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Encyclopedia > Bertrand's theorem

In classical mechanics, Bertrand's theorem states that only two types of potentials produce stable, closed orbits: an inverse-square force such as the gravitational or electrostatic potential Classical mechanics is a branch of physics which studies the deterministic motion of objects. ... It has been suggested that this article or section be merged with Scalar potential. ... This article is about the building; for another meaning, see stability. ... In the study of dynamical systems, an orbit is the sequence generated by iterating a map. ... Gravity is a force of attraction that acts between bodies that have mass. ... Electrostatics is the branch of physics that deals with the forces exerted by a static (i. ...

$V(mathbf{r}) = frac{-k}{r}$

and the radial harmonic oscillator potential A harmonic oscillator is either a mechanical system in which there exists a returning force F directly proportional to the displacement x, i. ...

$V(mathbf{r}) = frac{1}{2} kr^{2}$

1 General Preliminaries
2 Bertrand's theorem
3 Inverse-square force (Kepler problem)
4 Radial harmonic oscillator
5 Reference

General Preliminaries

All attractive central forces can produce circular orbits, which are naturally closed orbits. The only requirement is that the central force exactly equal the centripetal force requirement, which determines the required angular velocity for a given circular radius. Non-central forces (i.e., those that depend on the angular variables as well as the radius) are ignored here, since they do not produce circular orbits in general. A central force acting on an object is one whose magnitude depends only on the scalar distance r of the object from the origin and whose direction is along the position vector from the origin to the object. ... In Euclidean geometry, a circle is the set of all points in a plane at a fixed distance, called the radius, from a fixed point, the centre. ... In the study of dynamical systems, an orbit is the sequence generated by iterating a map. ... An object that moves in a circular path undergoes a continuous acceleration towards the center of the circle. ...

The equation of motion for the radius $r$ of a particle of mass $m$ moving in a central potential $V (r)$ is given by Lagrange's equations A central force acting on an object is one whose magnitude depends only on the scalar distance r of the object from the origin and whose direction is along the position vector from the origin to the object. ... The Euler-Lagrange Equation is the major formula of the Calculus of variations. ...

$mfrac{d^{2}r}{dt^{2}} - mr omega^{2} = mfrac{d^{2}r}{dt^{2}} - frac{L^{2}}{mr^{3}} = -frac{dV}{dr}$

where $omega equiv frac{dtheta}{dt}$ and the angular momentum $L = m r 2 ω$ is conserved. For illustration, the first term on the left-hand side is zero for circular orbits, and the applied inwards force $frac{dV}{dr}$ equals the centripetal force requirement $m r ω 2$ , as expected. Gyroscope. ... An object that moves in a circular path undergoes a continuous acceleration towards the center of the circle. ...

The definition of angular momentum allows a change of independent variable from $t$ to $θ$ Gyroscope. ...

$frac{d}{dt} = frac{L}{mr^{2}} frac{d}{dtheta}$

giving the new equation of motion that is independent of time

$frac{L}{r^{2}} frac{d}{dtheta} left( frac{L}{mr^{2}} frac{dr}{dtheta} right)- frac{L^{2}}{mr^{3}} = -frac{dV}{dr}$

This equation becomes quasilinear on making the change of variables $u equiv frac{1}{r}$ and multiplying both sides by $frac{mr^{2}}{L^{2}}$

$frac{d^{2}u}{dtheta^{2}} + u = -frac{m}{L^{2}} frac{d}{du} V(1/u)$

Bertrand's theorem

As noted above, all central forces can produce circular orbits given an appropriate initial velocity. However, if some radial velocity is introduced, these orbits need not be stable (i.e., remain in orbit indefinitely) nor closed (repeatedly returning to exactly the same path). Here we show that stable, exactly closed orbits can be produced only with an inverse-square force or radial harmonic oscillator potential (a necessary condition). In the following sections, we show that those force laws do produce stable, exactly closed orbits (a sufficient condition). A central force acting on an object is one whose magnitude depends only on the scalar distance r of the object from the origin and whose direction is along the position vector from the origin to the object. ... In the study of dynamical systems, an orbit is the sequence generated by iterating a map. ... In logic, the words necessary and sufficient describe problems that consist between propositions or states of affairs, if one is accidental on the other. ... In the study of dynamical systems, an orbit is the sequence generated by iterating a map. ... In logic, the words necessary and sufficient describe problems that consist between propositions or states of affairs, if one is accidental on the other. ...

For brevity, we introduce the function $J (u)$ into the equation for $u$

$frac{d^{2}u}{dtheta^{2}} + u = J(u) equiv -frac{m}{L^{2}} frac{d}{du} V(1/u) = -frac{m}{L^{2}u^{2}} f(1/u)$

where $f$ represents the radial force. The criterion for perfectly circular motion at a radius $r 0$ is that the first term on the left-hand side be zero In Euclidean geometry, a circle is the set of all points in a plane at a fixed distance, called the radius, from a fixed point, the centre. ...

u 0 = J (u 0)

where $u_{0} equiv 1/r_{0}$ .

