INTRODUCTION

This paper provides a survey of the research carried out for many years at the Department of Theoretical and Applied Mechanics of the Mathematics and Mechanics Faculty of St. Petersburg University devoted to the solution of the generalized Chebyshev problems. By the generalized Chebyshev problem we mean the problem in which the solution to the system of equations of motion must simultaneously satisfy an auxiliary system of higher-order (n\( \geqslant \) 3) differential equations. The novel class of control problems is therefore introduced. The generalized Chebyshev problem is considered an extension of the Chebyshev problem from the theory of synthesis of mechanisms, where it is necessary to construct a device whose links must perform the desired motions with a given accuracy. An example of such devices would be the well-known Chebyshev mechanisms with stops of certain links in the prescribed positions.

To solve the generalized Chebyshev problem, we propose applying two theories of motion of non-holonomic systems with the higher-order constraints developed at the Department of Theoretical and Applied Mechanics of St. Petersburg University. Here, the auxiliary system of differential equations is considered a set of higher-order program constraints whose reactions appear to be the sought control forces resolving the generalized Chebyshev problem. In the first theory we construct the joint system of differential equations for the unknown generalized coordinates and Lagrange multipliers. In the second theory we use the generalized Gauss principle. The application of the theories is demonstrated in the solution to the problem of the motion of an Earth satellite (spacecraft) with fixed acceleration.

In a later paper with the same title we will apply the second theory to solve one of the most important problems of the control theory on the transfer of a mechanical system from one phase state to another for a given time.

1 PROBLEM FORMULATION

The outstanding works of Chebyshev devoted to various areas of mathematics and mechanics are widely known [1]. In particular, he created the theory of synthesis of mechanisms [2], where he formulated the problem of creating such machines whose individual links must perform the prescribed motions with a given accuracy. For instance, among such devices we note the multilink mechanisms with stops of certain parts in prescribed positions.Footnote 1  We call such a problem the Chebyshev problem. We will propose the generalization of such a problem to the case when we require that the solution to the studied system satisfy an auxiliary system of higher-order differential equations.

Let us now formulate the following problem. Let the motion of a mechanical system under the generalized forces Q = (Q1, …, Qs) in the generalized coordinates q = (q1, …, qs) be described by the Lagrange equations of the second kind

$$\begin{gathered} \frac{d}{{dt}}\frac{{\partial T}}{{\partial {{{\dot {q}}}^{\sigma }}}} - \frac{{\partial T}}{{\partial {{q}^{\sigma }}}} = {{Q}_{\sigma }},\quad \sigma = \overline {1,s} , \\ T = \frac{M}{2}{{{\text{g}}}_{{\alpha \beta }}}{{{\dot {q}}}^{\alpha }}{{{\dot {q}}}^{\beta }},\quad \alpha ,\beta = \overline {0,s} ,\quad {{q}^{0}} = t,\quad {{{\dot {q}}}^{0}} = 1, \\ \end{gathered} $$
((1.1))

where M is the mass of the entire system.

We require that the motion of this mechanical system simultaneously satisfies the system of differential equations (with the order n\( \geqslant \) 3)

$$\begin{gathered} f_{n}^{\kappa } \equiv a_{{n\sigma }}^{{l + \kappa }}(t,q,\dot {q}, \ldots ,\mathop q\limits^{(n - 1)} )\mathop {{{q}^{\sigma }}}\limits^{(n)} + a_{{n0}}^{{l + \kappa }}(t,q,\dot {q}, \ldots ,\mathop q\limits^{(n - 1)} ) = 0, \\ \sigma = \overline {1,s} ,\quad \kappa = \overline {1,k} ,\quad k\;\leqslant \;s,\quad l = s - k, \\ \end{gathered} $$
((1.2))

where we have introduced the denotations of the type \(\mathop {{{q}^{\sigma }}}\limits^{(n)} \) = dnqσ/dtn. We call the formulated problem the generalized Chebyshev problem. It was already noted that the issue is made more complicated, compared to the above discussed Chebyshev problem from the synthesis of mechanisms, by the fact that we now need to find not only the motion of some links of the mechanism, but also the motion of the entire system satisfying the auxiliary system of higher-order differential equations (1.2).

To solve the formulated problem, we need to find the auxiliary forces R = (R1, …, Rs) which are a part of the closed problem formulation and appear in the right-hand side of Eqs. (1.1) in the form

$$\frac{d}{{dt}}\frac{{\partial T}}{{\partial {{{\dot {q}}}^{\sigma }}}} - \frac{{\partial T}}{{\partial {{q}^{\sigma }}}} = {{Q}_{\sigma }} + {{R}_{\sigma }},\quad \sigma = \overline {1,s} .$$
((1.3))

Thus, by the generalized Chebyshev problem we understand the solution of the joint system of Eqs. (1.2) and (1.3), where q = (q1, …, qs) and R = (R1, …, Rs) are unknown functions of time.

Note that the generalized Chebyshev problem is sometimes called the mixed problem of dynamics [4] following Grigoryan’s proposition, because it has the attributes of both direct and inverse problems of dynamics. Indeed, on the one side, we seek the motion under the given forces Q = (Q1, …, Qs), and, on  the other side, we seek the generalized forces R = (R1, …, Rs) under the given characteristics of motion (1.2).

We note the following circumstance. If the motion of a system is constrained by the holonomic constraints

$$f_{0}^{\kappa }(t,q) = 0,\quad \kappa = \overline {1,k} ,$$
((1.4))

by the first-order non-holonomic constraints

$$f_{1}^{\kappa }(t,q,\dot {q}) = 0,\quad \kappa = \overline {1,k} ,$$
((1.5))

or by the linear second-order non-holonomic constraints

$$\begin{gathered} f_{2}^{\kappa }(t,q,\dot {q},\ddot {q}) \equiv a_{{2\sigma }}^{{l + \kappa }}(t,q,\dot {q}){{{\ddot {q}}}^{\sigma }} + a_{{2,0}}^{{l + \kappa }}(t,q,\dot {q}) = 0, \\ \sigma = \overline {1,s} ,\quad \kappa = \overline {1,k} ,\quad l = s - k, \\ \end{gathered} $$
((1.6))

then, if these constraints are ideal, their reactions are as follows [5]

$${\mathbf{R}} = {{\Lambda }_{\kappa }}\nabla f_{0}^{\kappa },\quad {\mathbf{R}} = {{\Lambda }_{\kappa }}\nabla {\kern 1pt} '{\kern 1pt} f_{1}^{\kappa },\quad {\mathbf{R}} = {{\Lambda }_{\kappa }}\nabla {\kern 1pt} ''{\kern 1pt} f_{2}^{\kappa },\quad \kappa = \overline {1,k} .$$
((1.7))

The generalized Hamilton operators ∇' and ∇'' used in formulas (1.7) were introduced by N.N. Polya-khov [6, 7]. The classical Hamilton operator (the nabla operator) results from them as a special case when the holonomic constraints are imposed.

