KARLSTAD UNIVERSITY DEPARTMENT OF ENGINEERING AND PHYSICS Analytical mechanics RUNGE-LENZ-SYMMETRIES Author: Supervisor: Rasmus Laven Juirgen Fuchs January 21,2016
KARLSTAD UNIVERSITY DEPARTMENT OF ENGINEERING AND PHYSICS Analytical mechanics Runge-Lenz-Symmetries Author: Rasmus Lav´en Supervisor: J¨urgen Fuchs January 21, 2016
Abstract In this report we study the symmetries that correspond to the conservation of the Runge-Lenz vector in the Kepler problem.In section 2 we use Noether's theorem to define a Runge-Lenz vector as a consequence of an invariance of the action integral.It's shown that such a vector exists for all central potentials. In section 3 we describe the Kepler problem in space-time.By choosing a nice parametrization we show that the equations of motion and the conservation of energy describe a harmonic oscillator with an extra derivative in four dimensions and a four dimensional sphere,respectively.From this we define a conserved tensor.The components of this tensor correspond to the Runge-Lenz vector and angular momentum. 1
Abstract In this report we study the symmetries that correspond to the conservation of the Runge-Lenz vector in the Kepler problem. In section 2 we use Noether’s theorem to define a Runge-Lenz vector as a consequence of an invariance of the action integral. It’s shown that such a vector exists for all central potentials. In section 3 we describe the Kepler problem in space-time. By choosing a nice parametrization we show that the equations of motion and the conservation of energy describe a harmonic oscillator with an extra derivative in four dimensions and a four dimensional sphere, respectively. From this we define a conserved tensor. The components of this tensor correspond to the Runge-Lenz vector and angular momentum. 1
CONTENTS Contents 1 Introduction 3 2 Runge-Lenz-vector from a symmetry of the action integral 5 2.1 General aspects of conserved quantities 5 22 runge-Lenz-vector...:·················.····· 7 3 Extra symmetries in the Kepler problem 13 2
CONTENTS Contents 1 Introduction 3 2 Runge-Lenz-vector from a symmetry of the action integral 5 2.1 General aspects of conserved quantities . . . . . . . . . . . . . . . . 5 2.2 Runge-Lenz-vector . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3 Extra symmetries in the Kepler problem 13 2
Introduction 1 Introduction One of the most famous problems in classical mechanics is the Kepler problem. This is the problem of a point mass in a central force field of the form F(r)= 一k (1) A special thing about this problem is that there exists an extra conserved quantity besides the total energy and the angular momentum.This quantity is a vector called the Runge-Lenz vector.The Runge-Lenz vector A for a particle of mass m moving n a cra force fielddefine A=p×Lx (2) Here p is the momentum of the particle,L is the angular momentum,m the mass and r the position vector of the particle.In the Kepler-problem the angular momentum and energy are conserved.One might then think that there exist seven conserved quantities.This is not the case,because the variables are not independent of each other.From Figure 1 one sees that the Runge-Lenz-vector lies in the plane of motion and thus A.L=0.Further on by taking the dot product A.A one obtains A2 =m2k2+2mEL2.From this one can see that there are only five independent constants of motion in the Kepler-problem [1].The orbits in the Kepler-problem are conic-sections.A nice way to realize this is by using the Runge-Lenz-vector.By denoting 6 as the angle between the position vector and the Runge-Lenz-vector one has A·r=Arcost9=r·p×L-mkr=L·r×p-mkr=L2-mkr. (3) 3
Introduction 1 Introduction One of the most famous problems in classical mechanics is the Kepler problem. This is the problem of a point mass in a central force field of the form F(r) = −k r 2 er. (1) A special thing about this problem is that there exists an extra conserved quantity besides the total energy and the angular momentum. This quantity is a vector called the Runge-Lenz vector. The Runge-Lenz vector A for a particle of mass m moving in a central force field F = − k r 2 er is defined as A := p × L − mk |r| r. (2) Here p is the momentum of the particle, L is the angular momentum, m the mass and r the position vector of the particle. In the Kepler-problem the angular momentum and energy are conserved. One might then think that there exist seven conserved quantities. This is not the case, because the variables are not independent of each other. From Figure 1 one sees that the Runge-Lenz-vector lies in the plane of motion and thus A·L = 0. Further on by taking the dot product A · A one obtains A2 = m2k 2 + 2mEL2 . From this one can see that there are only five independent constants of motion in the Kepler-problem [1]. The orbits in the Kepler-problem are conic-sections. A nice way to realize this is by using the Runge-Lenz-vector. By denoting θ as the angle between the position vector and the Runge-Lenz-vector one has A · r = Arcosθ = r · p × L − mkr = L · r × p − mkr = L 2 − mkr. (3) 3
Introduction A pxL mkt pxL 1 A mkf 37 城≥ 4 Figure 1:Illustration of how the Runge-Lenz-vector is oriented in the Kepler orbits [3. Thus we can solve for r as mk (④) which is the equation of a conic section with eccentricity e=A/mk provided A is constant [3].Thus we see that the conservation of the Runge-Lenz-vector actually is the reason that the orbits of the Kepler-problem are closed.The Runge-Lenz- vector can in principle be generalized to any central potential as we shall see in Section 2.2.However these generalized Runge-Lenz-vectors are often complicated functions and usually not expressible in closed form[3].Since the conservation of the Runge-Lenz-vector implies closed orbits for the Kepler-problem one might expect that there exists some analogue of this derivation for the isotropic harmonic oscillator.This is indeed the case but we shall leave this question here [1]. 4
Introduction . Figure 1: Illustration of how the Runge-Lenz-vector is oriented in the Kepler orbits [3]. Thus we can solve for r as 1 r = mk L2 1 + A mk cosθ , (4) which is the equation of a conic section with eccentricity e = A/mk provided A is constant [3]. Thus we see that the conservation of the Runge-Lenz-vector actually is the reason that the orbits of the Kepler-problem are closed. The Runge-Lenzvector can in principle be generalized to any central potential as we shall see in Section 2.2. However these generalized Runge-Lenz-vectors are often complicated functions and usually not expressible in closed form[3]. Since the conservation of the Runge-Lenz-vector implies closed orbits for the Kepler-problem one might expect that there exists some analogue of this derivation for the isotropic harmonic oscillator. This is indeed the case but we shall leave this question here [1]. 4