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The motion of a point mass on the surface of a smooth funnel
Journal of Applied Mathematics and Mechanics 75 (2011) 171–175
Contents lists available at ScienceDirect
Journal of Applied Mathematics and Mechanics journal homepage: www.elsevier.com/locate/jappmathmech
The motion of a point mass on the surface of a smooth funnel夽 A.P. Blinov Moscow, Russia
a r t i c l e
i n f o
Article history: Received 4 May 2009
a b s t r a c t The motion of a point mass on a smooth concave surface (a funnel) under the action of a gravitational force is considered. The equations of motion are reduced to a form to which Lyapunov’s theorem on the representation of the solution in the form of power series in the initial conditions, which converge absolutely in a finite region of phase space is applied. In the non-local formulation of the problem, a procedure is described for estimating the libration periods, based on an analysis of geometric forms. A bilateral estimate of the region of possible motion of the point is given for rotational-type motions, when the funnel is a surface of revolution. © 2011 Elsevier Ltd. All rights reserved.
The problem considered here is a generalization of the well-known analysis of the motion of a point on a surface in a potential force field (Ref. 1, p. 60, Ref. 2, Ref. 3, p. 194, etc.). A method of reducing the system of equations of motion of a point on a plane in a potential force field, proposed by Zhukovskii (Ref. 1, p. 60) in the case of smooth surfaces, was generalized previously in Ref. 4. Below we construct a solution of the equations of motion of a heavy point in the form of series in the initial conditions and we prove their convergence in a limited region of phase space in a bounded time interval using Lyapunov’s theorem on non-linear equations (Ref. 5, p. 46). Non-local estimates are obtained using well-known results (Ref. 3, p. 194, Ref. 6, etc.). 1. The motion of a point mass on the bottom of a funnel. Consider the motion of a point mass of unit mass acted upon by a gravitational force on a smooth concave surface (a funnel), defined by the equation
where f1 (y, z) is a holomorphic function in the region |y| ≤ A1 , |z| ≤ A2 . The directions of the coordinate axes y and z will be assumed to coincide with the directions of the principal curvatures of the funnel k1 and k2 at its lowest point, and the x axis is directed vertically upwards. The equation of the surface of the funnel has the form
(1.1) where f(y, z) is a function, the expansion of which in series in powers of y and z begins with a power higher than the second. We will write the expressions for the kinetic and potential energy of the particle
(1.2) where
Lagrange’s equations (1.3)
夽 Prikl. Mat. Mekh. Vol. 75, No. 2, pp. 247-253, 2011. E-mail address: [email protected] 0021-8928/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.jappmathmech.2011.05.006
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A.P. Blinov / Journal of Applied Mathematics and Mechanics 75 (2011) 171–175
where
can be written in the form (1.4) The expansion of the function 1/ in powers of y and z converges when b21 + b22 < 1. A certain rectangle
is contained in the set of points which satisfy the last inequality. In view of the holomorphic nature of the function f(y, z) when |y| < A1 , |z| < A2 , the right-hand sides of Eqs (1.4) are also holomorphic, at least in the rectangle
Because of the choice of ls we can assume that A∗1 = A∗2 = A. Putting
we will write system (1.4) in the normal form (1.5) where
Below we will assume system (1.5) reduced to dimensionless form. By Lyapunov’s theorem on the representation of the solutions of non-linear equations (Ref. 5, p. 46) we will seek solutions of system (1.5) in the form of expansions in powers of the initial data xi (0) = ai , (i = 1, 2, 3, 4). We have
(1.6) The sum extends to integer non-negative numbers mi , which satisfy the condition m1 + m2 + m3 + m4 = m. The fundamental matrix of the solutions of the linear part of the system, which is transformed into a unit matrix when t = O, has the form
(1.7) √
O is a zero 2 × 2 matrix and w2s−1 = g2s−1 , s = 1, 2, det = 1. The moduli of the diagonal elements of the matrix and their cofactors do not exceed unity, while the moduli of the remaining elements and their cofactors do not exceed the value
The moduli of the solutions of the linear part of system (1.5) are estimated using the inequalities
Suppose a is the maximum of the right-hand sides of these inequalities. Following the well-known approach,5 we will denote by u˜ the result of the replacement in any function u of the quantities ai of all terms by their moduli. Then, for m > 1
(m)
where xkj are the elements of the matrix (1.7). In the same way we also obtain estimates for the remaining elements x˜ i
.
