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The ellipsoidal pendulum
Journal of Applied Mathematics and Mechanics 75 (2011) 655–659
Contents lists available at SciVerse ScienceDirect
Journal of Applied Mathematics and Mechanics journal homepage: www.elsevier.com/locate/jappmathmech
The ellipsoidal pendulum夽 A.P. Blinov Moscow, Russia
a r t i c l e
i n f o
Article history: Received 3 March 2010
a b s t r a c t The problem of the motion of a heavy particle on a weightless inextensible thread, attached at two points on a horizontal level with a slack is considered. The configuration space of this system is a spheroid. The equations of the trajectories of a heavy particle on the surface of the spheroid with a horizontal axis of revolution with a specified energy level are obtained in dimensionless form. A solution of the equations is given in quadratures for a fairly small value of the ratio of the maximum possible potential energy of the particle on the spheroid to its total energy. The conditions for the motion of the particle to be stable along the spheroid equator and the conditions for the particle release from the constraint are determined. © 2012 Elsevier Ltd. All rights reserved.
It is well known that the problem of the motion of a heavy particle on a surface of revolution with a vertical axis of revolution can be solved in quadratures. The general solution of the problem in the case of an elliptic paraboloid was obtained by Chaplygin using elliptic coordinates.1 In the case of a spheroid, the use of spheroidal coordinates is impossible due to their degeneracy. The geometrical reduction method2 is used here to solve the problem. The asymptotic representation of the solutions and the stability conditions are obtained by the Poincaré and Lyapunov methods.3,4 1. Derivation of the equation of the particle trajectory Consider the motion of a heavy particle on the surface of a spheroid with a horizontal axis of revolution. This motion is equivalent to the motion of a heavy particle, sliding without friction on a weightless inextensible thread. The ends of the thread are fastened on a horizontal level so that its length is greater than the distance between the fastening points. We will choose a fixed system of coordinates xyz so that its origin coincides with the centre of the spheroid, the x axis is directed vertically upwards, and the y axis is directed along the axis of revolution of the spheroid. We will take the mass of the particle to be unity, and the semi-axis of the spheroid, lying along the y axis, will be taken as the unit of length. The radius vector r of a point on the surface of the spheroid can be defined by two parameters: p = y and q, that is the angle between the negative direction of the x axis and the meridian plane. With this choice of parameters, the coordinate lines on the spheroid (1.1) where b is the second semi-axis of the spheroid orthogonal to the y axis, form an orthogonal net. The linear element of the surface has the form
(1.2) The equation of the particle trajectory when it moves along a smooth surface in a potential force field has the form2
(1.3)
夽 Prikl. Mat. Mekh. Vol. 75, No. 6, pp. 934-939, 2011. E-mail address: [email protected] 0021-8928/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.jappmathmech.2012.01.005
656
A.P. Blinov / Journal of Applied Mathematics and Mechanics 75 (2011) 655–659
where V is the force function, h is the total energy of the system, q’ = dq/dp, and the subscripts p and q denote partial derivatives with respect to p and q. In the problem considered V = gb( trajectory of motion takes the form
1 − p2 cos q − 1), where g is the gravitational acceleration. Then the equation describing the
(1.4) When there is no gravitational force, Eq. (1.4) allows of the solution q = const, which is one of the geodesic lines – the meridian of the spheroid. Note that in formula (1.3) misprints which occurred in Ref. 2 have been corrected. 2. Integration of the trajectory equations We will assume that the dimensionless quantity = gb/h is a small dimensionless parameter, in which case, from Eq. (1.4) with = 0, / const, Bernoulli’s equation, the solution of which we obtain, after making the replacement q’ = (q) when q = (2.1) enables us to represent the solution of Eq. (1.4) in the form of a quadrature (2.2) where C1 and C2 are constants. The solution of Eq. (1.4) will be sought in the form of a series
By Poincaré’s theorem4 this series converges for sufficiently small values of and | − q|. The equation for the first correction has the form (2.3) where
After reducing the order of Eq. (2.3) we obtain (2.4) The solution of the homogeneous equation corresponding to Eq. (2.4), has the form
(2.5) The constant C˜ 1 is cancelled below. We will write the solutions of Eqs (2.4) and (2.3)
In order that the whole structure of the force function should be represented in the solution, we must also have a solution of the equation for the second correction
