{1}(\alpha ), \\ z_{{n - 1}}^{ \bullet } = - {{f}_{n}}(\alpha )[2\Gamma _{1}^{{}}(\alpha ) + D{{f}_{{}}}(\alpha )]{{z}_{{n - 1}}}{{z}_{n}} \\ \, - f(\alpha )g_{{}}^{2}({{\beta }_{1}})\Gamma _{{22}}^{1}(\alpha ,\beta )z_{{n - 2}}^{2} \\ - \,\, \ldots - f(\alpha )g_{{}}^{2}({{\beta }_{1}})h_{{}}^{2}({{\beta }_{2}}) \ldots i_{{}}^{2}({{\beta }_{{n - 2}}}) \\ \times \,\Gamma _{{n - 1,n - 1}}^{1}(\alpha ,\beta )z_{1}^{2} + {{z}_{{n - 1}}}F_{{}}^{1}(\alpha ), \\ \end{gathered} $$
$$ \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots ,$$
(22)
$$\begin{gathered} z_{2}^{ \bullet } = - {{f}_{n}}(\alpha )[2\Gamma _{1}^{{}}(\alpha ) + Df(\alpha )]{{z}_{2}}{{z}_{n}} \\ - \,{{f}_{{}}}(\alpha )[2\Gamma _{2}^{{}}({{\beta }_{1}}) + Dg({{\beta }_{1}})]{{z}_{2}}{{z}_{{n - 1}}} \\ - \,\, \ldots - {{f}_{{}}}(\alpha ){{g}_{{}}}({{\beta }_{1}}){{h}_{{}}}({{\beta }_{2}}) \ldots s({{\beta }_{{n - 4}}}) \\ \times \,[2\Gamma _{{n - 2}}^{{}}({{\beta }_{{n - 3}}}) + Dr({{\beta }_{{n - 3}}})]{{z}_{2}}{{z}_{3}} \\ - f(\alpha )g({{\beta }_{1}})h({{\beta }_{2}}) \ldots r({{\beta }_{{n - 3}}})i_{{}}^{2}({{\beta }_{{n - 2}}}) \\ \times \,\Gamma _{{n - 1,n - 1}}^{{n - 2}}(\alpha ,\beta )z_{1}^{2} + {{z}_{2}}F_{{}}^{1}(\alpha ), \\ z_{1}^{ \bullet } = - {{f}_{n}}(\alpha )[2\Gamma _{1}^{{}}(\alpha ) + Df(\alpha )]{{z}_{1}}{{z}_{n}} \\ - \,f(\alpha )[2\Gamma _{2}^{{}}({{\beta }_{1}}) + Dg({{\beta }_{1}})]{{z}_{1}}{{z}_{{n - 1}}} \\ - f(\alpha )g({{\beta }_{1}})[2\Gamma _{3}^{{}}({{\beta }_{2}}) + Dh({{\beta }_{2}})]{{z}_{1}}{{z}_{{n - 2}}} \\ - \,\, \ldots - f(\alpha )g({{\beta }_{1}})h({{\beta }_{2}}) \ldots r({{\beta }_{{n - 3}}}) \\ \times \,[2\Gamma _{{n - 1}}^{{}}({{\beta }_{{n - 2}}}) + Di({{\beta }_{{n - 2}}})]{{z}_{1}}{{z}_{2}} + {{z}_{1}}F_{{}}^{1}(\alpha ), \\ \beta _{1}^{ \bullet } = z_{{n - 1}}^{{}}f(\alpha ),\;\beta _{2}^{ \bullet } = z_{{n - 2}}^{{}}f(\alpha )g({{\beta }_{1}}),\; \ldots , \\ \beta _{{n - 1}}^{ \bullet } = z_{1}^{{}}f(\alpha )g({{\beta }_{1}})h({{\beta }_{2}}) \ldots i({{\beta }_{{n - 2}}}), \\ \end{gathered} $$

and it is almost everywhere equivalent to a system for the second derivatives of α and β with explicitly extracted alternating dissipation [2, 3].