The next step is to consider the equation for $u$ under small perturbations $eta equiv u - u_{0}$ from perfectly circular orbits. On the right-hand side, the $J$ function can be expanded in a standard Taylor series Perturbation theory comprises mathematical methods that are used to find an approximate solution to a problem which cannot be solved exactly, by starting from the exact solution of a related problem. ... As the degree of the Taylor series rises, it approaches the correct function. ...

$J(u) approx u_{0} + eta J^{prime}(u_{0}) + frac{1}{2} eta^{2} J^{primeprime}(u_{0}) + frac{1}{6} eta^{3} J^{primeprimeprime}(u_{0}) + ldots$

Substituting this expansion into the equation for $u$ and subtracting the constant terms yields

$frac{d^{2}eta}{dtheta^{2}} + eta = eta J^{prime}(u_{0}) + frac{1}{2} eta^{2} J^{primeprime}(u_{0}) + frac{1}{6} eta^{3} J^{primeprimeprime}(u_{0}) ldots$

which can be written as

$frac{d^{2}eta}{dtheta^{2}} + beta^{2} eta = frac{1}{2} eta^{2} J^{primeprime}(u_{0}) + frac{1}{6} eta^{3} J^{primeprimeprime}(u_{0}) ldots$

where $beta^{2} equiv 1 - J^{prime}(u_{0})$ is a constant. $β 2$ must be non-negative; otherwise, the radius of the orbit would vary exponentially away from its initial radius. (The solution $β = 0$ corresponds to a perfectly circular orbit.) If the right-hand side may be neglected (i.e., for very small perturbations), the solutions are

η(θ) = h 1 cosβθ

where the amplitude $h 1$ is a constant of integration. For the orbits to be closed, $β$ must be a rational number. What's more, it must be the same rational number for all radii, since $β$ cannot change continuously; the rational numbers are totally disconnected from one another. Since the defining equations In mathematics, a rational number (commonly called a fraction) is a ratio or quotient of two integers, usually written as the vulgar fraction a/b, where b is not zero. ... In mathematics, a rational number (commonly called a fraction) is a ratio or quotient of two integers, usually written as the vulgar fraction a/b, where b is not zero. ... In topology and related branches of mathematics, a topological space X is said to be disconnected if it is the union of two disjoint nonempty open sets. ...

$J^{prime}(u_{0}) equiv -2 + frac{u_{0}}{f(1/u_{0})} frac{df}{du} = 1 - beta^{2}$

must hold for any value of $u 0$ , we can write

$frac{df}{dr} = left( beta^{2} - 3 right) frac{f}{r}$

which implies that the force must follow a power law See Also: Watt In physics, a power law relationship between two scalar quantities x and y is any such that the relationship can be written as where a (the constant of proportionality) and k (the exponent of the power law) are constants. ...

$f(r) = - frac{k}{r^{3-beta^{2}}}$

Hence, $J$ must have the general form

$J(u) = frac{mk}{L^{2}} u^{1-beta^{2}}$

For more general deviations from circularity (i.e., when we cannot neglect the higher order terms in the Taylor expansion of $J$ ), $η$ may be expanded in a Fourier series, e.g.,

$eta(theta) = h_{0} + h_{1} cos beta theta + h_{2} cos 2beta theta + h_{3} cos 3beta theta + ldots$

Substituting this solution into both sides of the equation for $η$ and equating the coefficients belonging to the same frequency yields the system of equations

$h_{0} = h_{1}^{2} frac{J^{primeprime}(u_{0})}{4beta^{2}}$

$h_{2} = -h_{1}^{2} frac{J^{primeprime}(u_{0})}{12beta^{2}}$

$h_{3} = -frac{1}{8beta^{3}} left[ h_{1}h_{2} frac{J^{primeprime}(u_{0})}{2} + h_{1}^{3} frac{J^{primeprimeprime}(u_{0})}{24} right]$

and, most importantly,

$left( 2 h_{1} h_{0} + h_{1} h_{2} right) frac{J^{primeprime}(u_{0})}{2} + h_{1}^{3} frac{J^{primeprimeprime}(u_{0})}{8} = 0$

This last equation, when combined with the equation for $J$ in terms of $β$ , yields the main result of Bertrand's theorem

$beta^{2} left( 1 - beta^{2} right) left( 4 - beta^{2} right) = 0$

Hence, the only potentials that can produce stable, closed, non-circular orbits are the inverse-square force law ( $β = 1$ ) and the radial harmonic oscillator potential ( $β = 2$ ). The solution $β = 0$ corresponds to perfectly circular orbits, as noted above. It has been suggested that this article or section be merged with Scalar potential. ...

Inverse-square force (Kepler problem)

For an inverse-square force law such as the gravitational or electrostatic potential, the potential can be written Gravity is a force of attraction that acts between bodies that have mass. ... Electrostatics is the branch of physics that deals with the forces exerted by a static (i. ... It has been suggested that this article or section be merged with Scalar potential. ...

