Uniform motions in central fields

We present a theoretical problem of uniform motions, i.e. motions with constant magnitude of the velocity in central fields as a nonholonomic system of one particle with a nonlinear constraint. The concept of the article is in analogy with the recent paper [ 21 ]. The problem is analysed from the kinematic and dynamic point of view. The corresponding reduced equation of motion in the Newtonian central gravitational field is solved numerically. Appropriate trajectories for suitable initial conditions are presented. Symmetries and conservation laws are investigated using the concept of constrained Noetherian symmetry [ 9 ] and the corresponding constrained Noetherian conservation law. Isotachytonic version of the conservation law of mechanical energy is found as one of the corresponding constraint Noetherian conservation law of this nonholonomic system.


1.
Introduction. From the mechanics it is known that motions are classified with respect to two basic aspects. The first aspect is the shape of a trajectory, we distinguish between rectilinear and curvilinear motions. The second one is the magnitude of the velocity, we distinguish between uniform and non-uniform motions. In the case of the uniform motion the magnitude of the velocity remains constant during the entire motion, |v| = const. In the case of non-uniform motion the magnitude of the velocity changes during the motion. Evidently, the motions in central fields are non-uniform motions (except the circular motions). To keep the motions in central fields uniform, it is necessary to consider these motions as constrained motions subjected to a certain additional condition called constraint, which ensures the desired character of the motions.
In this paper we deal with the correction of non-uniform motions in central fields on uniform motions. The issue is solved as a classical initial value problem for a 92 MARTIN SWACZYNA AND PETR VOLNÝ constrained mechanical system arisen from a system of one particle moving in a central force field subjected to one constraint called isotachytonic constraint. The word isotachytonic was introduced in [21] from Greek; iso = uniform; τ αχύτ ητ α = velocity. Since isotachytonic constraint represents the nonholonomic constraint nonlinear with respect to components of the velocity, it is necessary to adopt a modern approach based on the geometric concept of nonholonomic mechanical systems [6,7,9,11,13]. This problem posses some interesting aspects concerning proper kinematics (specific trajectories), understanding the role of the Chetaev constrained force and its active contribution to the energetic balance of the system, understanding of the constraint Noetherian conservation laws as a direct consequence of certain symmetries of the studied constrained system. It seems that the problem can be useful for some practical applications concerning regulation and optimal control of the space satellites and different maneuvers of spacecrafts as a suitable theoretical background [4].
The bibliography in a branch of nonholonomic mechanics is very extensive and includes many important contributions and alternative geometric concepts, see [1]- [21] and references therein. It should be stressed that almost all the papers deal with the case of linear nonholonomic constraints except the several ones, e.g. [10,12,20,21].
In this paper we apply a geometric theory of nonholonomic mechanical systems which was developed for the first order in [7] and then was generalized for the higher order case in [8]. The theory enables an investigation of different aspects of these systems; reduced equations of motion of constrained systems (equations of motion on the constraint submanifold), deformed equations of motion (deformation of original unconstrained motion equations by adding Chetaev forces), constraint symmetries and corresponding conservation laws [9] and represents a reasonable concept for an alternative mathematical treatment of concrete examples of such systems, either with linear constraints [5,20] or with nonlinear constraints [10,20,21] and even with the higher order constraints [19].
Theoretical problems concerning various modifications of the motion of a particle in the gravitational field have been considered and solved in the late of the 17th century using the infinitesimal calculus, which was recently discovered. The development of mathematical analysis and methods of solving differential equations enabled mathematicians and physicists to study various geometric curves of prescribed properties and to solve complicated problems from mechanics. They were looking for various trajectories along which the particle would move in the gravitational field, if the particle was subjected to some additional constraint.
In 1687 Gottfried Wilhelm Leibniz (1646-1716) called mathematics to find a shape of a curve along it the particle falls in the homogeneous gravitational field F = −mg = (0, −mg) such that the vertical component of the velocity remains constant during the motion. The solution was found by Jacob Bernoulli (1654-1705) and he showed that the solution is a semi-cubic parabola called Leibniz isochrone. The result was published in Acta Eruditorum Journal in 1690 in Leipzig and it is interesting to remark that in this article first appeared the word integral.
We briefly recall his original solution within the contemporary notation used in mechanics. Let is given the Cartesian coordinate system Oxy in the vertical plane, where the x-axis is horizontal and the y-axis is directed vertically down. The particle of the mass m is located in the homogeneous (constant) gravitational field, which is oriented vertically down, i.e. vector of the gravitational force F = (0, mg), where gis the constant gravitational acceleration. Without loss of generality, we assume that the particle starts from the origin O with the vector of the initial velocity v 0 , which has only the vertical component and is also oriented vertically down, i.e. v 0 = (0, v 0 ), v 0 > 0. The motion of the particle is subjected to one additional condition that the vertical component of the velocity of the particle remains constant during the motion, i.e. v y =ẏ = v 0 = const.
(2) The air resistance is neglected. Jacob Bernoulli based his solution on validity of the classical conservation law of the mechanical energy E. The potential energy of the particle in the homogeneous gravitational field at the height y is U (y) = −mgy. The mechanical energy E of the particle at the beginning of the motion has the value E 0 = (1/2)mv 2 0 , since the potential energy of the particle at the origin O is U (0) = 0. The conservation law of the mechanical energy has the form Hence v 2 − 2gy = v 2 0 . On the other hand v 2 = v 2 x + v 2 y =ẋ 2 +ẏ 2 =ẋ 2 + v 2 0 . Thus, x = 2gy. (4) After the integration of the constraint condition (2) and with respect the initial condition y(0) = y 0 = 0 we get y(t) = v 0 t. (5) By the substitution into (4) we have dx/dt = √ 2gv 0 t. After the integration and with respect to the initial condition x(0) = x 0 = 0 we get The equations (5) and (6) represents the parametric expression for searched trajectory, called Leibniz isochrone. After eliminating of the time parameter from (5) we obtain the analytic equation of Leibniz isochrone in the Cartesian coordinates Later, in 1699 Pierre Varignon (1654-1722) modified the problem of Leibniz isochrone on the problem of finding such a curve along it the particle would move in the uniform (constant) central gravitational field, i.e. the central field with the constant magnitude of the gravitational force, under the same constraint condition as Leibniz, i.e. that the vertical (radial) component of the velocity of the particle remains constant during the motion. Varignon analysed the problem within the Leibniz formulation of the infinitesimal calculus and found the curve called Varignon's isochrone. This result was published in Mémoires de l'Académie des Sciences de Paris in 1699.
