1. Energy

2. Oscillator in One Coordinate  Conservative system E = T + V(q)E = T + V(q)  Solve for the velocity.  Position can be found analytically for some V(q).  General solution is numeric.

3. Potential Energy  Turning points q 1, q 2 v q = 0 V(q 1 ) = V(q 2 ) = E T > 0, so V(q 1 < q < q 2 ) < E  Equilibrium q min  The period depends on E only when potential is not a parabola. V(q)V(q) q q1q1 q2q2 E q min

4. Plane Pendulum  Single variable is the angle s = l s = l  q = q =  V(q) = mgl(1 – cosq)V(q) = mgl(1 – cosq)  Small oscillation limit F = –(mg/l) sF = –(mg/l) s Harmonic motionHarmonic motion  Finite amplitude Make substitutionsMake substitutions Consider E 2Consider E 2

5. Bound Motion  Energy below threshold E < 2E < 2  Turning points exist  Solution is an elliptic integral  Approximate period

6. Critical Energy  Energy at threshold E = 2E = 2  Non-periodic motion  Non-circular motion  Reaches peak at infinite time

7. Unbound Motion  Energy above threshold E > 2E > 2  Non-uniform circular motion  Solution is an elliptic integral  Period depends on energy and acceleration of gravityand acceleration of gravity

8. Phase Portrait  A plot of position vs. velocity. Phase space is something more detailed.Phase space is something more detailed. E < 2 E = 2 E > 2

9. Damped Oscillator  Small damping factor  Small damping factor Depends on velocityDepends on velocity  Total energy is decreasing  Find q, q ’ by usual means  Compare periods One cycle TOne cycle T Energy loss in that timeEnergy loss in that time next