Now we integrate the 2nth-order system (22) with conditions (11) and

$$\begin{gathered} \Gamma _{{11}}^{\alpha }(\alpha ,\beta ) \equiv \Gamma _{{22}}^{\alpha }(\alpha ,\beta )g_{{}}^{2}({{\beta }_{1}}) \\ \equiv \ldots \equiv \Gamma _{{n - 1,n - 1}}^{\alpha }(\alpha ,\beta )g_{{}}^{2}({{\beta }_{1}})h_{{}}^{2}({{\beta }_{2}}) \ldots = \Gamma _{n}^{{}}(\alpha ). \\ \end{gathered} $$

Assume that the function \({{f}_{n}}(\alpha )\) satisfies the first equality in (7). Additionally (by analogy with (11)) \(f(\alpha )\) is assumed to satisfy the following transformed equality from (7):

$$f_{n}^{2}(\alpha )[2\Gamma _{1}^{{}}(\alpha ) + Df(\alpha )] + \Gamma _{n}^{{}}(\alpha )f_{{}}^{2}(\alpha ) \equiv 0.$$

In this case, an independent subsystem of order 2n – 1 decouples, namely,

$$\begin{gathered} {{\alpha }^{ \bullet }} = {{z}_{n}}{{f}_{n}}(\alpha ) + b\delta (\alpha ), \\ z_{n}^{ \bullet } = {{F}_{n}}(\alpha ){{f}_{n}}(\alpha ) \\ \, - \frac{{{{f}^{2}}(\alpha )}}{{{{f}_{n}}(\alpha )}}{{\Gamma }_{n}}(\alpha )(z_{{n - 1}}^{2} + \ldots + z_{1}^{2}) + {{z}_{n}}F_{n}^{1}(\alpha ), \\ z_{{n - 1}}^{ \bullet } = - {{f}_{n}}(\alpha )[2\Gamma _{1}^{{}}(\alpha ) + D{{f}_{{}}}(\alpha )]{{z}_{{n - 1}}}{{z}_{n}} \\ - \,f(\alpha )g_{{}}^{2}({{\beta }_{1}})\Gamma _{{22}}^{1}({{\beta }_{1}})z_{{n - 2}}^{2} \\ - \,\, \ldots - f(\alpha )g_{{}}^{2}({{\beta }_{1}})h_{{}}^{2}({{\beta }_{2}}) \ldots i_{{}}^{2}({{\beta }_{{n - 2}}}) \\ \times \,\Gamma _{{n - 1,n - 1}}^{1}({{\beta }_{1}}, \ldots ,{{\beta }_{{n - 2}}})z_{1}^{2} + {{z}_{{n - 1}}}F_{{}}^{1}(\alpha ), \\ \end{gathered} $$
$$ \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots ,$$
$$\begin{gathered} z_{2}^{ \bullet } = - {{f}_{n}}(\alpha )[2\Gamma _{1}^{{}}(\alpha ) + Df(\alpha )]{{z}_{2}}{{z}_{n}} \\ \, - {{f}_{{}}}(\alpha )[2\Gamma _{2}^{{}}({{\beta }_{1}}) + Dg({{\beta }_{1}})]{{z}_{2}}{{z}_{{n - 1}}} \\ - \,\, \ldots - {{f}_{{}}}(\alpha ){{g}_{{}}}({{\beta }_{1}}){{h}_{{}}}({{\beta }_{2}}) \ldots s({{\beta }_{{n - 4}}}) \\ \times \,[2\Gamma _{{n - 2}}^{{}}({{\beta }_{{n - 3}}}) + Dr({{\beta }_{{n - 3}}})]{{z}_{2}}{{z}_{3}} \\ - f(\alpha )g({{\beta }_{1}})h({{\beta }_{2}}) \ldots r({{\beta }_{{n - 3}}})i_{{}}^{2}({{\beta }_{{n - 2}}}) \\ \times \,\Gamma _{{n - 1,n - 1}}^{{n - 2}}({{\beta }_{1}}, \ldots ,{{\beta }_{{n - 2}}})z_{1}^{2} + {{z}_{2}}F_{{}}^{1}(\alpha ), \\ z_{1}^{ \bullet } = - {{f}_{n}}(\alpha )[2\Gamma _{1}^{{}}(\alpha ) + Df(\alpha )]{{z}_{1}}{{z}_{n}} \\ - \,f(\alpha )[2\Gamma _{2}^{{}}({{\beta }_{1}}) + Dg({{\beta }_{1}})]{{z}_{1}}{{z}_{{n - 1}}} \\ - f(\alpha )g({{\beta }_{1}})[2\Gamma _{3}^{{}}({{\beta }_{2}}) + Dh({{\beta }_{2}})]{{z}_{1}}{{z}_{{n - 2}}} \\ - \,\, \ldots - f(\alpha )g({{\beta }_{1}})h({{\beta }_{2}}) \ldots r({{\beta }_{{n - 3}}}) \\ \times \,[2\Gamma _{{n - 1}}^{{}}({{\beta }_{{n - 2}}}) + Di({{\beta }_{{n - 2}}})]{{z}_{1}}{{z}_{2}} + {{z}_{1}}F_{{}}^{1}(\alpha ), \\ \beta _{1}^{ \bullet } = z_{{n - 1}}^{{}}f(\alpha ),\quad \beta _{2}^{ \bullet } = z_{{n - 2}}^{{}}f(\alpha )g({{\beta }_{1}}),\; \ldots , \\ \beta _{{n - 1}}^{ \bullet } = z_{1}^{{}}f(\alpha )g({{\beta }_{1}})h({{\beta }_{2}}) \ldots i({{\beta }_{{n - 2}}}). \\ \end{gathered} $$

