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Analyzing Energy and Motion in Conservative Systems

This work explores the dynamics of conservative systems, focusing on potential energy (V(q)) and its relationship with mechanical energy (E). By deriving the general solution numerically and analyzing the position and velocity of oscillators in one-dimensional potential energy contexts, crucial insights into turning points and equilibrium positions are uncovered. The discussion extends to energy thresholds in various motion types: bounded, critical, and unbounded. The effect of damping in small oscillations is also addressed, providing a comprehensive overview of the oscillatory behavior of a plane pendulum.

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Analyzing Energy and Motion in Conservative Systems

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  1. Energy

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

  3. Potential Energy • Turning points q1, q2 • vq = 0 • V(q1) = V(q2) = E • T > 0, so V(q1 < q < q2) < E • Equilibrium qmin • The period depends on E only when potential is not a parabola. V(q) E q q1 qmin q2

  4. Single variable is the angle s = lq q = q V(q) = mgl(1 – cosq) Small oscillation limit F = –(mg/l) s Harmonic motion Finite amplitude Make substitutions Consider E < 2, E = 2, E > 2 Plane Pendulum

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

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

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

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

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

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