It is important that in the classical analytical mechanics the Lagrange multipliers Λκ, κ = \(\overline {1,k} \) for ideal constraints (1.4), (1.5), and (1.6) are determined as the known functions of variables t, q, and \(\dot {q}\) [5]

$${{\Lambda }_{\kappa }} = {{\Lambda }_{\kappa }}(t,q,\dot {q}),\quad \kappa = \overline {1,k} .$$
((1.8))

For the holonomic constraints the functions (1.8) were obtained at the beginning of the previous century by Lyapunov [8] and Suslov [9], and the non-holonomic constraints were first introduced in work [10] and then repeated in the first edition of textbook [11] (in 1985). After ten years, this result was obtained using other methods in the United States [12, 13], Italy [14], Poland [15], Sweden [16], and the Soviet Union [17, 18].

In contrast to that, if we treat the formulated generalized Chebyshev problem as a non-holonomic problem with (n\( \geqslant \) 3)th-order ideal linear non-holonomic constraints (1.2), then we have to seek the reactions of these constraints (which play the role of sought control forces from the point of view of the introduced novel class of the control theory problems) as unknown functions of time. It is clear that such set of these functions Rσ(t), σ = \(\overline {1,s} \) always exist, with whose action upon the mechanical system we may achieve not only the prescribed motion qσ(t), σ = \(\overline {1,s} \), but also the satisfaction of any their combinations together with the derivatives given in the form (1.2).

These considerations withdraw the question that arose for some well-known Soviet scientists with regard to the formulation of the generalized Chebyshev problem. The trouble is that, as Pars [19] and Rumyantsev [20] pointed out, the forces cannot depend on accelerations. Therefore, many researchers believed that the prescription of the higher-order constraints requires the prescription of the reaction forces of these constraints which depend not only on accelerations, but also on the higher derivatives of the generalized coordinates. However, as we saw, we have to consider these reaction forces in the generalized Chebyshev problem as well as the unknown functions of time to be determined simultaneously with the generalized coordinates.

To solve the formulated generalized Chebyshev problem, we constructed the two theories of motion of non-holonomic systems with higher-order constraints (see Sections 3 and 7).

2 ON THE MATHEMATICAL APPARATUS FOR SOLVING THE GENERALIZED CHEBYSHEV PROBLEMS

Textbook [11] (1985) for university students studying mathematics and mechanics played a key role in creating the theory of motion of non-holonomic systems with higher-order constraints. The writing of this textbook was initiated by the well-known mechanician Polyakhov (1906–1987). As befits a textbook intended for classical universities, it contained several recent scientific results of the authors (see, for instance, papers [4, 6, 7, 10, 21]). In the two central chapters of the textbook, it articulates the following in a novel manner: Motion under Constraints and Variational Principles of Mechanics. It is important that, based on the concept of the generalized Hamilton operator [6, 7], introduced by Polyakhov, they succeeded in presenting a uniform representation of the theory of motion of both holonomic and non-holonomic systems.

A natural extension of these chapters was the creation of the two theories of motion of non-holonomic systems with higher-order constraints, which are necessary for solving the above formulated generalized Chebyshev problem. These theories were presented in the doctoral thesis of Yushkov “Equations of motion of non-holonomic systems and variational principles of mechanics” and published in monography [22]. The theories were then developed and applied during the fruitful scientific cooperation between Yushkov and Zegzhda (1935–2015). In this paper we provide, in particular, a survey of their scientific papers and of the Bachelors, Masters, and PhD theses of their students [23–32]. Among these works we should note the doctoral thesis of Soltakhanov mentioned in his monography [33]. All of these studies were summarized in two monographs [5, 34]; the first was translated into Chinese [35] and the second one was published in English by the Springer publishing company [36]. The solution to practically important problems of one of the most important areas of the control theory was provided in monography [37] using the solution to the generalized Chebyshev problem. It is important that the St. Petersburg Publishing House is currently preparing the publication of a two-volume textbook for classical universities written by Polyakhov, Zegzhda, Tovstik, and Yushkov named Theoretical and Applied Mechanics, which will reflect the above-mentioned results.

We underline a crucial role in the idea of creating the theory of motion of non-holonomic systems with higher-order constraints of Polyakhov, who created the scientific school of analytical mechanics at the Department of Theoretical and Applied Mechanics. However, in this respect we should also mention the works of Novoselov (1926–2019) who graduated from the Mathematics and Mechanics Faculty in 1951 and after a year defended his thesis on the motion of systems with variable masses. He later published these investigations in his monography [38]. Further, Novoselov studied the issues of the non-holonomic mechanics and defended his PhD thesis in 1959 at Moscow University. His scientific works in analytical mechanics (see, e.g., [39, 40]) are highly regarded by mechanicians, and Polyakhov cites them on multiple occasions. From 1961 Novoselov directed the Department of Theoretical Astronomy (later called the Department of Celestial Mechanics) and from 1969 he headed the Department of Mechanics of Controlled Motion at the newly organized Faculty of Applied Mathematics and Control Processes of St. Petersburg University.

Various problems of analytical mechanics were studied at the department, for instance, the application of the Gauss principle in the dynamics of systems with random forces [41] or the application of the non-holonomic mechanics to the theory of electromechanical systems [42].

We proceed to present the first theory of motion of non-holonomic systems with higher-order constraints.

3 THE FIRST THEORY OF MOTION OF NON-HOLONOMIC SYSTEMS WITH HIGHER-ORDER CONSTRAINTS

Applying the apparatus of the non-holonomic mechanics to solve the formulated generalized Chebyshev problem, we will interpret the program of motion given by the system of differential Eqs. (1.2) of the order n\( \geqslant \) 3 as the program higher-order linear constraints and their reactions as the control forces providing program (1.2) to be followed. However, these forces are given by the control system generating the desired forces Rσ

$${{R}_{\sigma }} = {{\Lambda }_{\kappa }}b_{\sigma }^{\kappa },\quad \sigma = \overline {1,s} ,\quad \kappa = \overline {1,k} ,$$

where the coefficients \(b_{\sigma }^{\kappa }\) are prescribed and Λκ, κ = \(\overline {1,k} \) are the sought generalized control forces.