To integrate this system completely, we need to know, generally speaking, 2n – 1 independent first   integrals. However, after making the change of   variables \({{w}_{1}} = {{z}_{{n - 1}}}{\text{/}}\sqrt {z_{1}^{2} + \ldots + z_{{n - 2}}^{2}} \), …, wn – 3 = \({{z}_{3}}{\text{/}}\sqrt {z_{1}^{2} + z_{2}^{2}} \), \({{w}_{{n - 2}}} = {{z}_{2}}{\text{/}}{{z}_{1}}\), \({{w}_{{n - 1}}} = \sqrt {z_{1}^{2} + \ldots + z_{{n - 1}}^{2}} \), \({{w}_{n}} = {{z}_{n}}\), the last system splits into

$$\begin{gathered} {{\alpha }^{ \bullet }} = {{w}_{n}}{{f}_{n}}(\alpha ) + b\delta (\alpha ), \\ w_{n}^{ \bullet } = {{F}_{n}}(\alpha ){{f}_{n}}(\alpha ) - \frac{{{{f}^{2}}(\alpha )}}{{{{f}_{n}}(\alpha )}}\Gamma _{n}^{{}}(\alpha )w_{{n - 1}}^{2} + {{w}_{n}}F_{n}^{1}(\alpha ), \\ w_{{n - 1}}^{ \bullet } = \frac{{{{f}^{2}}(\alpha )}}{{{{f}_{n}}(\alpha )}}\Gamma _{n}^{{}}(\alpha ){{w}_{{n - 1}}}{{w}_{n}} + {{w}_{{n - 1}}}F_{{}}^{1}(\alpha ), \\ \end{gathered} $$
(23)
$$\begin{gathered} w_{s}^{ \bullet } = ( \pm ){{w}_{{n - 1}}}\sqrt {1 + w_{s}^{2}} f(\alpha )g({{\beta }_{1}}) \ldots [2\Gamma _{{s + 1}}^{{}}({{\beta }_{s}}) + Dj({{\beta }_{s}})], \\ \beta _{s}^{ \bullet } = ( \pm )\frac{{{{w}_{s}}{{w}_{{n - 1}}}}}{{\sqrt {1 + w_{s}^{2}} }}f(\alpha )g({{\beta }_{1}}) \ldots ,\;s = 1, \ldots ,n - 2, \\ \end{gathered} $$
(24)
$$\beta _{{n - 1}}^{ \bullet } = ( \pm )\frac{{{{w}_{{n - 1}}}}}{{\sqrt {1 + w_{{n - 2}}^{2}} }}f(\alpha )g({{\beta }_{1}})h({{\beta }_{2}}) \ldots i({{\beta }_{{n - 2}}}),$$
(25)

in system (24), the ellipsis denotes identical terms and \(j({{\beta }_{s}})\) is one of the functions g, h, …, depending on the corresponding angle \({{\beta }_{s}}\).