A.P. Blinov / Journal of Applied Mathematics and Mechanics 75 (2011) 655–659
657
where
using the solution for the first correction with a1 (p) replaced by a2 (p). The solution of the equation for the third and subsequent orders are constructed in the same way. The velocity of the particle at any point of the trajectory (including at the initial point) is determined from the energy integral 2 = 2V + 2h, while the parameters h, C1 and C2 remain arbitrary. The solution obtained can be continued so long as the value of p does not reach the value p0 , for which the radicand in Eq. (2.1) vanishes. In the neighbourhood of the value p = p0 in Eq. (1.4) we must interchange the roles of the coordinate q and p. This transition will be shown below in Section 3. The region of possible motion is estimated by the inequality −V ≤ h. Note that trajectories passing through the poles of the spheroid are not considered here due to the degeneracy of the coordinates (when p = ±1). Only the trajectory q ≡ 0 is an exception, i.e., the case of plane oscillations of the particle along an ellipse x2 /b2 + y2 = 1, when
3. Rotational-type motions To investigate rotational-type motions (h > 2gb) and for oscillations with a large amplitude along the equator, it is more convenient to use the coordinate q as the independent variable. Taking into account the fact that
we reduce Eq. (1.4) to the form
(3.1) Note that when g = 0 this equation allows of the following solutions: p 0 are motions of the particle along the equator and p 1 is the equilibrium position. If we consider rotational-type motions, similar to the motion along the equator of the spheroid, we can assume |p|, |dp/dq| 1. Discarding terms higher than the first order in p and p’ in Eq. (3.1), we obtain the equation
(3.2) where
The distance between the closest zeros q0 of the solutions of Eq. (3.2) can be estimated using the de la Vallée-Poussin theorem with the inequality (3.3) where |Q(q)| ≤ M1 , |R(q)| ≤ M2 , q ∈ [0, 2]. Using the replacement of variables3
(3.4)
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A.P. Blinov / Journal of Applied Mathematics and Mechanics 75 (2011) 655–659
we reduce Eq. (3.2) to the form
(3.5) For an equation of the form (3.5) Lyapunov constructed a fundamental system of solutions 1 and 2 , which, for the initial conditions,
can be represented by converging series. Sturm’s theorem5 gives the following bilateral and more accurate estimate than (3.3)
where M and m are positive numbers, such that m2 ≤ D(q) ≤ M2 , q ∈ [0, 2]. From the energy integral, taking into account that the particle velocity is given by the expression
we obtain the time dependence of the coordinate q in the form of a quadrature
To determine the stability of the solution p = 0 we can use the Lyapunov integral criterion, when the function D(q) is non-negative in the period 2.3 This solution is stable if
It can be seen from the expression for D(q) that the inequality will be satisfied for a fairly small value of = gb/h, i.e., at a fairly high 2. particle velocity along the equator of the spheroid and for a fairly small value of b sup D(q) − inf D(q) ≤ 1/2, q ∈ [0, 2]. By the Zhukovskii stability criterion, the solution p = 0 is stable if This inequality is satisfied for a high particle velocity for any finite value of b. Taking into account the fact that Eq. (3.4) describes a quasi-harmonic system,6 in which parametric resonance is possible, we conclude that other regions can also exist in the plane of the parameters (, b) in which the solution p = 0 is stable.6 Remark 1.
To find 2-periodic solutions we can use their representation in the form of Fourier series
Suppose
After substituting the expansions into Eq. (3.5) and equating coefficients of like harmonics, we obtain an infinite system of linear homogeneous equations in the coefficients a0 , a1 , a2 ,... and b0 , b1 , b2 ,.... Confining ourselves to an n-th order system, to obtain non-trivial solutions we must require that the determinant of the system should vanish as a result of changing the parameter . The boundary condition
closes the system of equations obtained. In a similar way we can also find the 2k-periodic solutions for k = 2, 3,.... Remark 2. For the motion of the particle in the upper part of the spheroid, losses at its contact with the surface may occur (or slackening of the supporting thread, i.e., the release from the constraint). Vanishing of the pressure of the particle on the surface, defined by the difference in the acceleration v2 / and gcos , precedes this, where is the radius of normal curvature of the trajectory and  is the angle between the normal to the surface at the contact point and the vertical. The value of lies between the values of R1 and R2 , where R1 and R2 are the principal radii of curvature of the surface at the contact point. 3/2
For a spheroid R1 = [1 + (b2 − 1)p2 ] /b is the radius of curvature of the meridian and R2 = b parallel. In particular, R1 = 1/b and R2 = b when p = 0, and R1 = R2 = b2 when p = 1.
