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Two Bodies

Two Bodies. A two-body system can be defined with internal and external forces. Center of mass R Equal external force. Add to get the CM motion Subtract for relative motion. Two-Body System. F 2 int. m 2. r = r 1 – r 2. F 2 ext. m 1. R. r 2. F 1 int. r 1. F 1 ext.

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Two Bodies

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  1. Two Bodies

  2. A two-body system can be defined with internal and external forces. Center of mass R Equal external force Add to get the CM motion Subtract for relative motion Two-Body System F2int m2 r = r1 – r2 F2ext m1 R r2 F1int r1 F1ext

  3. The internal forces are equal and opposite. Express the equation in terms of a reduced mass m. m less than either m1, m2 m approximately equals the smaller mass when the other is large. Reduced Mass for

  4. Use spherical coordinates. Makes r obvious from central force. Generalized forces Qq = Qf = 0. Central force need not be from a potential. Kinetic energy expression Central Force Equations

  5. T doesn’t depend on f directly. Constant angular momentum about the polar axis. Constrain motion to a plane Include the polar axis in the plane Two coordinates r, q. Coordinate Reduction constant

  6. T also doesn’t depend on q directly. Represents constant angular momentum Angular momentum J to avoid confusion with the Lagrangian Change the time derivative to an angle derivative. Angle Equation constant

  7. Central Motion • Central motion takes place in a plane. • Force, velocity, and radius are coplanar • Orbital angular momentum is constant. • If the central force is time-independent, the orbit is symmetrical about an apse. • Apse is where velocity is perpendicular to radius

  8. Orbit Equation Let u = 1/r

  9. Central Potential • The central force can derive from a potential. • Rewrite as differential equation with angular momentum. • Central forces have an equivalent Lagrangian. next

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