It can be seen that, for the complete integrability of system (23)–(25), it suffices to indicate two independent first integrals of system (23), one for each of the systems in (24) (with changed independent variables; there are n – 2 such systems), and an additional first integral “coupling” Eq. (25) (altogether n + 1 first integrals).

Assume also that, for some \(\kappa \in \) R, it is true that

$$\frac{{{{f}^{2}}\left( \alpha \right)}}{{f_{n}^{2}\left( \alpha \right)}}{{{{\Gamma}}}_{n}}\left( \alpha \right) = \kappa \frac{d}{{d\alpha }}\ln \left| {{{\Delta }}\left( \alpha \right)} \right|,\quad {{\Delta }}\left( \alpha \right) = \frac{{\delta \left( \alpha \right)}}{{{{f}_{n}}\left( \alpha \right)}},$$
(26)

and, for some \(\lambda _{n}^{0},~\lambda _{k}^{1} \in \) R,

$$\begin{gathered} {{F}_{n}}\left( \alpha \right) = \lambda _{n}^{0}\frac{d}{{d\alpha }}\frac{{{{{{\Delta }}}^{2}}\left( \alpha \right)}}{2}, \\ F_{k}^{1}\left( \alpha \right) = \lambda _{k}^{1}{{f}_{n}}\left( \alpha \right)\frac{d}{{d\alpha }}{{\Delta }}\left( \alpha \right),\quad k = 1, \ldots ,n. \\ \end{gathered} $$
(27)

Here, \(F_{1}^{1}(\alpha )\, = \ldots = \,F_{{n - 1}}^{1}(\alpha )\, = \,F_{{}}^{1}(\alpha )\), i.e., \(\lambda _{1}^{1}\, = \,...\, = \,\lambda _{{n - 1}}^{1}\) = λ1. Condition (26) will be referred to as geometric and conditions of group (27), as energy conditions.

Condition (26) is called geometric, because, among other things, it imposes a constraint on \(\Gamma _{n}^{{}}(\alpha )\) such that the corresponding coefficients of the system are reduced to a homogeneous form with respect to the function Δ(α). The conditions of group (27) are called energy ones, because, among other things, the forces become, in a sense, “potential” with respect to the functions \({{{{\Delta }}}^{2}}\left( \alpha \right){\text{/}}2\) and Δ(α), so that the corresponding coefficients of the system are reduced to a homogeneous form (again with respect to Δ(α)). Moreover, it is the function Δ(α) that introduces alternating dissipation into the system.

Theorem 3. Suppose that conditions (26) and (27) hold. Then system (23)–(25) has a complete set of (n + 1) independent, generally transcendental [10, 11] first integrals.

In the general case, the first integrals have a cumbersome form (since the Abel equation has to be integrated [12]). Specifically, if \(\kappa = - 1\) and \(\lambda _{n}^{1} = {{\lambda }^{1}}\), then an explicit expression for the key first integral is given by

$$\begin{gathered} {{{{\Theta }}}_{1}}({{w}_{n}},{{w}_{{n - 1}}};\alpha ) = {{G}_{1}}\left( {\frac{{{{w}_{n}}}}{{\Delta (\alpha )}},\frac{{{{w}_{{n - 1}}}}}{{\Delta (\alpha )}}} \right) \\ = \frac{{f_{n}^{2}(\alpha )(w_{n}^{2}\, + \,w_{{n - 1}}^{2})\, + \,(b\, - \,{{\lambda }^{1}}){{w}_{n}}\delta (\alpha ){{f}_{n}}(\alpha )\, - \,\lambda _{n}^{0}{{\delta }^{2}}(\alpha )}}{{{{w}_{{n - 1}}}\delta (\alpha ){{f}_{n}}(\alpha )}}\, = \,{{C}_{1}}. \\ \end{gathered} $$
(28)

An additional first integral of system (23) has the following structural form:

$$\begin{gathered} {{{{\Theta }}}_{2}}\left( {{{w}_{n}},{{w}_{{n - 1}}};\alpha } \right) \\ = {{G}_{2}}\left( {\Delta \left( \alpha \right),\frac{{{{w}_{n}}}}{{\Delta \left( \alpha \right)}},\frac{{{{w}_{{n - 1}}}}}{{\Delta \left( \alpha \right)}}} \right) = {{C}_{2}} = {\text{const}}{\text{.}} \\ \end{gathered} $$
(29)