A.P. Blinov / Journal of Applied Mathematics and Mechanics 75 (2011) 171–175
173
(m) Further, denoting by x˜ (m) a certain general upper limit for x˜ i in the interval (−, ), and by X˜ (m) the quantity which is the result of the (m) (m) (m) replacement in each of the functions X˜ , X˜ of the quantities x˜ by the quantities x˜ (m) , and of the functions xs (t)(m1 , m2 , m3 , m4 ) by the 2
4
i
upper limits of these functions, which are independent of the subscript, in the interval (−, ), we obtain
The strengthening series5 for any of the solutions of system (1.5) here takes the form
Since
where M is the maximum of the modulus of the non-linear components of the right-hand sides of system (1.4) when |y| < A, |z| < A, the quantity x˜ satisfies the equation
(1.8) Series (1.6) converge absolutely for all t ∈ (− , ), if |as | < a∗ , where a* is the solution of the system of equations F(a, x˜ ) = 0, and has the form
Fx˜ (a,
x˜ ) = 0,
2. The periodic libration-type periodic motions of a point mass. The problem of finding periodic solutions of the Lyapunov system considered in the local formulation has been thoroughly investigated. In the non-local case, corresponding to well-known theorems,6 periodic libration motion exists on the funnel. The trajectory of this motion has no self-intersections and its ends lie on the boundary of the region of possible motion. Hence, in particular, it follows that, if the region of possible motion possesses symmetry about the x axis, the trajectory of such motion passes through the origin of coordinates. We will prove the last assertion by assuming the opposite. We will assume that the particle released from the end of the trajectory has passed by the point O at the minimum distance ␦, i.e., this trajectory has touched the circle C of radius ␦ with centre at the point O. We will release the same particle from the other end of the libration trajectory. In view of the symmetry of the region of possible motion, the particle touches the circle C at a diametrically opposite point. In further motion, since self-intersection of the libration trajectory is impossible, the particle is incident on the opposite end of the trajectory along the line, different from the initial one. But this is impossible in view of the theorem of the uniqueness of the solutions of differential equations. If the funnel has a plane of symmetry, passing through the x axis, the section of the funnel by this plane is a line of curvature. A particle, situated on such a line at the initial instant of time with zero velocity, will always remain on this line, performing periodic oscillations. The motion of the particle along a known line can always be represented in the form of a quadrature.2 If there is no plane of symmetry, a particle beginning its motion from the point O along the line of least curvature (we will call it line A), will be deflected from the vertical plane containing the initial velocity vector. Suppose this is the plane z = 0, in which the x axis is directed vertically upwards. Suppose the surface of the funnel in the region of possible motion can be specified in the form (2.1) The function f0 (y, z) describes a smooth concave surface, symmetrical about the z = 0 plane, henceforward called the reference surface; > 0 is a small parameter, and f1 (y, z) is a smooth non-negative function, f1 (0, 0) = f0 (0, 0) = 0. The direction of minimum curvature, corresponding to this function, coincides with the direction of the y axis. If f1 (y, − z) = f1 (y, z), then, the long libration period 1 is the oscillation period of the particle on the line A
The oscillation period 0 on the more mildly sloping plane line x = f0 (y, 0) is greater than 1 , and the difference 0 − 1 gives some idea / f1 (y, z). of the order of smallness of the deviation of the period from 0 in the more general case, when f1 (y, − z) = We can give one more estimate of the long period , close to the upper estimate, by considering the limit possible deviation of the line A from the z = 0 plane (for the reference surface chosen earlier). It is a difficult problem to obtain the line A directly. When is small this line can be approximated by the projection of the funnel onto the reference surface in the direction of the z axis. The boundary of the shadow of the funnel on the reference surface (the edge) is the boundary of possible positions of the line A. The time of motion of the particle along the edge e in the forward and reverse directions gives the required upper estimate of the period , ignoring oscillations transverse to the line A. Hence, we have approximately ≤ max(0 , e ). Another libration with a shorter period s is possible on the funnel, for which we can give a lower estimate, if we calculate the time of motion of the particle tc along the arc of the cycloid (brachristochrone) connecting the boundary of the region of possible motion closest to the x axis with the point O with zero initial velocity, i.e., s ≥ 4tc .
