Two-Body Central-Force Two-Body Central-Force ProblemsProblems

Hasbun Ch 8Hasbun Ch 8

Taylor Ch 8Taylor Ch 8

Marion & Thornton Ch 8Marion & Thornton Ch 8

Central Force Central Force ExamplesExamples

Gravitational ForceGravitational Force

Conservative force, so Conservative force, so the force can be the force can be derived from a derived from a potential energy, U(potential energy, U(rr11, , rr22))

Coulomb ForceCoulomb Force

Also a conservative Also a conservative force, described by the force, described by the potential energy potential energy function:function:

O

r1

r = r1 - r2

r2

White BoardsWhite BoardsWrite the Lagrangian for a two-body Write the Lagrangian for a two-body central-force system.central-force system.

The Problem: find the possible motions The Problem: find the possible motions of these two bodiesof these two bodies

Center of Mass and Center of Mass and Relative CoordinatesRelative Coordinates

What generalized coordinates should we use What generalized coordinates should we use to solve the two-body central force problem?to solve the two-body central force problem?

Relative position, rRelative position, r

Position of center of mass, R, wherePosition of center of mass, R, where

O

r1

CM

r2

R

The total momentum of the system is:

B/c total momentum is constant, CM frame is an inertial reference frame.

White BoardsWhite BoardsWrite Write rr11 and and rr22 in terms of in terms of RR and and rr..

Write the kinetic energy T in terms of Write the kinetic energy T in terms of RR and and rr..

Write the Lagrangian L in terms of Write the Lagrangian L in terms of RR and and rr..

White BoardsWhite BoardsWrite the equations of motion from the Write the equations of motion from the LagrangianLagrangian(one equation for R, one equation for r)(one equation for R, one equation for r)

Center of Mass Center of Mass FrameFrame

If we choose the center-of-mass frame as our If we choose the center-of-mass frame as our reference frame (we can do this b/c it’s an reference frame (we can do this b/c it’s an intertial frame), thenintertial frame), then

The center of mass part of L is zero, so the The center of mass part of L is zero, so the Lagrangian becomes:Lagrangian becomes:

Note this is the Lagrangian for a single particle moving in a central potential!

Conservation of Angular Conservation of Angular MomentumMomentum

Angular momentum of the two particles is Angular momentum of the two particles is conserved.conserved.

Rewrite this in the COM frame Rewrite this in the COM frame

Because the direction of angular momentum is constant, the motion remains in a fixed plane, which we can take to be the x-y plane.

Always a 2D problem in COM frame

White BoardsWhite BoardsThe easiest coordinates for this 2-D problem The easiest coordinates for this 2-D problem are polar coordinates.are polar coordinates.

Remember that the velocity in polar Remember that the velocity in polar coordinates is given bycoordinates is given by

Write the Lagrangian in polar coordinates, Write the Lagrangian in polar coordinates, and find the equations of motion.and find the equations of motion.

Equations of Equations of MotionMotion

Angular EquationAngular Equation Radial EquationRadial Equation

CentrifugalForce

White BoardWhite BoardUse the phi equation to eliminate phi-Use the phi equation to eliminate phi-dot from the radial equationdot from the radial equation

White BoardWhite BoardUse the phi equation to eliminate phi-Use the phi equation to eliminate phi-dot from the radial equationdot from the radial equation

Bound & Unbound Bound & Unbound OrbitsOrbits

At rmin, the energy is equal to Ueff

Matlab ProblemMatlab ProblemWrite down the actual and effective Write down the actual and effective potential energies for a comet (or potential energies for a comet (or planet) moving in the gravitational field planet) moving in the gravitational field of the sun. Plot the 3 potential of the sun. Plot the 3 potential energies involved (U, Ucf, Ueff) and use energies involved (U, Ucf, Ueff) and use the graph of Ueff vs r to describe the the graph of Ueff vs r to describe the motion as r goes from large to small motion as r goes from large to small values.values.

UseUseG=1G=1

m1=m2=l=1m1=m2=l=1

Matlab ProblemMatlab Problem