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Understanding Two-Body Systems: Forces, Motion, and Central Potentials

This overview explores the principles of two-body systems, defining key concepts such as internal and external forces, center of mass (CM) motion, and relative motion. By examining reduced mass and its relationship to spherical coordinates, we illustrate how these elements interact. Topics include constant angular momentum, the significance of central forces, and the conditions for central motion in a plane. Furthermore, we highlight the role of potential energies and their relationship to the Lagrangian framework in analyzing two-body interactions.

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Understanding Two-Body Systems: Forces, Motion, and Central Potentials

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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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