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A.P. Blinov / Journal of Applied Mathematics and Mechanics 75 (2011) 171–175
Remark. An accurate but more difficult method of solution is possible in principle for the problem considered if, in the equation of the funnel surface, we change to curvilinear coordinates, representing the orthogonal network of lines of principal curvatures, one of which is the line A. We can then use the equations of the particle trajectory obtained earlier. 3. A bilateral estimate of the region of possible motion. We will define, for the funnel surface (2.1), a semigeodesic polar system of coordinates (p, q) with centre at the point O, taking the direction of the y axis as the origin of the angle p, and q as the length of the geodesic, emerging from the point O. The linear element of the surface in these coordinates has the form (Ref. 7, p. 117), (3.1) where E(p, q) is the coefficient of the first quadratic form of the surface, while the equation of the trajectory of the particle has the form4
where h is the total energy of the system. If the funnel is a surface of revolution, the motion of the particle can be interpreted as its motion on the plane in a central force field, which enables us to apply well-known results (Ref. 8, p. 61) to the investigation. For example, if U0 is the non-critical level of the potential energy, the region of possible motion is a ring 0< q1 ≤ q ≤ q2 < ∞. The distance of the particle from the point O then varies periodically with time. Further, if, in the position of relative equilibrium q = q0 , the reduced potential has a local minimum, the corresponding circular motion is orbitally stable. When the curvature of the funnel is reduced, the ring-shaped region of possible motion should shift to the periphery, by analogy with the weakening of the central force field. This can be used to estimate the size of the ring-shaped region of possible motion using standard surfaces of revolution, i.e., using surfaces on which the motion of the particles is known. It can be assumed that, in the more general case, when the quantities E and U also depend on the angular coordinate, the qualitative pattern of the change in the region of possible motion is preserved. This assumption is confirmed by the following example. We will estimate the ring-shaped region of possible motion of a point of unit mass on an elliptic paraboloid
considered as a funnel using an inscribed paraboloid of revolution, regarded as a standard surface
Suppose U = x and h = 2. We will write the equation of the funnel in the form
which was used previously (Ref. 3, p. 194). The closed trajectory of the point on an elliptic paraboloid coincides3 with the line of curvature, represented by the elliptic coordinate line w = , where is the double root of the equation
In the example considered, m = 1 and g = 1, and hence
and the closed trajectory can be represented in the parametric form
The lower horizontal, which touches this trajectory, is defined by the limit value of as → 0, i.e., = 7/8, or x = 15/16. Contact occurs when z = 0 and y = 0.968. The geodesic, which connects this point with the centre, is the parabola x = y2 , and the length of its corresponding portion is
and on the paraboloid of revolution x = 4y2 + 4z2 when h = 2 the particle will remain on a parallel of radius 1/2, and the distance from it to the centre O
A.P. Blinov / Journal of Applied Mathematics and Mechanics 75 (2011) 171–175
175
Hence, the inequality q0 < q1 is satisfied here. The time taken for the particle to travel along the closed curve is equal to3
where E is the elliptic integral of the second kind. The particle makes a rotation round the paraboloid of revolution in a time
The elliptic paraboloid and the paraboloid of revolution now change roles, i.e., we estimate the ring-shaped region of possible motion on the paraboloid of revolution x = y2 + z2 using the ring-shaped region of possible motion on the previous elliptic paraboloid x = y2 + 4z2 at the previous energy level h = 2. According to the condition of relative equilibrium on the paraboloid of revolution x = y2 + z2 = 1, the length of the geodesic from the point √ ˜ O to this horizontal is equal to 1.53 and the time for a single revolution = 2=4.4. References 1. 2. 3. 4. 5. 6. 7. 8.
Zhukovskii NYe. Complete Works. Vol. 1. General Mechanics. Moscow: Glav Red Aviats Lit; 1937. Whittaker ET. Treatise on the Analytical Dynamics of Particles and Rigid Bodies. Cambridge: Univ Press; 1927, 430p. Chaplygin SA. Complete Works. Vol. 1. Leningrad: Izd Akad Nauk SSSR; 1933. Blinov AP. The motion of a point mass on a surface. Izv Ross Akad Nauk MTT 2007;1:23–8. Duboshin GN. Celestial Mechanics. Analytical and Qualitative Methods. Moscow: Nauka; 1964. Kozlov VV. The principle of least action and periodic solutions in problems of classical mechanics. Prikl Mat Mekh 1976;40(3):399–407. Poznyak EG, Shikin YeV. Differential Geometry. Moscow: Izd MGU; 1990. Arnold VI, Kozlov VV, Neishtadt AI. Mathematical Aspects of Classical and Celestial Mechanics. Advances in Science and Technology. Current Problems in Mathematics. Fundamental Trends. Vol. 3. Moscow: VINITI; 1985.