The first integrals for systems (24) are given by

$$\begin{gathered} {{\Theta }_{{s + 2}}}({{w}_{s}};{{\beta }_{s}}) = \frac{{\sqrt {1 + w_{s}^{2}} }}{{{{\Psi }_{s}}({{\beta }_{s}})}} = {{C}_{{s + 2}}} = {\text{const}}, \\ s = 1, \ldots ,n--2, \\ \end{gathered} $$
(30)

for the functions \({{\Psi }_{s}}({{\beta }_{s}})\), s = 1, …, n – 2 (see (16), …, (18)). An additional first integral coupling Eq. (25) is found by analogy with (19):

$$\begin{gathered} \Theta _{{n + 1}}^{{}}({{\beta }_{{n - 2}}},{{\beta }_{{n - 1}}}) = {{\beta }_{{n - 1}}} \\ \pm \int\limits_{{{\beta }_{{n - 2,0}}}}^{{{\beta }_{{n - 2}}}} {\frac{{i(b)}}{{\sqrt {C_{n}^{2}\Psi _{{n - 2}}^{2}(b) - 1} }}db} = C_{{n + 1}}^{{}} = {\text{const}}, \\ \end{gathered} $$
(31)

where, after the integral in (31) has been evaluated, the constant Cn can be formally replaced by the left-hand sides of (30) for s = n – 2.

The expressions for the first integrals (28)–(31) as finite combinations of elementary functions depend not only on the computed quadratures, but also on the explicit form of the function \(\Delta \left( \alpha \right)\). Indeed, for \(\kappa = - 1\) and \(\lambda _{n}^{1} = {{\lambda }^{1}}\), an additional first integral of system (23) can be found from the quadrature

$$\begin{gathered} d\ln \left| {\Delta \left( \alpha \right)} \right| = \frac{{\left( {b + {{u}_{n}}} \right)d{{u}_{4}}}}{{2W\left( {{{u}_{n}}} \right) - {{C}_{1}}\{ {{C}_{1}} \pm \sqrt {C_{1}^{2} - 4W\left( {{{u}_{n}}} \right)} \} {\text{/}}2}}, \\ W\left( {{{u}_{n}}} \right) = u_{n}^{2} + (b - {{\lambda }^{1}}){{u}_{n}} - \lambda _{n}^{0},\quad {{u}_{n}} = \frac{{{{w}_{n}}}}{{\Delta \left( \alpha \right)}}. \\ \end{gathered} $$

Moreover, after the integration, C1 can be replaced by the left-hand side of (28). The right-hand side of this quadrature is expressed as a finite combination of elementary functions, while the left-hand side is expressed depending on the function \(\Delta \left( \alpha \right)\).

The following result holds, which is, in a sense, converse to Theorem 3.

Theorem 4. Conditions (26) and (27) (e.g., for \(\kappa = - 1\), \(\lambda _{n}^{1} = {{\lambda }^{1}}\)) are necessary conditions for the existence of the key first integral (28) for system (23)–(25).

4 STRUCTURE OF FIRST INTEGRALS FOR SYSTEMS WITH DISSIPATION

If α is a periodic coordinate with period 2π, then, under the conditions of Theorem 3, system (23)–(25) becomes a dynamical system having alternating dissipation with zero mean [2, 13]. Moreover, for \(b = - {{\lambda }^{1}}\), it turns into a conservative system with the following smooth first integrals:

$$\begin{gathered} {{{{\Phi }}}_{1}}\left( { - b;{{w}_{n}},{{w}_{{n - 1}}};\alpha } \right) \\ = w_{{n - 1}}^{2} + w_{n}^{2} + 2b{{w}_{n}}{{\Delta }}\left( \alpha \right) - \lambda _{n}^{0}{{{{\Delta }}}^{2}}\left( \alpha \right) = {\text{const}}, \\ \end{gathered} $$
(32)
$${{{{\Phi }}}_{2}}\left( {{{w}_{{n - 1}}};\alpha } \right) = {{w}_{{n - 1}}}{{\Delta }}\left( \alpha \right) = {\text{const}}{\text{.}}$$
(33)