We present the original Varignon's solution in the contemporary notation. Varignon considered the special central gravitational field, which is constant at every point and is oriented to the center of the Earth, which is understood as a single massive mass point. This force field is expressed mathematically as follows:

MARTIN SWACZYNA AND PETR VOLNÝ
where F (r) is the vector of the gravitational force at the point characterized by the position vector r, F is the magnitude of the gravitational force F . The magnitude F is constant at every point and equals to mg, where m is the mass of the particle and g is the constant gravitational acceleration. The gravitational force vector F (r) lies on the line connecting the center of the field (located in the center of Earth) with this point and is oriented directly to the center. The vector e r = r/r represents a unit vector in the direction of r and r denotes the distance from the center. Since the position vector r is oriented from the center, therefore we write the sign minus for the arbitrary attractive force field. The potential energy of the particle in such field at the distance r from the center is U (r) = mgr. It is convenient to solve this problem in the polar coordinates (r, ϕ). The particle of the mass m is located in the field described above and starts from the initial point (r 0 , ϕ 0 ). The vector of the initial velocity v 0 has only the radial component and is oriented to the center, i.e. v r (0) =ṙ(0) = −v 0 . The angular component of the initial velocity equals to zero, v ϕ (0) = 0. The motion of the particle is subjected to one additional condition that the radial component of the velocity remains constant during the motion, i.e.
The air resistance is neglected. Varignon as well as Bernoulli before based the solution of this problem on the validity of the classical conservation law of the mechanical energy E. The mechanical energy E of the particle at the beginning of the motion has the value E 0 = (1/2)mv 2 0 + mgr 0 . The conservation law of the mechanical energy has the form The square of the magnitude of the velocity vector in polar coordinates is v 2 = r 2 + r 2φ2 . Due to the constraint condition (9), we can writė After the integration of the constraint condition (9) we obtain Substituting into (12) we geṫ After separation of variables and the integration we obtain The equation (13) and (15) represent the parametric expression of the Varignon's isochrone in the polar coordinates. After the elimination of the time parameter from the equation (15) we obtain the analytic equation of Varignon's isochrone in the polar coordinates It arises questions, why did Varignon consider this special gravitational field? Did Varignon know the formulation of the Newtonian gravitational law in 1699? Or did Varignon intend to investigate only the full analogy of the Leibniz isochrone problem for the case of the most simple central field without any context with the Newtonian gravitational law? If we realize that the Newton's monograph Principia was first published in 1687 in London in a small edition and its second edition on the European continent was made in 1714, then it is possible, that Varignon was familiar with Newton's book until after 1699.
The aim of the paper is the complete treatment of a theoretical problem of uniform motions in central fields. At first we recall the important results known from the classical mechanics for motions of a particle in central fields. These facts will serve us for a comparison with the obtained results valid for uniform motions in central fields. Initially, the problem of uniform motion in central fields is formulated in the three dimensional situation. After derivation of a certain (isotachytonic) analog of the conservation law of the angular momentum, we reduce the problem as well as in the classical case on planar motions. Subsequently, the problem of uniform motions is investigated from the kinematic and dynamic point of view. We present the trajectories of uniform motions in the Newtonian gravitational filed with zero radial component of the initial velocity obtained by the numerical solutions of the reduced motion equation under the selected initial conditions. Corresponding trajectories of the uniform (constrained) and classical (unconstrained) motions under the same initial conditions are compared. Finally, we analyze the energetic balance of the system and we find the set of constraint Noetherian symmetries and corresponding Noetherian conservation laws. In the Sec. 8 we derive the analytical solution of the uniform motion problem in an inverse proportional gravitational field.
2. Nonholonomic Lagrangian systems on fibered manifolds. We briefly recall basic concepts of a theoretical background of nonholonomic mechanical systems on fibered manifolds. Let π : Y → X be a fibered manifold, dim X = 1, dim Y = m + 1, π 1 : J 1 Y → X and π 2 : J 2 Y → X its jet prolongations and π 1,0 : J 1 Y → Y the jet projection. Coordinate systems are denoted by (t, q σ ), 1 ≤ σ ≤ m on Y , (t, q σ ,q σ ) and (t, q σ ,q σ ,q σ ) on J 1 Y , J 2 Y, respectively. A mapping γ : X ⊇ I → Y is called a (local) section of π if π • γ = id I . Sections of π 1 are denoted by δ. The mappings J 1 γ, J 2 γ are called the first and second jet prolongations of the section γ, they are the sections of π 1 and π 2 respectively. A differential form η on J 1 Y is called contact if J 1 γ * η = 0 for every section γ of π. The 1-forms ω σ = dq σ −q σ dt generate a basis of contact forms on J 1 Y . Fibered manifolds are naturally endowed with the structure of projectable and vertical vector fields and horizontal and contact forms. For details we refer to [7].
A section γ of π is called a path of the Euler-Lagrange form E λ if E σ (L)•J 2 γ = 0. These equations are called the Euler-Lagrange equations or motion equations and can be written in an intrinsic geometric form J 1 γ * i ξ dθ λ = 0, where ξ is a π 1vertical vector field on J 1 Y , or quite equivalently in the form J 1 γ * i ξ α = 0, where α is any 2-form defined on an open subset W ⊂ J 1 Y, such that p 1 α = E λ (p 1 α is 1-contact part of α). The family of all such (local) 2-forms α = dθ λ + F = A σ ω σ ∧ dt + B σν ω σ ∧ dq ν + F , where F runs over the π 1,0 -horizontal 2-contact 2-forms, is called the first order Lagrangian system, and is denoted by [α].
From the physical point of view by constraints of a mechanical system we understand a given condition or set of conditions, which restrict the possible geometric positions of the mechanical system. We distinguish between holonomic and nonholonomic constraints. From the geometric point of view holonomic constraints represent submanifolds in the configuration space Y , while geometric concept of nonholonomic constraints is such that the nonholonomic constraints represent certain submanifolds in the phase space J 1 Y .
By a constraint submanifold Q ⊂ J 1 Y we shall mean a fibered submanifold π 1,0 | Q : Q → Y of the fibered manifold π 1,0 . We denote by ι : Q → J 1 Y the canonical embedding of Q into J 1 Y , and suppose dim Q = 2m + 1 − k. Locally, Q is given by equations or equivalently in the normal forṁ called a system of k nonholonomic constraints.
Recall, Q-admissible section is a local sectionγ of the fibered manifold π such that J 1γ (t) ∈ Q for every t ∈ domγ and Q-valued section is a local sectionδ of the fibered manifold π 1 such thatδ(t) ∈ Q for every t ∈ domδ.
The submanifold Q is naturally endowed with a distribution C, called canonical distribution [7]. The distribution C is generated by the following vector fields (19) or it is annihilated by the system of k linearly independent 1-forms called canonical constraint 1-forms The ideal in the exterior algebra of forms on Q generated by the annihilator C 0 of C is called the constraint ideal, and denoted by I; its elements are called constraint forms. The pair (Q, C) is then called a nonholonomic constraint structure.
Consider an unconstrained Lagrangian system whereF is a 2-contact 2-form on Q and φ 2 ∈ I is any constraint 2-form. The arising class [α Q ] is called the nonholonomic Lagrangian constrained system related to the Lagrangian system [α] on J 1 Y and the constraint structure (Q, C). In fibered coordinates [α Q ] is represented by the following 2-forms where 1 ≤ i, j ≤ k, 1 ≤ l, r, s ≤ m − k,F ls =F ls (t, q σ ,q r ) are arbitrary functions and A nonholonomic constrained system [α Q ] will be called regular if the matrix B ls is regular, i.e.