In the first theory of motion of non-holonomic systems with higher-order constraints, we assume the ideal character of the imposed constraints and compose the joint system of differential equations for the sought functions of time qσ, σ = \(\overline {1,s} \) and Λκ, κ = \(\overline {1,k} \):

$$\begin{gathered} {{{\ddot {q}}}^{\sigma }} = {{F}^{\sigma }}(t,q,\dot {q},\Lambda ),\quad \sigma = \overline {1,s} , \\ \mathop {{{\Lambda }_{\kappa }}}\limits^{(n - 2)} = C_{\kappa }^{n}(t,q,\dot {q},\Lambda ,\dot {\Lambda }, \ldots ,\mathop \Lambda \limits^{(n - 3)} ),\quad \kappa = \overline {1,k} ,\quad n\; \geqslant \;3. \\ \end{gathered} $$
((3.1))

The construction of the functions Fσ and \(C_{\kappa }^{n}\) is discussed in detail in monographs [5, 34]. To integrate system (3.1) we need to prescribe the initial conditions

$$\begin{gathered} {{\Lambda }_{\kappa }}({{t}_{0}}) = \Lambda _{\kappa }^{0},\,\,\,{{{\dot {\Lambda }}}_{\kappa }}({{t}_{0}}) = \dot {\Lambda }_{\kappa }^{0},\,\,\,\,...,\,\,\mathop {{{\Lambda }_{\kappa }}({{t}_{0}})}\limits^{(n - 3)} \,\,\,\,\, = \mathop {\Lambda _{\kappa }^{0}}\limits^{(n - 3)} , \\ {{q}^{\sigma }}({{t}_{0}}) = q_{0}^{\sigma },\,\,\,\,{{{\dot {q}}}^{\sigma }}({{t}_{0}}) = \dot {q}_{0}^{\sigma },\,\,\,\,\,\kappa = \overline {1,k} ,\,\,\,\sigma = \overline {1,s} . \\ \end{gathered} $$
((3.2))

4 APPLICATION OF THE FIRST THEORY TO SOLVING A GENERALIZED CHEBYSHEV PROBLEM

The number of examples of mechanical systems with higher-order constraints is rather limited. An example of such a system is mentioned by Hamel in his monography [43], where he imposed the constraint on the motion of a point by requiring the equality of the vertical component of acceleration to the product of two horizontal components of the same acceleration. Despite the elegance of this higher-order constraint, it is of less interest, because it does not contain any physical meaning.

The first example of the real mechanical system to whose motion the non-holonomic higher-order constraint was laid was formulated in papers [44, 45]. We consider it in detail, because it vividly demonstrates the application of the first theory of motion of non-holonomic systems with higher-order constraints to the solution to the generalized Chebyshev problem.

Consider the motion of an artificial Earth satellite (AES) in the field of the Earth’s gravity. We characterize the AES position by the polar coordinates q1 ≡ r, q2 ≡ φ and place the origin of coordinates to the center of the Earth. If a satellite moves around the Earth along an ellipse, its acceleration continuously varies. We formulate the problem of determining the satellite motion if, starting from some time instance t0, its acceleration absolute value is equal to w0 and in the subsequent motion this value will be constant and always equal to w0. This means that the condition must be fulfilled

$$w_{0}^{2} = {{(\ddot {r} - r{{\dot {\varphi }}^{2}})}^{2}} + {{(r\ddot {\varphi } + 2\dot {r}\dot {\varphi })}^{2}}.$$

This condition means that, beginning from the time instance t0, we impose the nonlinear non-holonomic second-order constraint on the satellite (spacecraft) motion

$${{f}_{2}}(q,\dot {q},\ddot {q}) \equiv {{(\ddot {r} - r{{\dot {\varphi }}^{2}})}^{2}} + {{(r\ddot {\varphi } + 2\dot {r}\dot {\varphi })}^{2}} - w_{0}^{2} = 0.$$
((4.1))

Thus, we have formulated the real mechanical problem from the astronautics, which is stated as the generalized Chebyshev problem: we need to find such a control force that, being applied to the satellite along with the gravity force of the Earth, results in the motion satisfying differential Eq. (4.1).

We have to explain: in the generalized Chebyshev problem the system of Eqs. (1.2) must be linear. However, we reduce constraint equation (4.1) to the linear equation by differentiating it with respect to time. As a result, we obtain

$${{f}_{3}} \equiv {{\dot {f}}_{2}} = (\ddot {r} - r{{\dot {\varphi }}^{2}})(\ddot {r} - \dot {r}{{\dot {\varphi }}^{2}} - 2r\dot {\varphi }\ddot {\varphi }) + (r\ddot {\varphi } + 2\dot {r}\dot {\varphi })(\dot {r}\ddot {\varphi } + r\dddot \varphi + 2\ddot {r}\dot {\varphi } + 2\dot {r}\ddot {\varphi }) = 0.$$
((4.2))

It is convenient to rewrite constraint equation (4.2) to the standard form

$${{f}_{3}}(q,\dot {q},\ddot {q},\dddot q) \equiv {{a}_{{3r}}}\dddot r + {{a}_{{3\varphi }}}\dddot \varphi + {{a}_{{3,0}}} = 0,$$
((4.3))

where the coefficients a3r, a3φ, and a3, 0 depend on r, φ, and their two first derivatives.

We will solve the formulated generalized Chebyshev problem using the apparatus of the motion of non-holonomic systems with higher-order constraints, consider constraint (4.3) the program constraint, and regard the reaction of this constraint as the control which provides that the program of motion, given by the third-order linear differential Eq. (4.3), is followed.

We write the vector Newton equation used when the ideal non-holonomic third-order constraint (4.3) is imposed by using the generalized Hamilton operator, introduced by Polyakhov (er and eφ are the vectors of the co-basis of the introduced polar system of coordinates):

$$m{\mathbf{w}} = {\mathbf{F}} + \Lambda \nabla {\kern 1pt} '''{\kern 1pt} {{f}_{3}} \equiv {\mathbf{F}} + \Lambda \left( {\frac{{\partial {{f}_{3}}}}{{\partial \dddot r}}{{{\mathbf{e}}}^{r}} + \frac{{\partial {{f}_{3}}}}{{\partial \dddot \varphi }}{{{\mathbf{e}}}^{\varphi }}} \right).$$
((4.4))

Here,

$${\mathbf{F}} = - \frac{{\mu m}}{{{{r}^{2}}}}{{{\mathbf{e}}}^{r}},$$

where m is the satellite mass and μ is the Gauss constant for the Earth gravity field.

