Central ForcesTwo-Body SystemReduced MassCentral MotionCentral Force EquationsCoordinate ReductionAngle EquationOrbit EquationCentral PotentialCentral ForcesCentral ForcesTwo-Body SystemTwo-Body SystemCenter of mass Center of mass RREqual external force on both Equal external force on both bodies.bodies.Add to get the CM motionAdd to get the CM motionSubtract for relative motionSubtract for relative motionext1int111FFrmm2r1F2intr2Rm1F1intF1extF2extr = r1 – r2ext2int222FFrmext2ext1FFRM2int21int121mFmFrrReduced MassReduced MassInternal forces are equal and Internal forces are equal and opposite.opposite.Express in terms of a Express in terms of a reduced mass reduced mass ..•  less than either less than either mm11, , mm22•  approximately equals the approximately equals the smaller mass when the smaller mass when the other is large.other is large.int212int1int21)11( FmmmFmFrrintint212121)(FFmmmmrr22121mmmmm21mm forCentral motion takes place in a plane.Central motion takes place in a plane.•Force, velocity, and radius are coplanar.Force, velocity, and radius are coplanar.Orbital angular momentum is constant.Orbital angular momentum is constant.If the central force is time-independent, the orbit is If the central force is time-independent, the orbit is symmetrical about an apse.symmetrical about an apse.•Apse is where velocity is perpendicular to radius Apse is where velocity is perpendicular to radius Central MotionCentral Motion0rrrrdtJdrrprJUse J to avoid confusion with Lagrangian LCentral Force EquationsCentral Force EquationsUse spherical coordinates.Use spherical coordinates.•Makes Makes rr obvious from central obvious from central force.force.•Generalized forces Generalized forces QQ = = QQ = 0. = 0.•Central force need not be from Central force need not be from a potential.a potential.Kinetic energy expressionKinetic energy expression)sin(22222221rrrT rQrTrTdtd0TTdtd0TTdtdCoordinate ReductionCoordinate ReductionTT doesn’t depend on doesn’t depend on  directly.directly.Constant angular momentum Constant angular momentum about the polar axis.about the polar axis.•Constrain the motion to a Constrain the motion to a plane plane •Include the polar axis in the Include the polar axis in the planeplaneTwo coordinates Two coordinates rr, , ..0TTdtd0Tdtd22sinrTconstant)(22221rrT Angle EquationAngle EquationTT doesn’t depend on doesn’t depend on  directly.directly.Also represents constant Also represents constant angular momentum.angular momentum.•A constant of the motionA constant of the motionChange the time derivative Change the time derivative to an angle derivative.to an angle derivative.0TTdtd0TdtdJrT2constantddrJdddtddtd2Orbit EquationOrbit EquationLet u = 1/rrQrTrTdtdrQrTrTddrJ2rQrrrrrrddrJ )]([)]([22221222212rQrJddrrJddrJrJrdrdrJ3222222)()(23221)1(1JQrddrrddrr2222uJQududrCentral PotentialCentral PotentialCentral force can derive from Central force can derive from a potential.a potential.Rewrite as differential Rewrite as differential equation with angular equation with angular momentum.momentum.Equivalent LagrangianEquivalent LagrangianrVQrTrTdtdrVrJrL