Let [α] be a Lagrangian system on J 1 Y , (Q, C) a nonholonomic constraint structure. A Q-admissible sectionγ : X ⊇ I → Y of π is a constrained path of the nonholonomic Lagrangian constrained system [α Q ] if and only if for every π 1 -vertical vector field ξ belonging to the canonical distribution C and every representative called reduced equations of motion of the constrained Lagrangian system [α Q ]. In fibered coordinates the reduced equations of motion (26) of the constrained system [α Q ] take the following form An alternative approach to dynamics of nonholonomic system is based on the existence of an additional force Φ, called Chetaev (constraint) force which replaces the influence of constraints, where µ i are Lagrange multipliers and f i are left-hand sides in (17). Equations of motion of the Lagrangian system [α] subjected to nonholonomic constraints (17) take the form and are called deformed equations of motion since they arise by a deformation of the original unconstrained system [α] by means of Chetaev force Φ (28).
3. Classical motions in central fields.
3.1. Central fields. In the Newtonian reference frame we consider two bodies, which are approximated by means of the two mass points P 1 and P 2 with the masses M and m, respectively. Denote r 1 , and r 2 the position vectors of the mass points P 1 and P 2 , respectively. Furthermore, denote r = r 2 − r 1 the vector of their relative position and its magnitude |r| = r represents their distance. One can suppose that the bodies interact by means of the force, which depends only on their relative position and their distance. Such a force, called central force of the interaction, it can be written as follows where the vector e r = r/r represents the unit vector oriented from P 1 to P 2 . Evidently, the force vector F lies on the line connecting P 1 and P 2 and its magnitude F depends only on their distance by means of the monotonic differentiable function F (r).
In accordance with the law of action and reaction one can state: if the particle P 2 exerts on the particle P 1 by the force F 21 = F (r)e r then also the particle P 1 exerts on the particle P 2 by the force F 12 = −F 21 = −F (r)e r , see the figure below. The motion equations for a system of two interacting particles in the Newtonian reference frame then takes a familiar form The interaction between particles is mediated through the central force fields generated by the particles. In the framework of the classical Newtonian mechanics, we assume that the interaction is spreading infinitely fast, i.e. immediately (action at distance). Therefore the particle P 1 generates a force field F 12 which immediately acts on the particle P 2 and causes a change of its motion state. Similarly, the particle P 2 is the source of the force field F 21 which causing immediately the change of the motion state of the particle P 1 . In consequence of this interaction both particles move and therefore also their force fields move. In particular, if the particle P 1 has much greater mass than the particle P 2 , i.e. M m, then the mas M of P 1 represents almost the mass of the entire system M + m, i.e. M + m ≈ M . The position of the mass center r C of the system of two interacting particles P 1 , P 2 approximately coincides with the position of the massive particle P 1 , The sum of the motion equations (31) gives us The relation m/M ≈ 0 enables us to omit the influence of the force F 21 on the particle P 1 , and we obtainr 1 ≈ 0, i.e. the massive particle P 1 is at rest or in the uniform rectilinear motion. Hence, we can investigate the motion of the particle P 2 in the Newtonian (inertial) reference frame with the origin that is located at the massive particle P 1 and is rigidly connected with it. In such a way we regard the massive particle P 1 as approximately fixed source of the force field F 12 in which the second lighter particles P 2 moves. Its motion is governed by the motion equation In the more general case when both particles have comparable masses it is well known from the classical mechanics, that the problem of two interacting particles one can separate into the uniform rectilinear motion of the mass center of the system and into the relative motions of particles P 1 and P 2 in the Newtonian (inertial) reference frame connected with the mass center of the system. The relative motion of two particles in this reference frame is mathematically equivalent with the motion of a single particle with the reduced massm = M m/(M + m) in the central force field (30). The reduced massm is approximately equals to m, when M m. A central field is such a vector field that in arbitrary point (characterized by the position vector r) the force vector F lies on the line connecting the center of the field (located in the origin) with this point. The center of the field is placed at the origin, that coincides with the massive particle or with the mass center of the system and is rigidly connected with it.
The force F is oriented directly from the center (repulsive field) or to the center (attractive field).
Since the position vector r is oriented from the center then for an attractive field we have F (r) = −F (r)e r and F (r) = F (r)e r for a repulsive field respectively. The most typical examples of central fields in the classical physics are: • Newtonian gravitational field α = −κM m, M is the mass of the center, m is the mass of the moving particle and κ is the universal gravitational constant. • Coulomb's electrostatic field a = (Qq)/(4πε), Q is the electric charge of the center, q is the electric charge of the particle and ε is the permittivity of a medium.
k > 0 is the spring constant. It is also possible to consider the following force field which do not have the physical interpretation. These fields have only the theoretical meaning.
• Constant central gravitational field The field was taken into account by the Varignon in the context of the Varignon's isochrone problem. • Inverse proportional gravitational field i.e. the gravitational field which is proportional to the reciprocal value of the distance r, b > 0. A force field F (r) is called conservative (or potential) if rotF (r) = 0. If F (r) is a conservative field then there exists a scalar function U (r) such that To every point X (characterized by a position vector r) of a conservative field F (r) one can assign a real number U (r), called the potential energy or the potential, given by the following curve integral which represents the work of the field needed to displace a particle along some trajectory from the reference pointX to X. The reference pointX is usually some significant point of the field. Since in the conservative fields the work is independent of an integration path, one can integrate simply along the lineXX. Recall that any central field is a conservative field. For the central fields given by F (r) ∼ r n we setX = 0 (center) and for the central fields given by F (r) ∼ 1/r n we setX = ∞ (infinity). The potential energy of a central field is given by where we integrate along the lineXX (dr = e r dr ). Hence, the potential energy of the particle in a central field depends on a position of the particle only by means of its distance from the origin (center), thus U (r) = U (r). Finally, a relation between the potential energy U (r) and the magnitude F (r) of the central force F (r) can be also expressed by the formula In particular, the potential energy of the Newtonian gravitational field (35), the Coulomb's electrostatic field (36) and the elastic force field (37) are expressed by respectively. The potential energy of the constant gravitational field (38) and the inverse proportional gravitational field (39) are expressed by respectively. A Lagrange function of a particle in a central field characterized by the potential energy U (r) represented in the Cartesian coordinates with the origin at the center has the standard form where v =ṙ = (ẋ,ẏ,ż) is a vector of the velocity of the particle. The motion equations of the particle (Euler-Lagrange equations) one can write in the vector form 3.2. Conservation laws. Let us consider the Lagrangian L = L(t, q σ ,q σ ) of the first order in term of generalized coordinates q σ and generalized velocitiesq σ in the classical form where E k is the kinetic energy and U is the generalized potential. The most general expression for the kinetic energy E k in terms of generalized coordinates is i.e. the kinetic energy E k is a homogeneous at most quadratic function of the generalized velocitiesq σ . The T 0 , T 1 and T 2 are those parts of E k that are homogeneous functions of degree 0, 1 and 2 in the generalized velocities, respectively, [2]. The expression (49) for the kinetic energy simplifies if the mechanical system does not explicitly depend on time, e.g. if considered constraints are scleronomic. In this case the kinetic energy E k reduces only into i.e. the kinetic energy E k is a homogeneous function of degree 2 in generalized velocities. Suppose that the potential U does not depend on the generalized velocities, and does not depend on time, i.e.