We multiply Eq. (4.4) by the vectors of the main basis er and eφ and obtain

$$\begin{gathered} {{\Lambda }_{*}}(\ddot {r} - r{{{\dot {\varphi }}}^{2}}) = \frac{\mu }{{{{r}^{2}}}}, \\ {{\Lambda }_{*}}(r\dddot \varphi + 2r\dot {\varphi }) = 0, \\ {{\Lambda }_{*}} = \frac{\Lambda }{m} - 1. \\ \end{gathered} $$
((4.5))

From system (4.5) we derive the first group of differential equations of the form (3.1):

$$\ddot {r} = r{{\dot {\varphi }}^{2}} + \frac{\mu }{{{{\Lambda }_{*}}{{r}^{2}}}},$$
((4.6))
$$\ddot {\varphi } = - \frac{{2\dot {r}\dot {\varphi }}}{r}.$$
((4.7))

To obtain the auxiliary differential equation for \({{\Lambda }_{*}}\) (as a special case of the second group of differential equations in formulas (3.1)), we differentiate the first and the second equations of system (4.5) with respect to time

$${{\dot {\Lambda }}_{*}}(\ddot {r} - r{{\dot {\varphi }}^{2}}) + {{\Lambda }_{*}}(\dddot r - \dot {r}{{\dot {\varphi }}^{2}} - 2r\dot {\varphi }\ddot {\varphi }) + 2\frac{\mu }{{{{r}^{3}}}}\dot {r} = 0,$$
((4.8))
$${{\dot {\Lambda }}_{*}}(r\ddot {\varphi } + 2\dot {r}\dot {\varphi }) + {{\Lambda }_{*}}(3\dot {r}\ddot {\varphi } + r\dddot \varphi + 2\ddot {r}\dot {\varphi }) = 0.$$
((4.9))

To eliminate the third derivatives and satisfy the constraints f2 = 0 and f3 = 0, we multiply Eqs. (4.8) and (4.9) by the coefficients at \(\dddot r\) and \(\dddot \varphi \) in Eq. (4.3) and sum up the results

$${{\dot {\Lambda }}_{*}} = \frac{{2\mu \dot {r}(r{{{\dot {\varphi }}}^{2}} - \ddot {r})}}{{{{r}^{3}}({{{(\ddot {r} - r{{{\dot {\varphi }}}^{2}})}}^{2}} + {{{(r\ddot {\varphi } + 2\dot {r}\dot {\varphi })}}^{2}})}}.$$

Using constraint (4.1) and the first equation of system (4.5), we may write

$${{\dot {\Lambda }}_{*}} = - \frac{{2{{\mu }^{2}}}}{{w_{0}^{2}}}\frac{{\dot {r}}}{{{{\Lambda }_{*}}{{r}^{5}}}}.$$
((4.10))

Differential Eqs. (4.6), (4.7), and (4.10) provide the system whose integration gives the solution to the formulated problem.

5 STUDY OF KOSMOS- AND MOLNIYA-TYPE SATELLITES MOTIONS WITH CONSTANT ACCELERATIONS USING THE FIRST THEORY OF MOTION OF NONHOLONOMIC SYSTEMS WITH HIGHER-ORDER CONSTRAINTS

To obtain the initial data for the system of differential Eqs. (4.6), (4.7), and (4.10), we make use of the well-known formulas (see [11], e is the eccentricity of the orbit and p is the focal parameter)

$$r = \frac{p}{{1 + e\cos \varphi }},\quad \dot {\varphi } = \frac{{\sqrt {p\mu } }}{{{{r}^{2}}}},\quad \dot {r} = \frac{{pe\dot {\varphi }\sin \varphi }}{{{{{(1 + e\cos \varphi )}}^{2}}}},\quad \ddot {\varphi } = - \frac{{2\dot {r}\dot {\varphi }}}{r},\quad \ddot {r} = r{{\dot {\varphi }}^{2}} - \frac{\mu }{{{{r}^{2}}}}.$$
((5.1))

Consider the motion of a USSR Kosmos-type satellite with the perigee altitude Hπ = 183 km and apogee altitude Hα = 244 km over the Earth’s surface (from now on, the numerical data are taken from the web). We take that the Earth’s radius equals RE = 6371 km and the gravity acceleration on the Earth’s surface equals g0 = 9.82 × 10–3 km/s2 (that is, we assume that the Earth is a ball with the uniformly distributed mass). Hence, we obtain

$${{r}_{\pi }} = 6554\;{\text{km}},\quad {{r}_{\alpha }} = 6615\;{\text{km}},\quad \mu = {{g}_{0}}R_{3}^{2} = 398\,590\;{\text{k}}{{{\text{m}}}^{{\text{3}}}}{\text{/}}{{{\text{s}}}^{{\text{2}}}},$$
$$e = \frac{{{{r}_{\alpha }} - {{r}_{\pi }}}}{{{{r}_{\alpha }} + {{r}_{\pi }}}} = 0.004632,\quad p = {{r}_{\pi }}(1 + e) = 6584.36\;{\text{km}}.$$

We consider the case when the satellite acceleration is fixed by its position in the apogee. Then, the initial data at t0 = 0 for the numerical integration of the system of Eqs. (4.6), (4.7), and (4.10) with formulas (5.1) have the following values (we supplement them according to formulas (3.2) by the condition Λ(0) = 0, which is equivalent to the condition \({{\Lambda }_{*}}\)(0) = –1):

$$\begin{gathered} r(0) = 6554\;{\text{km}},\quad \varphi (0) = 0,\quad \dot {r}(0) = 0, \\ \dot {\varphi }(0) = 0.00119263\;{{{\text{s}}}^{{ - 1}}},\quad \ddot {r}(0) = 0.0000429824\;{\text{km/}}{{{\text{s}}}^{2}}, \\ \ddot {\varphi }(0) = 0,\quad \Lambda (0) = 0. \\ \end{gathered} $$
(                                         (5.2))

The results of the numerical integration with initial data (5.2) are presented in Fig. 1. It is seen in Fig. 1a that the studied satellite moves practically along the circle in the formulation of the considered problem. We note that, initially, most Earth satellites operated exactly in this manner, which allowed using the smallness of eccentricity e of their orbits in computation. In Fig. 1b we draw the time dependence of the Lagrange multiplier forming the control force, providing that the program of motion is followed, given in the form of differential Eq. (4.3). Note that, from the technical point of view, generation of the needed control force is easy to carry out by mounting an additional jet engine on the satellite.

Fig. 1.
figure 1

Motion of Kosmos-type satellite with constant acceleration when the first theory of motion is used.

Full size image

If we look at the Kosmos-type satellite trajectory in our problem on a large scale, then we notice that the satellite begins to rotate after fixing and alternatively touches two coaxial circles with the centers in the Earth’s center (the dashed arcs of circles in Fig. 2). After the beginning of motion from the inner circle, the first approach to the outer circle is visible at the sequence of trajectory fragments (the solid line) depicted in Fig. 2.

Fig. 2.
figure 2

Segments of motion of Kosmos-type satellite with constant acceleration when the first theory of motion is used.

Full size image

The motion of the satellite between two coaxial circles is more evident for the satellites launched to highly elliptic orbits. For that purpose we consider the motion of a Molniya-type satellite. The perigee of such an orbit was situated over Moscow, and the apogee was situated over Vladivostok. Due to validity of the area law, such satellites moved quickly over Moscow and moved slowly over Vladivostok, with a sort of hovering. When a series of such satellites was launched, there was always a satellite over Vladivostok which provided a good transmission of a television signal from Moscow.