Translated by R.C.G.
Contents lists available at ScienceDirect
Journal of Applied Mathematics and Mechanics journal homepage: www.elsevier.com/locate/jappmathmech
The motion of a point mass on the surface of a smooth funnel夽 A.P. Blinov Moscow, Russia
a r t i c l e
i n f o
Article history: Received 4 May 2009
a b s t r a c t The motion of a point mass on a smooth concave surface (a funnel) under the action of a gravitational force is considered. The equations of motion are reduced to a form to which Lyapunov’s theorem on the representation of the solution in the form of power series in the initial conditions, which converge absolutely in a finite region of phase space is applied. In the non-local formulation of the problem, a procedure is described for estimating the libration periods, based on an analysis of geometric forms. A bilateral estimate of the region of possible motion of the point is given for rotational-type motions, when the funnel is a surface of revolution. © 2011 Elsevier Ltd. All rights reserved.
The problem considered here is a generalization of the well-known analysis of the motion of a point on a surface in a potential force field (Ref. 1, p. 60, Ref. 2, Ref. 3, p. 194, etc.). A method of reducing the system of equations of motion of a point on a plane in a potential force field, proposed by Zhukovskii (Ref. 1, p. 60) in the case of smooth surfaces, was generalized previously in Ref. 4. Below we construct a solution of the equations of motion of a heavy point in the form of series in the initial conditions and we prove their convergence in a limited region of phase space in a bounded time interval using Lyapunov’s theorem on non-linear equations (Ref. 5, p. 46). Non-local estimates are obtained using well-known results (Ref. 3, p. 194, Ref. 6, etc.). 1. The motion of a point mass on the bottom of a funnel. Consider the motion of a point mass of unit mass acted upon by a gravitational force on a smooth concave surface (a funnel), defined by the equation
where f1 (y, z) is a holomorphic function in the region |y| ≤ A1 , |z| ≤ A2 . The directions of the coordinate axes y and z will be assumed to coincide with the directions of the principal curvatures of the funnel k1 and k2 at its lowest point, and the x axis is directed vertically upwards. The equation of the surface of the funnel has the form
(1.1) where f(y, z) is a function, the expansion of which in series in powers of y and z begins with a power higher than the second. We will write the expressions for the kinetic and potential energy of the particle
(1.2) where
Lagrange’s equations (1.3)
夽 Prikl. Mat. Mekh. Vol. 75, No. 2, pp. 247-253, 2011. E-mail address: [email protected] 0021-8928/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.jappmathmech.2011.05.006
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A.P. Blinov / Journal of Applied Mathematics and Mechanics 75 (2011) 171–175
where
can be written in the form (1.4) The expansion of the function 1/ in powers of y and z converges when b21 + b22 < 1. A certain rectangle
is contained in the set of points which satisfy the last inequality. In view of the holomorphic nature of the function f(y, z) when |y| < A1 , |z| < A2 , the right-hand sides of Eqs (1.4) are also holomorphic, at least in the rectangle
Because of the choice of ls we can assume that A∗1 = A∗2 = A. Putting
we will write system (1.4) in the normal form (1.5) where
Below we will assume system (1.5) reduced to dimensionless form. By Lyapunov’s theorem on the representation of the solutions of non-linear equations (Ref. 5, p. 46) we will seek solutions of system (1.5) in the form of expansions in powers of the initial data xi (0) = ai , (i = 1, 2, 3, 4). We have
(1.6) The sum extends to integer non-negative numbers mi , which satisfy the condition m1 + m2 + m3 + m4 = m. The fundamental matrix of the solutions of the linear part of the system, which is transformed into a unit matrix when t = O, has the form
(1.7) √
O is a zero 2 × 2 matrix and w2s−1 = g2s−1 , s = 1, 2, det = 1. The moduli of the diagonal elements of the matrix and their cofactors do not exceed unity, while the moduli of the remaining elements and their cofactors do not exceed the value
The moduli of the solutions of the linear part of system (1.5) are estimated using the inequalities
Suppose a is the maximum of the right-hand sides of these inequalities. Following the well-known approach,5 we will denote by u˜ the result of the replacement in any function u of the quantities ai of all terms by their moduli. Then, for m > 1
(m)
where xkj are the elements of the matrix (1.7). In the same way we also obtain estimates for the remaining elements x˜ i
.