then the Lagrangian L takes the form i.e. the Lagrangian L does not depend explicitly on time, One can find ∂L ∂q µq Consequently, the function ∂L ∂q µq represents the mechanical energy, which remains constant. Indeed, since the Euler-Lagrange equations hold. Evidently, the kinetic part of the Lagrangian (46) of the particle moving in the central field is a homogeneous quadratic function in Cartesian componentsẋ,ẏ,ż of the velocity v, thus the conservation law of the mechanical energy E of the particle holds v 0 is the magnitude of the initial velocity v 0 and r 0 is the initial distance of the particle from the center.
Moreover, it is well known that the motion of the particle in an arbitrary central field is always planar. This fact is a direct consequence of the conservation law of the angular momentum l of the particle, r 0 represents a starting position of the particle, v 0 is a vector of the initial velocity of the particle. Thus, at each time the position vector r and the velocity vector v of the particle lie in the plane perpendicular to the constant angular momentum vector l 0 . Without loss of generality we consider that the motion of the particle takes place e.g. in the xy-plane (z = 0). The position vector r and the velocity vector v can be expressed in the polar coordinates (r, ϕ) as follows Consequently for the angular momentum vector l we obtain l = r×mv = re r ×m(ṙe r +rφe ϕ ) = mr 2φ (e r ×e ϕ ) = mr 2φ e z = (0, 0, mr 2φ ), (60) and with respect to (58) one can write where l 0 = ±|l 0 |. On the other side the conservation law (58) can be rewritten in the scalar form for the magnitude of the instantaneous angular momentum vector as follows where β(t) ∈ 0, π is the instantaneous angle between vectors r(t) and v(t), and |l 0 | is the magnitude of the initial angular momentum l 0 , β 0 ∈ 0, π is the initial angle between vectors r 0 and v 0 .

Motions in central fields.
From the previous subsection we know that a particle in a central field can be treated as a simple mechanical system with the corresponding Lagrangian L expressed in the polar coordinates (r, ϕ) with the origin placed in the center of the force field, Evidently, the kinetic part of the Lagrangian (63) is a homogeneous quadratic function of time derivativesṙ,φ, of the polar coordinates r, ϕ, therefore the conservation law of the mechanical energy in the form holds. The movement of the particle is then represented by the solution of the corresponding Euler-Lagrange equations where l 0 = mr 2 0φ (0) = mr 0 v ϕ (0) = mr 0 v 0 sin β 0 . Using the conservation law of angular momentum (66) the equation (65) describing the radial component of the motion takes the following form It means that the influence of the "centrifugal" term mv 2 ϕ /r onto the radial component of the motion one can replace by means of the action of an additional repulsive central force where The differential equations (65) and (66) can be solved numerically for the concrete force field F (r) and for the given initial conditions. The particle can start from the initial position one can change e.g. only the angular component v ϕ (0) and consequently v r (0) is In particular, if v ϕ (0) = 0 then v r (0) =ṙ(0) = v 0 and it is a radial motion.
If v r (0) = 0 then v ϕ (0) = v 0 and it is a motion with zero radial component of the initial velocity. Investigated motions with non-zero angular component of the initial velocity will be considered only for the case v ϕ (0) > 0, i.e. anticlockwise motions, thus l 0 > 0. As an alternative starting point for the investigation of motions of the particle in central fields can serve the conservation laws. In the polar coordinates the conservation laws (57) and (62) take the following form v ϕ = v sin β = rφ is the angular component of v. The system (72) represents the system of two first order differential equations for unknown functions r = r(t), ϕ = ϕ(t). We substituteφ,φ into the first equation of the system (72), and after the separation of variables we geṫ The sign plus is valid in the time interval in which the particle moves away from the center and the sign minus is valid for the approaching to the center. In general, integration of (76) leads to a numerical quadrature. If it is possible (after the integration of (76)) to express r = r(t) as the inverse function to t = t(r) given by (76), then we get the time dependence of the angle coordinate ϕ = ϕ(t), where for l 0 > 0 is ϕ(t) an increasing function of time (anticlockwise motion) and for l 0 < 0 is ϕ(t) an decreasing function of time (clockwise motion). Relations (76) and (77) represent a parametric expression of the trajectory of a motion in a central field.
The explicit expression r = r(ϕ) of the trajectory in the polar coordinates can be obtained immediately from (75) using the formula mr 2φ = l 0 . Sincė we obtain and after the separation and integration Evidently, the relation (80) represents two functions ϕ = ϕ(r), i.e. the motion of the particle is reversible. The particle can move anticlockwise or clockwise. The last integral is suitable to compute using the substitution r = 1/u, The presented formulas can be found almost in every classical books of theoretical mechanics, e.g. in [2]. It is known that the integral (81) can be expressed by means of elementary functions for power function types potentials U (r) = kr n for which n = −2, n = −1 (Newtonian or Coulomb's potential), n = 0 (constant potential) and n = 2 (potential of the elastic force).

3.4.
Motions in Newtonian gravitational field, Kepler's problem. In particular, for the Newtonian gravitational potential U (r) = α/r, α = −κM m, the integral (81) leads to the well known Kepler's trajectories represented in polar coordinates by the formula where The equation (82)represents the conic sections with the focus placed in the center of the Newtonian gravitational field with the focal parameter p and with the numerical eccentricity . Geometric parameters p and given by (83) are determined only by the initial value E 0 of the mechanical energy E, and by means of the initial value l 0 of the angular momentum l, The appropriate trajectory is given by the relations between the initial values E 0 and l 0 as follows: for the circular trajectory = 0 for the elliptic trajectory for the parabolic trajectory = 1 Conditions (86) are specific conditions for particular trajectories of the moving particle in the Newtonian gravitational field. If the initial velocity has zero angular component, v ϕ = 0, then with respect to (85) the magnitude of the angular momentum l = l 0 = 0. Hence the motion is realized only in the radial direction. If v r (0) = v 0 > 0, this is the radial motion from the center (upward throw), if v r (0) = −v 0 < 0, this is the radial motion to the center (drop) and if v r (0) = v 0 = 0, this is the free fall in the Newtonian gravitational field. Especially, if the particle starts with zero radial velocity component i.e. v ϕ (0) = v 0 , then l 0 = mr 0 v 0 . Moreover, if the initial energy E 0 is given by the first relation in (86), one can derive the formula for the circular velocity (the first space velocity), If the initial energy E 0 = 0 (the parabolic trajectory) then one can derive the formula for the escape velocity, 3.5. Effective potential energy. The conditions (86) for the values of the initial mechanical energy E 0 can also be derived graphically from the so-called diagram of the effective potential energy. A diagram of the effective potential energy enables us to make a qualitative analysis of the trajectories of motions in the central fields for arbitrary course of the potential energy U (r) of the given force field [2].