Thus, for a Molniya-type satellite we have

$${{r}_{\pi }} = 6871\;{\text{km}},\quad {{r}_{\alpha }} = 46\,371\;{\text{km}},\quad \mu = {{g}_{0}}R_{3}^{2} = 398\,590\;{\text{k}}{{{\text{m}}}^{{\text{3}}}}{\text{/}}{{{\text{s}}}^{{\text{2}}}},$$
$$e = \frac{{{{r}_{\alpha }} - {{r}_{\pi }}}}{{{{r}_{\alpha }} + {{r}_{\pi }}}} = 0.741895,\quad p = {{r}_{\pi }}(1 + e) = 11968.6\;{\text{km;}}$$

therefore, the initial conditions for the numerical integration of the system of differential equations are as follows

$$\begin{gathered} r(0) = 6871\;{\text{km}},\quad \varphi (0) = 0,\quad \dot {r}(0) = 0,\quad \Lambda (0) = 0, \\ \dot {\varphi }(0) = 0.001463\;{{{\text{s}}}^{{ - 1}}},\quad \ddot {r}(0) = 0.00626368\;{\text{km/}}{{{\text{s}}}^{2}},\quad \ddot {\varphi }(0) = 0. \\ \end{gathered} $$
((5.3))

The results of integration of the equations of motion with initial data (5.3) are presented in Fig. 3.

Fig. 3.
figure 3

Motion of Molniya-type satellite with constant acceleration when the first theory of motion is used.

Full size image

We compare the computational results obtained for the two considered AESs. The orbit of the Kosmos-type satellite is almost circular; therefore, its acceleration varies slightly. Consequently, to create the motion with constant acceleration, we need just an insignificant generalized force Λ = Λ(t). In contrast to that, when the acceleration value is fixed in the perigee, the Molniya-type satellite begins to move along the almost elliptic trajectory, which is markedly different from the initial orbit. This requires a significantly larger control force, increased by two orders of magnitude in comparison to the control force in the previous example (cf. the graphs in the right parts of Figs. 2 and 3).

In the dimensionless form the study of the motion of a satellite with acceleration constant in absolute value was considered in detail in the monographs [5, 34]. In addition to that, the problem of the smooth transition of a satellite from a circular orbit to another circular orbit was discussed as an example of motion with non-holonomic third-order constraint. The smoothness of the flight was characterized by the parameters of the specially constructed generalized Sears equation.

6 TANGENT SPACE. CONSTRAINT REACTIONS

We introduce the differentiable manifold of all positions of the mechanical system which it may have at the current time instance. The Euclidean structure of the tangent space to this manifold [46] is determined by the matrix (gστ) when we use the curvilinear coordinate system q = (q1, …, qs). The elements of this matrix are equal to the coefficients of the positive definite quadratic form

$${{T}^{{(2)}}} = \frac{M}{2}{{{\text{g}}}_{{\sigma \tau }}}{{\dot {q}}^{\sigma }}{{\dot {q}}^{\tau }}.$$

The chosen curvilinear coordinate system generates the basis and the co-basis in the tangent space

$$\{ {{{\mathbf{e}}}_{1}}, \ldots ,{{{\mathbf{e}}}_{s}}\} ,\quad \{ {{{\mathbf{e}}}^{1}}, \ldots ,{{{\mathbf{e}}}^{s}}\} .$$
((6.1))

Now, the Lagrange equations of the second kind (1.1) may be represented in the tangent space by the vector equation with the form of the second Newton’s law (see a detailed discussion in monographs [5, 34])

$$M{\mathbf{W}} = {\mathbf{Y}},\quad {\text{where}}\quad {\mathbf{W}} = {{W}_{\sigma }}{{{\mathbf{e}}}^{\sigma }},\quad {\mathbf{Y}} = {{Q}_{\sigma }}{{{\mathbf{e}}}^{\sigma }}.$$
((6.2))

Let us proceed to the constrained motion of a mechanical system when the holonomic constraints (1.4), non-holonomic first-order constraints (1.5), and linear non-holonomic second-order constraints (1.6) are imposed. Note that constraint equations (1.6) may be prescribed independently; however, only one example [47] of such a constraint conducted mechanically is known. In this example we study the coiling of a thread carrying a heavy point on the surface of a vertical circular cylinder. In addition to that, in the form (1.6) we may write both constraints (1.5), by differentiating them with respect to time, and constraints (1.4), by differentiating them twice with respect to time. Therefore, the constraints of the holonomic mechanics and of the classical non-holonomic mechanics may be represented in the form (1.6).

We point out that constraint Eqs. (1.6) may be transformed to the scalar products

$${{{\mathbf{\varepsilon }}}^{{l + \kappa }}} \cdot {{{\mathbf{W}}}^{K}} = \chi _{2}^{\kappa }(t,q,\dot {q}),\quad \chi _{2}^{\kappa }(t,q,\dot {q}) = a_{{2\sigma }}^{{l + \kappa }}\Gamma _{{\alpha \beta }}^{\sigma }{{\dot {q}}^{\alpha }}{{\dot {q}}^{\beta }} - a_{{20}}^{{l + \kappa }},\quad \kappa = \overline {1,k} .$$
((6.3))

We underline that the right-hand sides \(\chi _{2}^{\kappa }\) in formulas (6.3) are the given functions of variables t, q, and \(\dot {q}\), and the introduced vectors \({{{\mathbf{\varepsilon }}}^{{l + \kappa }}}\), κ = \(\overline {1,k} \), have the form

$${{{\mathbf{\varepsilon }}}^{{l + \kappa }}} = a_{{2\sigma }}^{{l + \kappa }}{{{\mathbf{e}}}^{\sigma }},\quad \kappa = \overline {1,k} .$$

The covariant components of these vectors are determined by the coefficients \(a_{{2\sigma }}^{{l + \kappa }}\), σ = \(\overline {1,s} \) given by constraint Eqs. (1.6). We recall that, according to formulas (6.1) the bases of the used curvilinear coordinate system are regarded as prescribed.

Hence, the imposition of constraints (1.6) on the motion of a mechanical system separates the k‑dimensional K-space with the basis {εl + 1, …, εs} in the tangent s-dimensional space. It is convenient to introduce the l-dimensional L-space with the basis {ε1, …, εl} orthogonal to it

$${{{\mathbf{\varepsilon }}}_{\lambda }} \cdot {{{\mathbf{\varepsilon }}}^{{l + \kappa }}} = 0,\quad \lambda = \overline {1,l} ,\quad l = s - k,\quad \kappa = \overline {1,k} .$$

Thus, the imposition of constraints (1.6) divides the tangent space into the direct sum of the subspaces K and L with the dimensions k and l. It is easy to follow that the basis {εl + 1, …, εs} transforms to {\(\nabla {\kern 1pt} '{\kern 1pt} f_{1}^{{l + 1}}\), …, \(\nabla {\kern 1pt} '{\kern 1pt} f_{1}^{s}\)} when non-holonomic constraints (1.5) are imposed and to {\(\nabla f_{0}^{{l + 1}}\), …, \(\nabla f_{0}^{s}\)} when holonomic constraints (1.4) are imposed.