A.P. Blinov / Journal of Applied Mathematics and Mechanics 75 (2011) 171–175
173
(m) Further, denoting by x˜ (m) a certain general upper limit for x˜ i in the interval (−, ), and by X˜ (m) the quantity which is the result of the (m) (m) (m) replacement in each of the functions X˜ , X˜ of the quantities x˜ by the quantities x˜ (m) , and of the functions xs (t)(m1 , m2 , m3 , m4 ) by the 2
4
i
upper limits of these functions, which are independent of the subscript, in the interval (−, ), we obtain
The strengthening series5 for any of the solutions of system (1.5) here takes the form
Since
where M is the maximum of the modulus of the non-linear components of the right-hand sides of system (1.4) when |y| < A, |z| < A, the quantity x˜ satisfies the equation
(1.8) Series (1.6) converge absolutely for all t ∈ (− , ), if |as | < a∗ , where a* is the solution of the system of equations F(a, x˜ ) = 0, and has the form
Fx˜ (a,
x˜ ) = 0,
2. The periodic libration-type periodic motions of a point mass. The problem of finding periodic solutions of the Lyapunov system considered in the local formulation has been thoroughly investigated. In the non-local case, corresponding to well-known theorems,6 periodic libration motion exists on the funnel. The trajectory of this motion has no self-intersections and its ends lie on the boundary of the region of possible motion. Hence, in particular, it follows that, if the region of possible motion possesses symmetry about the x axis, the trajectory of such motion passes through the origin of coordinates. We will prove the last assertion by assuming the opposite. We will assume that the particle released from the end of the trajectory has passed by the point O at the minimum distance ␦, i.e., this trajectory has touched the circle C of radius ␦ with centre at the point O. We will release the same particle from the other end of the libration trajectory. In view of the symmetry of the region of possible motion, the particle touches the circle C at a diametrically opposite point. In further motion, since self-intersection of the libration trajectory is impossible, the particle is incident on the opposite end of the trajectory along the line, different from the initial one. But this is impossible in view of the theorem of the uniqueness of the solutions of differential equations. If the funnel has a plane of symmetry, passing through the x axis, the section of the funnel by this plane is a line of curvature. A particle, situated on such a line at the initial instant of time with zero velocity, will always remain on this line, performing periodic oscillations. The motion of the particle along a known line can always be represented in the form of a quadrature.2 If there is no plane of symmetry, a particle beginning its motion from the point O along the line of least curvature (we will call it line A), will be deflected from the vertical plane containing the initial velocity vector. Suppose this is the plane z = 0, in which the x axis is directed vertically upwards. Suppose the surface of the funnel in the region of possible motion can be specified in the form (2.1) The function f0 (y, z) describes a smooth concave surface, symmetrical about the z = 0 plane, henceforward called the reference surface; > 0 is a small parameter, and f1 (y, z) is a smooth non-negative function, f1 (0, 0) = f0 (0, 0) = 0. The direction of minimum curvature, corresponding to this function, coincides with the direction of the y axis. If f1 (y, − z) = f1 (y, z), then, the long libration period 1 is the oscillation period of the particle on the line A
The oscillation period 0 on the more mildly sloping plane line x = f0 (y, 0) is greater than 1 , and the difference 0 − 1 gives some idea / f1 (y, z). of the order of smallness of the deviation of the period from 0 in the more general case, when f1 (y, − z) = We can give one more estimate of the long period , close to the upper estimate, by considering the limit possible deviation of the line A from the z = 0 plane (for the reference surface chosen earlier). It is a difficult problem to obtain the line A directly. When is small this line can be approximated by the projection of the funnel onto the reference surface in the direction of the z axis. The boundary of the shadow of the funnel on the reference surface (the edge) is the boundary of possible positions of the line A. The time of motion of the particle along the edge e in the forward and reverse directions gives the required upper estimate of the period , ignoring oscillations transverse to the line A. Hence, we have approximately ≤ max(0 , e ). Another libration with a shorter period s is possible on the funnel, for which we can give a lower estimate, if we calculate the time of motion of the particle tc along the arc of the cycloid (brachristochrone) connecting the boundary of the region of possible motion closest to the x axis with the point O with zero initial velocity, i.e., s ≥ 4tc .