$V(mathbf{r}) = frac{-k}{r} = -ku$

The orbit $u (θ)$ can be derived from the general equation

$frac{d^{2}u}{dtheta^{2}} + u = -frac{m}{L^{2}} frac{d}{du} V(1/u) = frac{km}{L^{2}}$

whose solution is the constant $frac{km}{L^{2}}$ plus a simple sinusoid

$u equiv frac{1}{r} = frac{km}{L^{2}} left[ 1 + e cos left( theta - theta_{0}right) right]$

where $e$ (the eccentricity) and $θ 0$ (the phase offset) are constants of integration.

This is the general formula for a conic section that has one focus at the origin; $e = 0$ corresponds to a circle, $e < 1$ corresponds to an ellipse, $e = 1$ corresponds to a parabola, and $e > 1$ corresponds to a hyperbola. The eccentricity $e$ is related to the total energy $E$ (cf. the Laplace-Runge-Lenz vector) Types of conic sections Table of conics, Cyclopaedia, 1728 In mathematics, a conic section (or just conic) is a curve formed by intersecting a cone (more precisely, a right circular conical surface) with a plane. ... In Euclidean geometry, a circle is the set of all points in a plane at a fixed distance, called the radius, from a fixed point, the centre. ... Wikisource has an original article from the 1911 EncyclopÃ¦dia Britannica about: Parabola A parabola The parabola (from the Greek: Ï€Î±ÏÎ±Î²Î¿Î»Î®) is a conic section generated by the intersection of a right circular conical surface and a plane parallel to a generating straight line of that surface. ... A graph of a hyperbola. ... In classical mechanics, for a central force with potential, the Laplace-Runge-Lenz vector is a conserved vector of motion. ...

$e = sqrt{1 + frac{2EL^{2}}{k^{2}m}}$

Comparing these formulae shows that $E < 0$ corresponds to an ellipse, $E = 0$ corresponds to a parabola, and $E > 0$ corresponds to a hyperbola. In particular, $E=-frac{k^{2}m}{2L^{2}}$ for perfectly circular orbits. Wikisource has an original article from the 1911 EncyclopÃ¦dia Britannica about: Parabola A parabola The parabola (from the Greek: Ï€Î±ÏÎ±Î²Î¿Î»Î®) is a conic section generated by the intersection of a right circular conical surface and a plane parallel to a generating straight line of that surface. ... A graph of a hyperbola. ... In Euclidean geometry, a circle is the set of all points in a plane at a fixed distance, called the radius, from a fixed point, the centre. ...

Radial harmonic oscillator

To solve for the orbit under a radial harmonic oscillator potential, it's easier to work in components $mathbf{r} = (x, y, z)$ . The potential energy can be written A harmonic oscillator is either a mechanical system in which there exists a returning force F directly proportional to the displacement x, i. ... In physics and in vector calculus, a spatial vector is a concept characterized by a magnitude, which is a scalar, and a direction (which can be defined in a 3-dimensional space by the Euler angles). ...

$V(mathbf{r}) = frac{1}{2} kr^{2} = frac{1}{2} k left( x^{2} + y^{2} + z^{2}right)$

The equation of motion for a particle of mass $m$ is given by three independent Lagrange's equations The Euler-Lagrange Equation is the major formula of the Calculus of variations. ...

$frac{d^{2}x}{dt^{2}} + omega_{0}^{2} x = 0$

$frac{d^{2}y}{dt^{2}} + omega_{0}^{2} y = 0$

$frac{d^{2}z}{dt^{2}} + omega_{0}^{2} z = 0$

where the constant $omega_{0}^{2} equiv frac{k}{m}$ must be positive (i.e., $k > 0$ ) to ensure bounded, closed orbits; otherwise, the particle will fly off to infinity. The solutions of these simple harmonic oscillator equations are all similar The word infinity comes from the Latin infinitas or unboundedness. It refers to several distinct concepts which arise in theology, philosophy, mathematics and everyday life. ... A harmonic oscillator is either a mechanical system in which there exists a returning force F directly proportional to the displacement x, i. ...

$x = A_{x} cos left(omega_{0} t + phi_{x} right)$

$y = A_{y} cos left(omega_{0} t + phi_{y} right)$

$z = A_{z} cos left(omega_{0} t + phi_{z} right)$

where the positive constants $A x$ , $A y$ and $A z$ represent the amplitudes of the oscillations and the angles $φ x$ , $φ y$ and $φ z$ represent their phases. The resulting orbit $mathbf{r}(t) = left[ x(t), y(y), z(t) right]$ is closed because it repeats exactly after a period

$T equiv frac{2pi}{omega_{0}}$

The system is also stable because small perturbations in the amplitudes and phases cause correspondingly small changes in the overall orbit.

Reference

Goldstein H. (1980) Classical Mechanics, 2nd. ed., Addison-Wesley. ISBN 0-201-02918-9

Category: Classical mechanics

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Contents

General Preliminaries var ACE_AR = {Site: '756743', Size: '300250'};

Bertrand's theorem

Inverse-square force (Kepler problem)

Radial harmonic oscillator

Reference

General Preliminaries