The division of the tangent space by the constraint equations into the two orthogonal subspaces allows us to represent the vector Newton equation of the constrained motion of a mechanical system

$$M{\mathbf{W}} = {\mathbf{Y}} + {\mathbf{R}},$$
((6.4))

where R is the vector of the tangent space which characterizes the effect of the imposed constraints, as the system of the two equations

$$M{{{\mathbf{W}}}^{K}} = {{{\mathbf{Y}}}^{K}} + {{{\mathbf{R}}}^{K}},$$
((6.5))
$$M{{{\mathbf{W}}}_{L}} = {{{\mathbf{Y}}}_{L}} + {{{\mathbf{R}}}_{L}}.$$
((6.6))

Since the left-hand side of Eq. (6.5) is a given function of variables t, q, and \(\dot {q}\) due to the satisfaction of constraints (6.3), to satisfy the constraints in this equation, we need to add the component of the constraint reactions

$${{{\mathbf{R}}}^{K}} = {{\Lambda }_{\kappa }}{{{\mathbf{\varepsilon }}}^{{l + \kappa }}}.$$
((6.7))

At the same time, it is now seen that with the known function Y(t, q, \(\dot {q}\)) we obtain the Lagrange multipliers as the known functions of the same variables from Eq. (6.5) accounting for formula (6.7).

The component WL in Eq. (6.6) is not restricted by the mathematical imposition of constraints (1.6) (or (6.3)); therefore, this equation may be valid for any vector RL, for instance, for

$${{{\mathbf{R}}}_{L}} = 0.$$
((6.8))

It is the constraints for which Eq. (6.8) is valid that are called ideal. Under such constraints the vector equation of motion in the L-space (6.6) becomes the equation of motion of a free mechanical system (6.2). If RL ≠ 0, then the formation of this vector must be explained from the physical realization of material imposition of the constraints. Thus, we see that the reactions of ideal constraints have the form (1.7).

7 GAUSS PRINCIPLE. GENERALIZED GAUSS PRINCIPLE. THE SECOND THEORY OF MOTION OF NON-HOLONOMIC SYSTEMS WITH HIGHER-ORDER CONSTRAINTS

The classical Gauss principle, which is true for ideal constraints (1.4), (1.5), and (1.6), is written in the form (see, e.g. [5, 34])

$$\delta {\kern 1pt} ''{\kern 1pt} Z = 0,$$
((7.1))

where we have applied the usual denotation for the Gauss constraint

$$Z = \frac{M}{2}{{\left( {{\mathbf{W}} - \frac{{\mathbf{Y}}}{M}} \right)}^{2}}.$$

Double prime in formula (7.1) underlines that we vary just the second derivatives of generalized coordinates. By varying, we obtain the representation of the Gauss principle in another notation:

$$(M{\mathbf{W}} - {\mathbf{Y}}) \cdot \delta {\kern 1pt} ''{\kern 1pt} {\mathbf{W}} = 0.$$
((7.2))

Because the constraints are regarded as ideal, the vector equation of the constrained motion (6.4) becomes

$$M{\mathbf{W}} = {\mathbf{Y}} + {{{\mathbf{R}}}^{K}};$$

therefore, we may rewrite Eq. (7.2) as

$${{{\mathbf{R}}}^{K}} \cdot \delta {\kern 1pt} ''{\kern 1pt} {\mathbf{W}} = 0.$$
((7.3))

We may represent condition (7.3) as

$$\delta {\kern 1pt} ''{\kern 1pt} {{({\mathbf{R}})}^{2}} = 0,$$
((7.4))

which we may interpret as the requirement that the reaction is minimum when ideal constraints (1.6) are imposed.

It is of interest to give the geometric interpretation of the Gauss principle. We recall that constraints (1.6) divide the tangent space into the two orthogonal subspaces K and L with the bases {εl + 1, …, εs} and {ε1, …, εl}. The constraints themselves prescribe the l-dimensional plane \(\mathbb{T}\)(t, q, \(\dot {q}\), \(\ddot {q}\)) in the space of generalized accelerations; in this plane t, q, and \(\dot {q}\) are the given parameters. The ends of the acceleration vectors W of the mechanical system must be in this plane. There must also be the vectors ελ, λ = \(\overline {1,l} \). We recall [5, 34] that the variation of acceleration δ''W is determined as the vector δ''W = δ''\({{\ddot {q}}^{\sigma }}\)eσ which may be decomposed in the basis of the L space and therefore is subject to the conditions

$$\nabla {\kern 1pt} ''{\kern 1pt} f_{2}^{\kappa } \cdot \delta {\kern 1pt} ''{\kern 1pt} {\mathbf{W}} = 0,\quad \kappa = \overline {1,k} .$$
((7.5))

From formulas (7.2) and (7.5) we may easily obtain that the reaction of ideal non-holonomic second-order constraints is expressed by formula (6.7), that is, it belongs to the K space.

The notation of the Gauss principle in the form (7.4) shows that, in the case of imposition of ideal linear non-holonomic second-order constraints, their reaction R/M = W – Y/M constrains the mechanical system to move with minimum value of this reaction. Therefore, the Gauss principle is also called the principle of least constraint.

Figure 4 illustrates all these considerations in the case of the motion of a single material point when the ideal non-holonomic constraint is imposed

$$f_{2}^{1}(t,y,\dot {y},\ddot {y}) \equiv a_{{2\sigma }}^{3}(t,y,\dot {y}){{\ddot {y}}_{\sigma }} + a_{{2,0}}^{3}(t,y,\dot {y}) = 0,\quad \sigma = \overline {1,3} ,$$
((7.6))

where y = (y1, y2, y3) are the Cartesian coordinates of the studied point. In this figure, in the space of accelerations we depict the plane (7.6) determined by the vector WK; the acceleration vector W of the material point ends on this plane in the point M1, and the co-basis vector ε3 = \(\nabla {\kern 1pt} ''{\kern 1pt} f_{2}^{1}\), along which the reaction RK of ideal constraint (7.6) should be directed, emerges from the point M1 perpendicularly to the plane. The reaction RK/M itself is depicted in Fig. 4 by the vector \(\overrightarrow {{{N}_{1}}{{M}_{1}}} \) drawn perpendicularly to the constraint plane from the end of the force vector YK/M acting upon the material point. From the construction of this vector, it follows that it has the shortest length; that is, according to the Gauss principle written in notation (7.4), the reaction, providing that ideal constraint (7.6) is followed, indeed appears to be minimum.

Fig. 4.
figure 4

Acceleration of a point with second-order constraint.

Full size image

We now proceed to discuss the generalized Gauss principle. It was first formulated in 1974 by  Chuev, in the little-known paper [48]. The same principle was rigorously presented in 1983 in work [21] by using the introduction of a linear transformation of forces. Let us explain the essence of this principle [37] extending the geometric illustration of the classical Gauss principle provided above.