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A.P. Blinov / Journal of Applied Mathematics and Mechanics 75 (2011) 171–175
Remark. An accurate but more difficult method of solution is possible in principle for the problem considered if, in the equation of the funnel surface, we change to curvilinear coordinates, representing the orthogonal network of lines of principal curvatures, one of which is the line A. We can then use the equations of the particle trajectory obtained earlier. 3. A bilateral estimate of the region of possible motion. We will define, for the funnel surface (2.1), a semigeodesic polar system of coordinates (p, q) with centre at the point O, taking the direction of the y axis as the origin of the angle p, and q as the length of the geodesic, emerging from the point O. The linear element of the surface in these coordinates has the form (Ref. 7, p. 117), (3.1) where E(p, q) is the coefficient of the first quadratic form of the surface, while the equation of the trajectory of the particle has the form4
where h is the total energy of the system. If the funnel is a surface of revolution, the motion of the particle can be interpreted as its motion on the plane in a central force field, which enables us to apply well-known results (Ref. 8, p. 61) to the investigation. For example, if U0 is the non-critical level of the potential energy, the region of possible motion is a ring 0< q1 ≤ q ≤ q2 < ∞. The distance of the particle from the point O then varies periodically with time. Further, if, in the position of relative equilibrium q = q0 , the reduced potential has a local minimum, the corresponding circular motion is orbitally stable. When the curvature of the funnel is reduced, the ring-shaped region of possible motion should shift to the periphery, by analogy with the weakening of the central force field. This can be used to estimate the size of the ring-shaped region of possible motion using standard surfaces of revolution, i.e., using surfaces on which the motion of the particles is known. It can be assumed that, in the more general case, when the quantities E and U also depend on the angular coordinate, the qualitative pattern of the change in the region of possible motion is preserved. This assumption is confirmed by the following example. We will estimate the ring-shaped region of possible motion of a point of unit mass on an elliptic paraboloid
considered as a funnel using an inscribed paraboloid of revolution, regarded as a standard surface
Suppose U = x and h = 2. We will write the equation of the funnel in the form
which was used previously (Ref. 3, p. 194). The closed trajectory of the point on an elliptic paraboloid coincides3 with the line of curvature, represented by the elliptic coordinate line w = , where is the double root of the equation
In the example considered, m = 1 and g = 1, and hence
and the closed trajectory can be represented in the parametric form
The lower horizontal, which touches this trajectory, is defined by the limit value of as → 0, i.e., = 7/8, or x = 15/16. Contact occurs when z = 0 and y = 0.968. The geodesic, which connects this point with the centre, is the parabola x = y2 , and the length of its corresponding portion is
and on the paraboloid of revolution x = 4y2 + 4z2 when h = 2 the particle will remain on a parallel of radius 1/2, and the distance from it to the centre O
A.P. Blinov / Journal of Applied Mathematics and Mechanics 75 (2011) 171–175
175
Hence, the inequality q0 < q1 is satisfied here. The time taken for the particle to travel along the closed curve is equal to3
where E is the elliptic integral of the second kind. The particle makes a rotation round the paraboloid of revolution in a time
The elliptic paraboloid and the paraboloid of revolution now change roles, i.e., we estimate the ring-shaped region of possible motion on the paraboloid of revolution x = y2 + z2 using the ring-shaped region of possible motion on the previous elliptic paraboloid x = y2 + 4z2 at the previous energy level h = 2. According to the condition of relative equilibrium on the paraboloid of revolution x = y2 + z2 = 1, the length of the geodesic from the point √ ˜ O to this horizontal is equal to 1.53 and the time for a single revolution = 2=4.4. References 1. 2. 3. 4. 5. 6. 7. 8.
Zhukovskii NYe. Complete Works. Vol. 1. General Mechanics. Moscow: Glav Red Aviats Lit; 1937. Whittaker ET. Treatise on the Analytical Dynamics of Particles and Rigid Bodies. Cambridge: Univ Press; 1927, 430p. Chaplygin SA. Complete Works. Vol. 1. Leningrad: Izd Akad Nauk SSSR; 1933. Blinov AP. The motion of a point mass on a surface. Izv Ross Akad Nauk MTT 2007;1:23–8. Duboshin GN. Celestial Mechanics. Analytical and Qualitative Methods. Moscow: Nauka; 1964. Kozlov VV. The principle of least action and periodic solutions in problems of classical mechanics. Prikl Mat Mekh 1976;40(3):399–407. Poznyak EG, Shikin YeV. Differential Geometry. Moscow: Izd MGU; 1990. Arnold VI, Kozlov VV, Neishtadt AI. Mathematical Aspects of Classical and Celestial Mechanics. Advances in Science and Technology. Current Problems in Mathematics. Fundamental Trends. Vol. 3. Moscow: VINITI; 1985.
Translated by R.C.G.