1 + (b2 − 1)p2 is the radius of the
A.P. Blinov / Journal of Applied Mathematics and Mechanics 75 (2011) 655–659
659
If b < 1, the radii R1 and R2 increase monotonically from the pole to the equator of the spheroid, and hence the least pressure is reached at the upper point of the spheroid, when the directions of the particle velocity and the least principal curvature coincide. Consequently, g/b. This velocity release from the constraints is impossible if the particle velocity at the upper point of the spheroid is no less than defines the critical level of the energy h. gb, the upper point If b > 1, then, in addition to the upper part of the spheroid, where the particle velocity must now be no less than of one of its parallels may also be unsafe for the release from the constraint if the particle velocity is directed along it. Since the velocity for the chosen energy level depends solely on p, to avoid a release from the constraint the following inequality must be satisfied
This inequality can be replaced by the rougher but simpler inequality: > gb. This paper was prepared from the material of a paper presented at a conference devoted to the memory of V. V. Rumyantsev. References 1. 2. 3. 4. 5. 6.
Chaplygin SA. The Paraboloidal Pendulum. Complete Collected Papers. Leningrad: Izd Akad Nauk SSSR; 1933, 194–9. Blinov AP. The motion of a point mass on a surface. Izd Ross Akad Nauk HTT 2007;1:23–8. Demidovich BP. Lectures on the Mathematical Theory of Stability. Moscow: Izd Nauka; 1967. Moiseyev NN. Asymptotic Methods of Non-linear Mechanics. Moscow: Izd Nauka; 1981. Tricomi FG. Differential Equations. London: Blackie; 1961. Malkin IG. Some Problems in the Theory of Nonlinear Oscillations. Oak Ridge, TN: US Atomic Energy Commision. Technical Information Service; 1959.
Translated by R.C.G.
Contents lists available at SciVerse ScienceDirect
Journal of Applied Mathematics and Mechanics journal homepage: www.elsevier.com/locate/jappmathmech
The ellipsoidal pendulum夽 A.P. Blinov Moscow, Russia
a r t i c l e
i n f o
Article history: Received 3 March 2010
a b s t r a c t The problem of the motion of a heavy particle on a weightless inextensible thread, attached at two points on a horizontal level with a slack is considered. The configuration space of this system is a spheroid. The equations of the trajectories of a heavy particle on the surface of the spheroid with a horizontal axis of revolution with a specified energy level are obtained in dimensionless form. A solution of the equations is given in quadratures for a fairly small value of the ratio of the maximum possible potential energy of the particle on the spheroid to its total energy. The conditions for the motion of the particle to be stable along the spheroid equator and the conditions for the particle release from the constraint are determined. © 2012 Elsevier Ltd. All rights reserved.