Consider the case when the motion of mechanical system is restricted by the linear non-holonomic third-order constraints

$$f_{3}^{\kappa }(t,q,\dot {q},\ddot {q},\dddot q) \equiv a_{{3\sigma }}^{{l + \kappa }}(t,q,\dot {q},\ddot {q}){{\dddot q}^{\sigma }} + a_{{3,0}}^{{l + \kappa }}(t,q,\dot {q},\ddot {q}) = 0,\quad \kappa = \overline {1,k} .$$
((7.7))

These equations define the l-dimensional plane in the space of vectors \({\mathbf{\dot {W}}}\); in this plane the ends of these vectors must be located to satisfy constraints (7.7). If we now substitute the values \({{\ddot {y}}_{1}}\), \({{\ddot {y}}_{2}}\), and \({{\ddot {y}}_{3}}\) by \({{\dddot y}_{1}}\), \({{\dddot y}_{2}}\), and \({{\dddot y}_{3}}\) and the vectors W, WK, WL, Y, and RK by the vectors \({\mathbf{\dot {W}}}\), \({{{\mathbf{\dot {W}}}}^{K}}\), \({{{\mathbf{\dot {W}}}}_{L}}\), \({\mathbf{\dot {Y}}}\), and \({{{\mathbf{\dot {R}}}}^{K}}\) in Fig. 4, then we may, similarly to the previous discussion, state that the value \({\mathbf{\dot {R}}}\)/M = \({{{\mathbf{\dot {R}}}}^{K}}\)/M is minimized when constraints (7.7) are imposed, and, therefore, the following statement is satisfied analogously to formula (7.1)

$$\delta {\kern 1pt} '''{\kern 1pt} {{Z}_{{(1)}}} = 0,$$
((7.8))

where we have introduced the denotation

$${{Z}_{{(1)}}} = \frac{M}{2}{{\left( {{\mathbf{\dot {W}}} - \frac{{\mathbf{Y}}}{M}} \right)}^{2}}.$$
((7.9))

We may consider notation (7.8) the generalized Gauss principle which is valid when constraints (7.7) are imposed. The subscript (1) in formulas (7.8) and (7.9) denotes the order of the generalized principle with respect to the classical Gauss principle, and the triple prime in notation (7.8) underlines that we vary only the third derivatives of generalized coordinates. Similarly to the relation of formulas (7.1) and (7.2), we may rewrite the generalized Gauss principle (7.8) in the form

$$(M{\mathbf{\dot {W}}} - {\mathbf{\dot {Y}}}) \cdot \delta {\kern 1pt} '''{\kern 1pt} {\mathbf{\dot {W}}} = 0.$$

The obtained generalized first-order Gauss principle in the case with (n + 2)th-order linear non-holonomic constraints is easily generalized to the generalized nth-order Gauss principle

$${{\delta }^{{(n + 2)}}}{{Z}_{{(n)}}} = 0,$$
((7.10))

where we have introduced the denotation

$${{Z}_{{(n)}}} = \frac{M}{2}{{\left( {\mathop {\mathbf{W}}\limits^{(n)} - \frac{{\mathop {\mathbf{Y}}\limits^{(n)} }}{M}} \right)}^{2}}.$$
((7.11))

In formulas (7.10) and (7.11) the subscript (n) denotes the order of the time derivative of the vector, and the superscript (n + 2) denotes that the partial differential is computed with fixed t, qσ, \({{\dot {q}}^{\sigma }}\), …, \(\mathop {{{q}^{\sigma }}}\limits^{(n + 1)} \).

When we use principle (7.10), we must prescribe the initial conditions

$${{\Lambda }_{\kappa }}({{t}_{0}}) = \Lambda _{\kappa }^{0},\quad {{\dot {\Lambda }}_{\kappa }}({{t}_{0}}) = \dot {\Lambda }_{\kappa }^{0},\quad \ldots ,\quad \mathop {{{\Lambda }_{\kappa }}}\limits^{(n - 3)} ({{t}_{0}}) = \mathop {\Lambda _{\kappa }^{0}}\limits^{(n - 3)} ,$$
$${{q}^{\sigma }}({{t}_{0}}) = q_{0}^{\sigma },\quad {{\dot {q}}^{\sigma }}({{t}_{0}}) = \dot {q}_{0}^{\sigma },\quad \kappa = \overline {1,k} ,\quad \sigma = \overline {1,s} .$$

The vector \(\vec {\Re }\) ≡ \(\mathop {\mathbf{R}}\limits^{(n)} \) = \(M\mathop {\mathbf{W}}\limits^{(n)} \) – \(\mathop {\mathbf{Y}}\limits^{(n)} \) minimized by the absolute value in this section may be conventionally called the reaction of (n + 2)th-order linear non-holonomic constraints.

The construction of the equations of motion according to the second theory of motion of non-holonomic systems with higher-order constraints is based on the application of the generalized Gauss principle.

8 STUDY OF KOSMOS- AND MOLNIYA-TYPE SATELLITES MOTIONS WITH CONSTANT ACCELERATIONS USING THE SECOND THEORY OF MOTION OF NON-HOLONOMIC SYSTEMS WITH HIGHER-ORDER CONSTRAINTS [44, 45, 49]

We write the generalized Gauss principle for the Earth satellite with the imposition of linear non-holonomic third-order constraint (4.3)

$$(m{\mathbf{\dot {w}}} - {\mathbf{\dot {F}}}) \cdot \delta {\kern 1pt} '''{\kern 1pt} {\mathbf{\dot {w}}} = 0.$$
((8.1))

We rewrite principle (8.1)

$$(m{{U}_{\rho }} - {{P}_{\rho }})\delta '''{{U}^{\rho }} = 0,\quad \rho = 1,2,\quad {\mathbf{U}} = {\mathbf{\dot {w}}},\quad {\mathbf{P}} = {\mathbf{\dot {F}}}.$$
((8.2))

We now use the known formulas (see [5, 34]) for the covariant components of the vectors U and P

$${{U}_{\rho }} = {{\dot {w}}_{\rho }} - \Gamma _{{\rho \sigma }}^{\tau }{{w}_{\tau }}{{\dot {q}}^{\sigma }},\quad {{P}_{\rho }} = {{\dot {F}}_{\rho }} - \Gamma _{{\rho \sigma }}^{\tau }{{F}_{\tau }}{{\dot {q}}^{\sigma }},\quad \rho ,\sigma ,\tau = 1,2,$$
((8.3))

where \(\Gamma _{{\rho \sigma }}^{\tau }\) are the Christoffel symbols of the second kind. The covariant components of the acceleration and force in formula (8.3) has the form

$$\begin{gathered} {{w}_{1}} = {{w}_{r}} = \ddot {r} - r{{{\dot {\varphi }}}^{2}},\quad {{w}_{2}} = {{w}_{\varphi }} = r(r\ddot {\varphi } + 2\dot {r}\dot {\varphi }), \\ {{F}_{1}} = - \frac{{m\mu }}{{{{r}^{2}}}},\quad {{F}_{2}} = 0. \\ \end{gathered} $$