In the equation (74) for total mechanical energy of the particle moving in the central field the first term (1/2)mṙ 2 represents the kinetic energy of the radial motion, the second term l 2 0 /(2mr 2 ) = (1/2)mv 2 ϕ is called the centrifugal energy. The third term U (r) represents the potential energy of the given force field F (r).
The function is called the effective potential energy of the particle. Notice that initial angular momentum l 0 = mr 0 v ϕ (0) = mr 0 v 0 sin β 0 plays here the role of a parameter, therefore we sometimes write U ef (r, l 0 ). The centrifugal energy U c (r) represents the potential energy of an additional repulsive central force F c (90), In particular in the case of the radial motions the parameter l 0 = 0, i.e. v ϕ (0) = 0, there is no repulsive central force, and the effective potential energy coincides with the potential energy of the central field, i.e. U ef (r) = U (r). On the contrary the parameter l 0 is maximal, l 0 = mr 0 v 0 , if the radial component v r (0) = 0, i.e. v ϕ (0) = v 0 , then the influence of the repulsive central force F c is maximal.
Introducing the effective potential (90) the equation (75) takes the forṁ Sinceṙ 2 (t) is a non-negative function the right-hand side of the equation (92) must also be non-negative, i.e. it holds the relation This condition is principal for a qualitative analysis of trajectories in a central field, since it restricts admissible intervals of radial distances from the center. It is useful to solve the inequality (93) graphically. In the diagram of the effective potential energy one can draw a constant function corresponding to a certain value of the initial energy E 0 . Intersection points with the graph U ef (r) determine the limits of possible radial distances from the center. These points are called turning points and since they are solutions of the equation E 0 − U ef = 0, the radial component v r =ṙ of the velocity at these points is null. In these points the functionṙ =ṙ(t) changes its sign, i.e. either the approaching of the particle to the center turns into the motion away from the center, such turning point is called the pericenter (the particle is located in the closest distance from the center), or the motion away from the center turns into the approaching to the center, such turning point called the apocenter (the particle is located in the maximal distance from the center).
If there exist two such points, say r 1 , r 2 , then the motion is realized in the bounded region around the center and is called the finite motion. If the admissible interval of distances of the motion is bounded only from the left, the particle can approach the center up to a distance r 1 and then the particle escapes away from the center, the motion is called infinite or escaping motion. If the admissible interval of distances of the motion is bounded only from the right, e.g. by r 2 , the particle can move away from the center up to a distance r 2 and then it approaches the center.
In particular case if the line of the constant value E 0 is a tangent line to the graph of the effective potential energy then the appropriate contact point represents an extremum r ext and the motion proceeds in a constant distance r = r ext from the center, and is called circular motion. If at the point r ext is minimum then the corresponding circular orbit is stable, i.e. a small change of the initial conditions leads to a finite motion in the interval of possible distances r 1 , r 2 . If at the point r ext is maximum then the corresponding circular orbit is unstable, i.e. a small change of the initial conditions leads either to an approaching to the center or to an infinite motion.
3.6. Classification of trajectories in Newtonian gravitational field from the diagram of the effective potential energy. The effective potential energy for the Newtonian gravitational field takes the form and the course of this function is drawn on the following figure. . Notice that r min = 2r 0 . In the Fig. 3 the restricted area in which the particle cannot be located is displayed. Now we perform the complete classification of motions in the Newtonian gravitational field with zero radial component of the initial velocity using the diagram of the effective potential energy U ef . In the following figures one can see the typical situations concerning selected values of the initial energy E 0 related to the appropriate courses of the effective potential energy U ef (r, l 0 ) depending on the parameter l 0 = mr 0 v 0 and the corresponding trajectories. The testing particle starts in the fixed distance r 0 from the center with the initial velocity vector v 0 in the "horizontal" direction, i.e. in the direction perpendicular to the connecting line of the starting point and the center (motions with zero radial component of the initial velocity). The only variable quantity is the magnitude of the initial velocity v 0 . However the value of the initial velocity v 0 determines the value of the initial energy E 0 (v 0 ) = (1/2)mv 2 0 + α/r 0 , the value of the initial angular momentum l 0 (v 0 ) = mr 0 v 0 and consequently also the graph of the effective potential energy U ef (r, v 0 ) = −(κmM )/r + mr 2 0 v 2 0 /(2r) 2 . Recall that in this case the effective potential energy U ef takes the null value at the point r 0 = r 2 0 v 2 0 /(2κM ) and has minimum U min

Figure 4. Infinite motions in the Newtonian gravitational field
For E 0 > 0 (i.e. the corresponding magnitude of the initial velocity v 0 > v esc ) there exists just one intersection with the graph, the motion is infinite and proceeds in the interval of the possible distances r ∈ r 0 , ∞), and the trajectory of the particle is hyperbolic. The case E 0 = 0 is formally the same as the preceding situation but it represents a limit case when v 0 = v esc (88), for which the motion becomes infinite. The particle escapes away from the distance r 0 along the parabolic trajectory, see Fig. 4.
For E 0 < 0 and E 0 > U min ef (i.e. v circ < v 0 < v esc ) there exist two intersection points, r 0 , r 1 . The motion is finite and proceeds in the interval of distances r ∈ r 0 , r 1 , the trajectory is elliptical with the focus at the center and the starting point r 0 is the pericenter of this trajectory, see Fig. 5.
If E 0 = U min ef (i.e. v 0 = v circ (87)) then the motion proceeds in a constant distance r 0 = r min from the center, the trajectory is circular, Fig. 6.   It would seem that in the case E 0 < U min ef (i.e. when v 0 < v circ ) the particle is located in the restricted area, thus the motion would not have been possible. However, it is necessary to realize that for these values of the initial energy E 0 resp. the magnitude of the initial velocity v 0 the course of effective potential changes in such a way that minimum is at the smaller distance r min than in the previous case and its absolute value |U min ef | increases. Consequently, the line of the constant value E 0 has in this case again two intersection points, r 1 , r 0 . The motion is finite and proceeds again along the elliptical trajectory with the focus at the center. Unlike Fig. 5 the initial distance r 0 is the apocenter of this trajectory, see Fig. 7. By that the classification of the trajectories of motions in the Newtonian gravitational field with zero radial component of the initial velocity is completed.

Uniform motions in central fields -three dimensional situation.