Obviously, the ratio of two first integrals (32) and (33) is also a first integral of system (23)–(25) for \(b = - {{\lambda }^{1}}\). However, for \(b \ne - {{\lambda }^{1}}\), each of the functions

$$\begin{gathered} {{{{\Phi }}}_{1}}({{\lambda }^{1}};{{w}_{n}},{{w}_{{n - 1}}};\alpha ) \\ = w_{{n - 1}}^{2} + w_{n}^{2} + (b - {{\lambda }^{1}}){{w}_{n}}{{\Delta }}\left( \alpha \right) - \lambda _{n}^{0}{{{{\Delta }}}^{2}}\left( \alpha \right) \\ \end{gathered} $$
(34)

and (33) taken separately is not a first integral of system (23)–(25). Nevertheless, the ratio of functions (34) and (33) is a first integral of system (23)–(25) (at \(\kappa = - 1\) and \(\lambda _{n}^{1} = {{\lambda }^{1}}\)) for any b.

As was noted above, for dissipative systems of any order, transcendence of functions (in the sense of the presence of essential singularities) as first integrals is inherited from the existence of attracting or repelling limit sets in the system [14, 15].

CONCLUSIONS: SYSTEMS ON THE BUNDLE OF A FINITE-DIMENSIONAL SPHERE AND APPLICATIONS

Earlier, we identified, as examples, two classes of manifolds (multidimensional surfaces of revolution and Lobachevsky spaces) to which the proposed technique of integrating dissipative systems is applicable. Now we note a one-parameter family of functions \(f\left( \alpha \right)\) and \({{f}_{n}}\left( \alpha \right)\) defining a metric on the finite-dimensional sphere:

$$\begin{gathered} f\left( \alpha \right) = \frac{{{\text{cos}}\alpha }}{{{\text{sin}}\alpha \sqrt {1 + {{\mu }_{1}}{\text{si}}{{{\text{n}}}^{2}}\alpha } }}, \\ {{\mu }_{1}} \in {\mathbf{R}},\quad {{f}_{n}}\left( \alpha \right) \equiv - 1; \\ \end{gathered} $$

moreover, we distinguish between two important subcases:

$${{\mu }_{1}} = 0,$$
(35)
$${{\mu }_{1}} = - 1.$$
(36)

Case (35) forms a class of systems corresponding to a dynamically symmetric (n + 1)-dimensional rigid body moving at zero levels of cyclic integrals in a generally nonconservative force field in the case when the field additionally depends on (second-rank tensor of) angular velocity [2, 13]. Case (36) forms a class of systems corresponding to the motion of a point on an n-dimensional sphere with the natural metric induced by the metric of the ambient (n + 1)-dimensional Euclidean space. In particular, for \(\delta \left( \alpha \right) = {{F}_{n}}\left( \alpha \right) \equiv 0,\) the considered system describes a geodesic flow on the n-dimensional sphere. In the case of (35), if δ(α) = \({{F}_{n}}(\alpha ){{f}_{n}}(\alpha ){\text{/cos}}\alpha \), then the system describes the motion of an (n + 1)-dimensional rigid body in the force field \({{F}_{n}}\left( \alpha \right){{f}_{n}}\left( \alpha \right)\) under the action of a follower force [2, 3]. Specifically, if \({{F}_{n}}\left( \alpha \right) = {\text{sin}}\alpha {\text{cos}}\alpha \) and \(\delta \left( \alpha \right) = {\text{sin}}\alpha \), then the system describes a generalized spherical pendulum placed in a material flow in (n + 1)-dimensional space and has a complete set of transcendental first integrals that can be expressed as a finite combination of elementary functions [2, 3, 13, 14].

If the function \(\delta \left( \alpha \right)\) is not periodic, then the considered dissipative system has alternating dissipation with a nonzero mean (i.e., it is actually dissipative). Nevertheless, in this case (due to Theorems 3 and 4) closed-form expressions in terms of a finite combination of elementary functions can also be obtained for transcendental first integrals. This result also determines new nontrivial cases of closed-form integrability of dissipative dynamical systems on the tangent bundle of a smooth finite-dimensional manifold.