It is well known that the problem of the motion of a heavy particle on a surface of revolution with a vertical axis of revolution can be solved in quadratures. The general solution of the problem in the case of an elliptic paraboloid was obtained by Chaplygin using elliptic coordinates.1 In the case of a spheroid, the use of spheroidal coordinates is impossible due to their degeneracy. The geometrical reduction method2 is used here to solve the problem. The asymptotic representation of the solutions and the stability conditions are obtained by the Poincaré and Lyapunov methods.3,4 1. Derivation of the equation of the particle trajectory Consider the motion of a heavy particle on the surface of a spheroid with a horizontal axis of revolution. This motion is equivalent to the motion of a heavy particle, sliding without friction on a weightless inextensible thread. The ends of the thread are fastened on a horizontal level so that its length is greater than the distance between the fastening points. We will choose a fixed system of coordinates xyz so that its origin coincides with the centre of the spheroid, the x axis is directed vertically upwards, and the y axis is directed along the axis of revolution of the spheroid. We will take the mass of the particle to be unity, and the semi-axis of the spheroid, lying along the y axis, will be taken as the unit of length. The radius vector r of a point on the surface of the spheroid can be defined by two parameters: p = y and q, that is the angle between the negative direction of the x axis and the meridian plane. With this choice of parameters, the coordinate lines on the spheroid (1.1) where b is the second semi-axis of the spheroid orthogonal to the y axis, form an orthogonal net. The linear element of the surface has the form
(1.2) The equation of the particle trajectory when it moves along a smooth surface in a potential force field has the form2
(1.3)
夽 Prikl. Mat. Mekh. Vol. 75, No. 6, pp. 934-939, 2011. E-mail address: [email protected] 0021-8928/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.jappmathmech.2012.01.005
656
A.P. Blinov / Journal of Applied Mathematics and Mechanics 75 (2011) 655–659
where V is the force function, h is the total energy of the system, q’ = dq/dp, and the subscripts p and q denote partial derivatives with respect to p and q. In the problem considered V = gb( trajectory of motion takes the form
1 − p2 cos q − 1), where g is the gravitational acceleration. Then the equation describing the
(1.4) When there is no gravitational force, Eq. (1.4) allows of the solution q = const, which is one of the geodesic lines – the meridian of the spheroid. Note that in formula (1.3) misprints which occurred in Ref. 2 have been corrected. 2. Integration of the trajectory equations We will assume that the dimensionless quantity = gb/h is a small dimensionless parameter, in which case, from Eq. (1.4) with = 0, / const, Bernoulli’s equation, the solution of which we obtain, after making the replacement q’ = (q) when q = (2.1) enables us to represent the solution of Eq. (1.4) in the form of a quadrature (2.2) where C1 and C2 are constants. The solution of Eq. (1.4) will be sought in the form of a series
By Poincaré’s theorem4 this series converges for sufficiently small values of and | − q|. The equation for the first correction has the form (2.3) where
After reducing the order of Eq. (2.3) we obtain (2.4) The solution of the homogeneous equation corresponding to Eq. (2.4), has the form
(2.5) The constant C˜ 1 is cancelled below. We will write the solutions of Eqs (2.4) and (2.3)
In order that the whole structure of the force function should be represented in the solution, we must also have a solution of the equation for the second correction
A.P. Blinov / Journal of Applied Mathematics and Mechanics 75 (2011) 655–659
657
where
using the solution for the first correction with a1 (p) replaced by a2 (p). The solution of the equation for the third and subsequent orders are constructed in the same way. The velocity of the particle at any point of the trajectory (including at the initial point) is determined from the energy integral 2 = 2V + 2h, while the parameters h, C1 and C2 remain arbitrary. The solution obtained can be continued so long as the value of p does not reach the value p0 , for which the radicand in Eq. (2.1) vanishes. In the neighbourhood of the value p = p0 in Eq. (1.4) we must interchange the roles of the coordinate q and p. This transition will be shown below in Section 3. The region of possible motion is estimated by the inequality −V ≤ h. Note that trajectories passing through the poles of the spheroid are not considered here due to the degeneracy of the coordinates (when p = ±1). Only the trajectory q ≡ 0 is an exception, i.e., the case of plane oscillations of the particle along an ellipse x2 /b2 + y2 = 1, when
3. Rotational-type motions To investigate rotational-type motions (h > 2gb) and for oscillations with a large amplitude along the equator, it is more convenient to use the coordinate q as the independent variable. Taking into account the fact that
we reduce Eq. (1.4) to the form
(3.1) Note that when g = 0 this equation allows of the following solutions: p 0 are motions of the particle along the equator and p 1 is the equilibrium position. If we consider rotational-type motions, similar to the motion along the equator of the spheroid, we can assume |p|, |dp/dq| 1. Discarding terms higher than the first order in p and p’ in Eq. (3.1), we obtain the equation
(3.2) where
The distance between the closest zeros q0 of the solutions of Eq. (3.2) can be estimated using the de la Vallée-Poussin theorem with the inequality (3.3) where |Q(q)| ≤ M1 , |R(q)| ≤ M2 , q ∈ [0, 2]. Using the replacement of variables3
(3.4)
658
A.P. Blinov / Journal of Applied Mathematics and Mechanics 75 (2011) 655–659
we reduce Eq. (3.2) to the form
(3.5) For an equation of the form (3.5) Lyapunov constructed a fundamental system of solutions 1 and 2 , which, for the initial conditions,
can be represented by converging series. Sturm’s theorem5 gives the following bilateral and more accurate estimate than (3.3)
where M and m are positive numbers, such that m2 ≤ D(q) ≤ M2 , q ∈ [0, 2]. From the energy integral, taking into account that the particle velocity is given by the expression
we obtain the time dependence of the coordinate q in the form of a quadrature
To determine the stability of the solution p = 0 we can use the Lyapunov integral criterion, when the function D(q) is non-negative in the period 2.3 This solution is stable if
It can be seen from the expression for D(q) that the inequality will be satisfied for a fairly small value of = gb/h, i.e., at a fairly high 2. particle velocity along the equator of the spheroid and for a fairly small value of b sup D(q) − inf D(q) ≤ 1/2, q ∈ [0, 2]. By the Zhukovskii stability criterion, the solution p = 0 is stable if This inequality is satisfied for a high particle velocity for any finite value of b. Taking into account the fact that Eq. (3.4) describes a quasi-harmonic system,6 in which parametric resonance is possible, we conclude that other regions can also exist in the plane of the parameters (, b) in which the solution p = 0 is stable.6 Remark 1.