Let us consider the mechanical system of one particle moving in a central field. The Lagrangian of the system is given by (46). The particle is subject to to isotachytonic constraint i.e. the magnitude of the velocity v remains constant during the motion. The constraint (95) can be also represented by The existence of the constraint (95) or (96) gives rise the constrained force called Chetaev force From the modifications of the equation (98) follows By the differentiating the constraint condition (95) we obtain v ·v = 0, consequently for the Lagrange multiplier µ we get Since the motion equations (98) (after the left vector multiplication by r [4]) take the form and finallyl thus vectors l andl are collinear. Hence, uniform motions in central fields are always planar, the motions proceed in the plane perpendicular to the angular momentum l. In particular, if the initial value l 0 of the angular momentum l is zero (initial vectors r 0 and v 0 are collinear) then the angular momentum remains null during the motion (the uniform radial motion either from the center or to the center). Notice that in the caseṙ(t) = 0, i.e. when the particle moves in the constant distance r 0 from the center (the uniform circular motion) Lagrange multiplier (100) becomes null and (103) gives us the classical conservation law of the angular momentum l = l 0 = const. As we will see at the beginning of the subsection 5.3 in this extraordinary case the Chetaev force (97) disappears and therefore the uniform circular motion can be considered as well as the classical (non constrained) circular motion or the constrained circular motion in the Newtonian gravitational field.
The differential equation (103) can be easily integrated and we obtain Now, we substitute the Lagrange multiplier (100) into (104) and after some rearrangements we get the dependence of the angular momentum l of the particle on its distance r from the center during the uniform motion in a central field. The last relation in (105) can be rewritten in the form The conservation law (106) of the angular momentum l can be rewritten in the scalar form as follows which represents the isotachytonic version of the conservation law of the angular momentum instead (62).

Motion equations.
Similarly as in the Subsection 3.3, we will assume that the uniform motions proceed in the xy-plane, i.e. l 0 = (0, 0, l 0 ) and the problem will be solved in the polar coordinates (r, ϕ). For Lagrangian we have and the isotachytonic constraint is now expressed by the equation

MARTIN SWACZYNA AND PETR VOLNÝ
The reduced (constrained) LagrangianL has the expression The motions we are going to study will be considered as the anticlockwise motions (φ ≥ 0). Hence the equation (109) can be rewritten into the explicit forṁ The reduced motion equation (27) of the reduced Lagrangian (110) takes the form in particular in the Newtonian gravitational field which is in accordance with the motion equation presented in [4]. The second term on the right-hand side of (112), respective (113) we interpret as a certain additional repulsive central forceF An alternative approach to the dynamics offers us deformed equations of motion, which give rise from the original unconstrained motion equations by adding Chetaev force. The deformed equations of the uniform motions in the Newtonian gravitational field in the polar coordinates are expressed by Previous equations can be rewrite as follows where a r =r − rφ 2 , a ϕ = rφ + 2ṙφ, Φ r = 2µṙ, Φ ϕ = 2µrφ, F = α r 2 e r , (117) a r , a ϕ , Φ r and Φ ϕ are radial and angular components of the acceleration a and radial and angular components of the Chetaev force Φ, respectively.
The deformed equations (116) can be reformulated in the classical Newtonian vector form ma = F + Φ = F (r)e r + Φ = α r 2 e r + 2µv.
The procedure of the elimination of the Lagrange multiplier µ from (115) together with the equation of constraint (111) leads to the reduced equation (113). By the lifting of the constraint equation (95) written in the vector form v·v−v 2 0 = 0 onto the space of accelerations J 2 Y by means of the total time differentiating, we get a · 2v = 0, i.e. a ⊥ v. Such accelerationã which satisfies this condition, i.e. the constraint admissible acceleration, is called in [18] partial non-holonomic acceleration. The conditionã · 2v = 0, where v = v r e r + v ϕ e ϕ =ṙe r + rφe ϕ , can be satisfied for the following expression of the accelerationã, a =ã r e r +ã ϕ e ϕ = e r −ṙ rφ e ϕ .
By the scalar multiplication of the deformed equation (118) in the vector form by the non-holonomic partial acceleration (119) and by the application of the constraint condition (111) and its lifẗ we derive after some manipulations the reduced equation (112) or (113). The same result can be obtained by the scalar multiplication the undeformed equations of the motion (47) by the partial non-holonomic acceleration (119). If we suppose thatφ = 0, then after certain manipulations of the second deformed equation (115) we get the expression for the Lagrange multiplier µ in the form of the total time derivative, Dynamical equations for uniform motions in central fields admit special solutions, which are uniform radial motions (v ϕ (0) = 0), i.e. when the particle moves uniformly rectilinearly in a radial direction even from the center or to the center and uniform circular motions (v r (0) = 0), when the particle moves uniformly around the center at the constant distance r 0 . Except these special motions, when l 0 = l = 0 orṙ(t) = 0, the problem of uniform motions in central fields one can solve formally analytically by means of quadratures.
Let us suppose that l 0 = 0 andṙ = 0. Using the isotachytonic version of the conservation law of the angular momentum (107), where l = mr 2φ , we get, Taking into account the equation of the constraint (111) we havė After the separation of variables and integration we obtain which especially in the Newtonian gravitational field gives us mr dr in our case we set t 0 = 0. Sinceφ = (dϕ/dr)ṙ then In the case of the Newtonian gravitational field we get after the integration the following However, presented expression of the solution, which can be found also in [4], is not suitable for the visualization of the trajectories. Therefore the differential equations (111) and (113) are solved numerically by the Runge-Kutta method. The possible motions can be classified with respect to initial conditions, especially with respect to different values of the radial and angular components of the initial velocity as follows: uniform motions with zero angular component of the initial velocity -uniform radial motions, uniform motions with zero radial component of the initial velocity and uniform motion with non-zero radial and non-zero angular component of the initial velocity. The above mentioned special solutions, uniform circular motions, evidently belong to the second class. We are going to investigate only the first two classes because the third class is just a composition of them.

5.2.
Uniform radial motions -motions with zero angular component of the initial velocity. Let the initial conditions of the particle are the following: Then the magnitude l 0 = mr 2 0φ (0) of the initial angular momentum l 0 is 0. It is easy to see from (103) that the magnitude l remains null during the motion. So,φ(t) = 0 for every t, thus ϕ(t) = ϕ 0 = const. Hence, the motion is radial. Furthermore, the constraint equation (109) reduces tȯ and can be easily integrated, The particle moves uniformly rectilinearly in the radial direction either from the center (+v 0 ) or to the center (−v 0 ). The reduced equation of motion (113) degenerates. There is only one non-zero component of the Chetaev force Φ, the radial component Φ r = 2µṙ(0) = 2µv 0 .
(132) While the second deformed equation of (115) becomes trivial, the first one represents the equilibrium condition between the central gravitational and Chetaev force, |F | = |Φ|: which is provided by the Lagrange multiplier µ, for the motion from the center, for the motion to the center.

Uniform motions with zero radial component of the initial velocity.