The variations δ'''U1 and δ'''U2 in our problem are expressed as

$$\delta {\kern 1pt} '''{\kern 1pt} {{U}^{1}} = \delta '''\dddot r,\quad \delta {\kern 1pt} '''{\kern 1pt} {{U}^{2}} = \delta {\kern 1pt} '''\dddot \varphi ,$$

and, according to (4.3), they are related by

$$\delta {\kern 1pt} '''\dddot \varphi = - \frac{{\ddot {r} - r{{{\dot {\varphi }}}^{2}}}}{{r(r\ddot {\varphi } + 2\dot {r}\dot {\varphi })}}\delta {\kern 1pt} '''\dddot r.$$

Therefore, we may rewrite principle (8.2) in the form

$$\left( {m{{U}_{1}} - {{P}_{1}} - \frac{{\ddot {r} - r{{{\dot {\varphi }}}^{2}}}}{{r(r\ddot {\varphi } + 2\dot {r}\dot {\varphi })}}(m{{U}_{2}} - {{P}_{2}})} \right)\delta {\kern 1pt} '''\dddot r = 0.$$
((8.4))

Because only the following Christoffel symbols are nonzero

$$\Gamma _{{12}}^{2} = \Gamma _{{21}}^{2} = \frac{1}{r},\quad \Gamma _{{22}}^{1} = - r,$$

formulas (8.3) may be represented as

$$\begin{gathered} {{U}_{1}} = \dddot r - 3\dot {r}{{{\dot {\varphi }}}^{2}} - 3r\dot {\varphi }\ddot {\varphi },\quad {{U}_{2}} = 3r\ddot {r}\dot {\varphi } + 3r\dot {r}\ddot {\varphi } + {{r}^{2}}\dddot \varphi - {{r}^{2}}{{{\dot {\varphi }}}^{3}}, \\ {{P}_{1}} = \frac{{2\mu m\dot {r}}}{{{{r}^{3}}}},\quad {{P}_{2}} = - \frac{{\mu m\dot {\varphi }}}{r}. \\ \end{gathered} $$

Now, because the variations δ'''\(\dddot r\) are arbitrary, from the notation of principle (8.4) we obtain

$$\dddot r - \frac{{2\mu \dot {r}}}{{{{r}^{3}}}} - 3\dot {r}{{\dot {\varphi }}^{2}} - \frac{{(\ddot {r} - r{{{\dot {\varphi }}}^{2}})\left( {3r\ddot {r}\dot {\varphi } + 3r\dot {r}\ddot {\varphi } + \frac{{\mu \dot {\varphi }}}{r} + {{r}^{2}}\dddot \varphi - {{r}^{2}}{{{\dot {\varphi }}}^{3}}} \right)}}{{r(2\dot {r}\dot {\varphi } + r\ddot {\varphi })}} - 3r\dot {\varphi }\ddot {\varphi } = 0.$$
((8.5))

We solve the system of Eqs. (4.3) and (8.5) for \(\dddot r\) and \(\dddot \varphi \)

$$\begin{gathered} \dddot r = \frac{{\dot {r}(2\mu + 3{{r}^{2}}{{{\dot {\varphi }}}^{2}}) - \frac{{\mu (r{{{\dot {\varphi }}}^{2}} - \ddot {r})(2\dot {r}(2r{{{\dot {\varphi }}}^{2}} - \ddot {r}) + {{r}^{2}}\dot {\varphi }\ddot {\varphi })}}{{{{{(\ddot {r} - r{{{\dot {\varphi }}}^{2}})}}^{2}} + 4{{{\dot {r}}}^{2}}{{{\dot {\varphi }}}^{2}} + 4r\dot {r}\dot {\varphi }\ddot {\varphi } + {{r}^{2}}{{{\ddot {\varphi }}}^{2}}}} + 3{{r}^{4}}\dot {\varphi }\ddot {\varphi }}}{{{{r}^{3}}}}, \\ \dddot \varphi = \frac{{{{r}^{3}}\dot {\varphi }(r{{{\dot {\varphi }}}^{2}} - 3\ddot {r}) - 3{{r}^{3}}\dot {r}\ddot {\varphi } - \frac{{\mu (r{{{\dot {\varphi }}}^{2}} - \ddot {r})(\dot {\varphi }(r(r{{{\dot {\varphi }}}^{2}} - \ddot {r}) - 4{{{\dot {r}}}^{2}}) - 2r\dot {r}\ddot {\varphi })}}{{{{{(\ddot {r} - r{{{\dot {\varphi }}}^{2}})}}^{2}} + 4{{{\dot {r}}}^{2}}{{{\dot {\varphi }}}^{2}} + 4r\dot {r}\dot {\varphi }\ddot {\varphi } + {{r}^{2}}{{{\ddot {\varphi }}}^{2}}}}}}{{{{r}^{4}}}}. \\ \end{gathered} $$
((8.6))

We integrate the derived system of differential Eqs. (8.6) with initial data (5.2) and (5.3) and obtain the solution to the formulated problem.

In Fig. 5 we draw the trajectories of motion of Kosmos- and Molniya-type satellites computed with the differential equations derived with the second theory for the case when the acceleration absolute value is fixed in the perigee. As we see in these trajectories, when the satellite acceleration is fixed, the satellite begins asymptotically tending to the straight-line motion after some rotation around the Earth.

Fig. 5.
figure 5

Motion of Kosmos- and Molniya-type satellites with constant acceleration when the second theory of motion is used.

Full size image

9 DISCUSSION OF RESULTS

Using the two theories of motion of non-holonomic systems with higher-order constraints in the computations (in the solution of the generalized Chebyshev problems derived by different methods), we compare the trajectories of the satellites when their accelerations are fixed in the perigee and see their principal distinction.

From the point of view of the mechanics of non-holonomic systems, the difference between the obtained solutions may be explained as follows. The first theory is constructed on the transformation of the vector Newton equation in the case of application of ideal linear non-holonomic higher-order constraints (in our example on the transformations of Eq. (4.4)). Such equations of motion are characterized by the Mangeron–Deleanu principle (see, e.g., [5, 34]) which provides the minimum absolute value of the constraint reaction force. In the second theory we use the generalized Gauss principle [21] providing the minimum absolute value of the corresponding derivative (in our case, of the first derivative) of the reaction vector of the imposed higher-order constraints.

However, it is interesting to supplement the obtained results with the following considerations. It is well-known that the motion of a point with constant acceleration occurs either in the uniform rotation along a circle, or in the case of the straight-line uniformly accelerated motion. The elements of the first such motion in the rotation of the satellite between the two coaxial circles were obtained with the first theory, and the asymptotic tendency of the satellite to the straight-line uniformly accelerated motion was obtained with the second theory. Thus, we see that both distinct theories of motion of non-holonomic systems with higher-order constraints successfully supplement each other in our example.