To find 2-periodic solutions we can use their representation in the form of Fourier series
Suppose
After substituting the expansions into Eq. (3.5) and equating coefficients of like harmonics, we obtain an infinite system of linear homogeneous equations in the coefficients a0 , a1 , a2 ,... and b0 , b1 , b2 ,.... Confining ourselves to an n-th order system, to obtain non-trivial solutions we must require that the determinant of the system should vanish as a result of changing the parameter . The boundary condition
closes the system of equations obtained. In a similar way we can also find the 2k-periodic solutions for k = 2, 3,.... Remark 2. For the motion of the particle in the upper part of the spheroid, losses at its contact with the surface may occur (or slackening of the supporting thread, i.e., the release from the constraint). Vanishing of the pressure of the particle on the surface, defined by the difference in the acceleration v2 / and gcos , precedes this, where is the radius of normal curvature of the trajectory and  is the angle between the normal to the surface at the contact point and the vertical. The value of lies between the values of R1 and R2 , where R1 and R2 are the principal radii of curvature of the surface at the contact point. 3/2
For a spheroid R1 = [1 + (b2 − 1)p2 ] /b is the radius of curvature of the meridian and R2 = b parallel. In particular, R1 = 1/b and R2 = b when p = 0, and R1 = R2 = b2 when p = 1.
1 + (b2 − 1)p2 is the radius of the
A.P. Blinov / Journal of Applied Mathematics and Mechanics 75 (2011) 655–659
659
If b < 1, the radii R1 and R2 increase monotonically from the pole to the equator of the spheroid, and hence the least pressure is reached at the upper point of the spheroid, when the directions of the particle velocity and the least principal curvature coincide. Consequently, g/b. This velocity release from the constraints is impossible if the particle velocity at the upper point of the spheroid is no less than defines the critical level of the energy h. gb, the upper point If b > 1, then, in addition to the upper part of the spheroid, where the particle velocity must now be no less than of one of its parallels may also be unsafe for the release from the constraint if the particle velocity is directed along it. Since the velocity for the chosen energy level depends solely on p, to avoid a release from the constraint the following inequality must be satisfied
This inequality can be replaced by the rougher but simpler inequality: > gb. This paper was prepared from the material of a paper presented at a conference devoted to the memory of V. V. Rumyantsev. References 1. 2. 3. 4. 5. 6.
Chaplygin SA. The Paraboloidal Pendulum. Complete Collected Papers. Leningrad: Izd Akad Nauk SSSR; 1933, 194–9. Blinov AP. The motion of a point mass on a surface. Izd Ross Akad Nauk HTT 2007;1:23–8. Demidovich BP. Lectures on the Mathematical Theory of Stability. Moscow: Izd Nauka; 1967. Moiseyev NN. Asymptotic Methods of Non-linear Mechanics. Moscow: Izd Nauka; 1981. Tricomi FG. Differential Equations. London: Blackie; 1961. Malkin IG. Some Problems in the Theory of Nonlinear Oscillations. Oak Ridge, TN: US Atomic Energy Commision. Technical Information Service; 1959.
Translated by R.C.G.