Let v r (0) =ṙ(0) = 0, hence v ϕ (0) = r 0φ (0) = v 0 . First of all we analyze one special motion -uniform circular motion. As it will be shown in the Section 7 this motion is unstable. In the unconstrained case the uniform circular motion arises directly as the solution of differential equations (72) with respect to the specific initial conditions typical for this case: where v circ is given by (87). On the other side, the uniform circular motion can also be treated as constrained motion. Indeed, it can be obtained as the solution of the reduced equation (113) or deformed equations (116) with the same specific initial conditions (135). It shows that the presence of the isotachytonic constraint is redundant in this case, i.e. the Chetaev force vanishes. The uniform circular motion is described by the parametric equations hence the isotachytonic constraint (109) becomes (112) and will be discussed the energetic balance of the uniform motions. We omit uniform circular motions, when the energetic balance is trivial, and the radial motions since the reduced equation degenerates. At first, we multiply the reduced equation (112) byṙ,ṙ = 0. We obtain which can be expressed (after the multiplying the both sides of the previous equation by −1) in the form of the total time derivative Hence, the function is the first integral, i.e. it remains constant along every isotachytonic trajectory (except the circular and radial ones). First term in (143), By the relationŨ ef (r) = U (r) +Ũ c (r), (147) the modified effective potential energy is defined. The last term in (143), replaces the constant kinetic energy E k = mv 2 0 /2 in the energetic balance and is called the isotachytonic compensation of the kinetic energy. The function is called the isotachytonic compensation coefficient. Finally, the expression (143) becomes where κ(0) = ln(1/(v 2 0 sin 2 β 0 )), and can be interpreted as the isotachytonic version of the conservation law of the mechanical energy in central fields. The function ψ 1 is called the modified mechanical energy and is denoted byẼ.
The conservation law (150) can be reformulated in the following way: The conservation law (152) can be expressed in the following differential form along every isotachytonic trajectory except the circular and radial ones. If we join logarithm terms in (143) and add the conservation law (106), we get the isotachytonic analog of the classical conservation laws (72). If we substitute the expression (111) of the constraint into the second equation of (154) then after some arrangements we obtain the isotachytonic conservation law of the mechanical energy (the first equation of (154)), thus on the constraint manifold both conservation laws (154) merge into the each other. Evidently, the work W Φ of the Chetaev constraint force Φ along constrained path J 1γ balances the changes of the potential energy U (r) of the moving particle instead of the kinetic energy E k , which remains constant. A computation of the work W Φ of the Chetaev force along a small piece of the constrained trajectory J 1γ from the given position r 1 to r 2 in the infinitesimal small time interval ∆t = t 2 − t 1 in the polar coordinates gives us, where h r = 1 and h ϕ = r are the Lamé coefficients for the polar coordinates. After the integration we get Finally, the isotachytonic version of the conservation law of the mechanical energy along the entire trajectory J 1γ can be expressed by the following alternatives: We conclude this section by the discussion concerning the energetic balance for concrete particular cases of the uniform motions in the Newtonian gravitational field: • general motions (the isotachytonic version of conservation law of energy)

119
• circular motions (the classical version of conservation law of energy) • radial motions 7. Classification of uniform motions in Newtonian gravitational field from the diagram of the modified effective potential energy. Recall, that for uniform non-radial motions (φ(0) = 0 i.e.ṙ(0) = v 0 ) in central fields it holds the isotachytonic version of the conservation law of the mechanical energy, which has the formẼ whereŨ is the modified effective potential energy, which is the sum of the potential energy U (r) of the considered central field and the modified centrifugal potential energỹ U c (r) = mv 2 0 ln(1/r) and where the functioñ is the isotachytonic compensation of the constant kinetic energy E k = (1/2)mv 2 0 . Notice that the value of the magnitude of the initial velocity v 0 plays here the role of a parameter, therefore we sometimes writeŨ ef (r, v 0 ).
The constantẼ 0 occurring on the right-hand side of (161) represents the initial value of the modified mechanical energyẼ, and is equal tõ Notice that in the case of motions with zero radial component of the initial velocity we obtainẼ After the standard manipulations with the equation (161) we geṫ Sinceṙ 2 (t) is non-negative function, the right-hand side of (166) must be also nonnegative, After further manipulations one can derive the conditioñ which is a certain analog of the condition (93). It enables us to make a qualitative analysis of the uniform non-radial motions in central fields as well as the condition (93) provides a possibility for the qualitative analysis of the classical motions in central fields. In particular, for motions with zero radial component of the initial velocity the relation (168) simplifies toŨ ef (r 0 ) ≥Ũ ef (r), wherẽ Like in the Subsection 3.6 the graphic solution of the inequality (168) respective (169) provides us admissible intervals of radial distances from the center in which the particle can be located under the given initial valueẼ 0 of the modified mechanical energy, respective the magnitude of the initial velocity v 0 . The modified effective potential energy for the Newtonian gravitational field is the functionŨ The graph of this function is shown on the following figure. We see that the functionŨ ef (r) is negative at its domain of definition. At the point r max = (κM )/v 2 0 has maximum valueŨ max ef = −mv 2 0 (1 − ln(v 2 0 /(κM ))). Restricted area in which the particle cannot be located is displayed. If we consider only motions with zero radial component of the initial velocity for which it is relevant (169), we obtain using the diagram of the modified potential energyŨ ef (r) the following classification of uniform motions in the Newtonian gravitational field.
IfŨ ef (r 0 ) =Ũ max ef , it represents a limit case, when the line of the constant valuẽ U ef (r 0 ) is a tangent line to the graph ofŨ ef (r). The motion proceeds at the constant distance r = r max and since it is maximum, the appropriate motion is the unstable circular motion.
IfŨ ef (r) <Ũ ef (r 0 ) <Ũ max ef , in this case there exist two intersection points r 1 , r 2 of the line of the constant valueŨ ef (r 0 ) with the graphŨ ef (r). Then interval of admissible distances is either 0, r 1 or r 2 , ∞). In the first subcase it is a finite motion to the center (ending in the center). In the second subcase it is an infinite (escaping) motion. The case which is realized depends on the starting distance r 0 from the center.
In the following figures one can see these three typical situations for the case of uniform motions with zero radial component of the initial velocity, i.e. when the particle starts in the fixed distance r 0 from the center with the magnitude of the initial velocity v 0 (v 0 < v circ , v 0 = v circ , v 0 > v circ ) in the direction perpendicular to the connecting line of the starting point and the center.  The fact that the circular trajectory is unstable, is illustrated on the following figure. A small change of the value of the magnitude of the initial velocity v 0 with respect to the value v circ , i.e. v 0 = v circ ± ε, causes a perturbation of the circular trajectory. The motion begins almost as circular one. A deviation of the trajectory of the particle from the circular orbit will increase in time and particle either will approach to the center or will escape from the center. The rate of the deviation of the trajectory is proportional to the value of deviation ε.
with the potential U (r) = b ln r.
(172) The modified effective potential for this central gravitational field is the functioñ Notice that for b > mv 2 0 the potential U (r) of the attractive central field (171) is dominant and on the contrary in the case b < mv 2 0 the modified centrifugal potential U c (r) prevails. The course of the modified effective potential for the both cases is illustrated on the Fig. 12. The particle starts from the initial position r(0) = r 0 , ϕ(0) = ϕ 0 with the initial velocity vector v 0 =ṙ(0)e r + r 0φ (0)e ϕ . We consider only non-radial motions (φ(0) = 0, i.e.ṙ(0) = v 0 ). The magnitude of the initial angular momentum vector is l 0 = mr 0 v 0 sin β 0 , where β 0 ∈ (0, π) is the initial angle between the vector of the initial position r 0 and the initial velocity vector v 0 .
The equation (201) together with the first integrals ψ 1 , ψ 3 and ψ 4 formally represents the set of all constraint Noetherian symmetries of the considered nonholonomic constrained system.
Notice that the conserved function ψ 1 is Noetherian conservation law of the form (199). If we take then from (199) we directly get the isotachytonic version of the conservation law of mechanical energy ψ 1 , respectiveẼ. Evidently, the function A 1 is the particular solutions of PDE (195), since it satisfy the implicit relation −ψ 1 + ψ 2 = 0. Finally, the conservation law (198) corresponds to the following constraint symmetry where A 1 is given by (204) and the corresponding B 1 , C 1 one can compute by means of (194) substituting A = A 1 .

10.
Conclusions. The modification of the classical motions of the particle in the central fields on uniform motions means substantial changes from the point of view of kinematics, dynamics and energetic balance of the considered constrained motions. While the trajectories of classical motions in the Newtonian gravitational field one can get in the explicit analytical form (82) describing the Kepler's trajectories, in the case of uniform motions in this field it is not possible to express the formula (128) describing their trajectories by means of some reasonable elementary functions. The expression (128) is presented also in [4], but it is not suitable for the visualization of the trajectories. Therefore we solve reduced motion equation (113) together with the constraint equation (111) numerically with respect to suitable initial conditions.
In the paper the trajectories of the uniform motions in the Newtonian gravitational field for the typical values of the initial velocity 0 < v 0 < v circ , v 0 = v circ , v circ < v 0 < v esc , v 0 = v esc = v circ √ 2, v 0 > v esc was presented and compared with the corresponding classical unconstrained motions under the same initial conditions, Fig. 8. From the analysis of trajectories of the uniform motions follows that they have quite different qualitative character in the correspondence with the trajectories of the classical motions in the Newtonian gravitational field. There is only one common special case for both kinds of motions -uniform circular motions with the initial velocity v 0 = v circ (87). But there is a significant difference: while in the classical case the circular motion is stable, in the constrained (isotachytonic) case the circular motion is unstable, i.e. that it represents a border between the falling motions to the center (for v 0 < v circ ) and the escaping motions (for v 0 > v circ ). Described situation is a direct consequence of the course of the modified effective potential energyŨ ef (r), Fig. 9, which posses a global maximum and therefore it does not permit the existence of stable periodic trajectories as the closed Kepler's elliptical trajectories in the classical case for v 0 < v circ or v circ < v 0 < v esc . The dynamics of the uniform motions in the central fields is explained by the presence of the Chetaev constrained force Φ, which ensures the requirement of the uniformity of the motion. Recall the expression of the Chetaev force, Φ = Φ r e r + Φ ϕ e ϕ = 2µ(t) ṙe r + r 2φ e ϕ = 2µ(t)v(t) = 2µ(t)v 0 e v(t) , where µ(t) is the Lagrange multiplier, v(t) is the velocity vector at the time t, e v(t) is the unit vector in the direction of v(t). It is evident that during the motion the Chetaev constraint force has to change its magnitude |Φ| = 2v 0 |µ(t)| and also its direction e v(t) . However, in this case the time dependence of the Chetaev force can not be explicitly expressed unlike the uniform projectile motion in the homogeneous gravitational field [21]. From the point of view of the conservations laws one can state firstly, that the conservation law of angular momentum l has in this case another modified form than the classical conservation law of the angular momentum, i.e. its magnitude |l| is not constant but it changes during the uniform motions in central fields according to (107). However what remains valid is the fact, that during the uniform motions in central fields the angular momentum l lies still in the same line determined by the initial angular momentum l 0 , cf. (103) respective (104). This fact allows us to reduce uniform motions in central fields (as well as classical motions) into the planar motions proceeding in the initial plane perpendicular to the initial angular momentum l 0 [4].
Secondly, the requirement for uniformity of the motion in central fields completely disturbs the classical conservation law of the mechanical energy, which obviously can not be satisfied. However, one can formulate the isotachytonic version of the conservation law of mechanical energy, in which the role of the standard kinetic energy E k plays the isotachytonic compensation of the kinetic energyẼ k (148), which balances changes of the modified effective potential energyŨ ef (147) (the sum of the standard potential energy U of a central field and the modified centrifugal energyŨ c (145)) instead of the classical kinetic energy E k , which remains constant during the uniform motions. This modified version of the conservation law of energy was identified as one (ψ 1 , (198)) of the particular conservation laws ψ 1 , ψ 2 , ψ 3 , ψ 4 in the set of Noetherian conservation laws (200) for uniform motions in the Newtonian gravitational field arising from the constraint Noetherian symmetry (205) of the considered mechanical system applying the general definition [9] of the constraint Noetherian symmetry of nonholonomic constrained Lagrangian systems.
As an interesting particular result we present in the Sec. 8 an explicit analytical form of trajectories (182) (sinusoidal spirals) of the uniform motions in the special gravitational field, called inverse proportional gravitational field (39). Note that trajectories of classical (unconstrained) motions in this field can not be expressed by means of reasonable elementary functions.
As the main result we obtained in the Sec. 9 a formal expression of all Noetherian conservation laws (200) by means of three base conservation laws ψ 1 , ψ 3 and ψ 4 derived as a certain general solution of PDE (195) using the method of characteristics. Physical interpretation of these conversation laws in the context of the solution of the problem of uniform motions in the Newtonian gravitational field was found. The first base conservation law ψ 1 represents the isotachytonic version of the conservation law of mechanical energy, as it was mentioned above. The base conservation laws ψ 3 and ψ 4 represent directly the parametric expressions t = t(r) and ϕ = ϕ(r) of the trajectories of the uniform motions in the Newtonian gravitational field. These conservation laws one can understand as trivial conservation laws, which carry only an information about some initial data of the motion. Indeed, when we evaluate ψ 3 and ψ 4 along some concrete trajectory of the uniform motion then we obtain just the value t 0 , the instant of the beginning of the motion (in our case we set t 0 = 0) and the value ϕ(t 0 ) = ϕ(0) = ϕ 0 , the polar angle of the starting point of the motion.
The results described above contribute to the issue of the kinematics, dynamics, Noetherian symmetries and conservation laws of nonholonomic mechanical systems.
We are aware that the studied problem is only a kind of a theoretical modification of the real situation, whose technical realization would seem difficult to implement.