Harmonic Oscillator

Hooke’s Law  The Newtonian form of the spring force is Hooke’s Law. Restoring forceRestoring force Linear with displacement.Linear with displacement.  The Lagrangian form uses the potential energy. L+ x L - x L

Energy Curve  The spring force has a potential energy V = ½ kx 2. Minimum energy at equilibrium. No velocity, K = ½ mv 2 = 0  A higher energy has two turning points. Corresponds to K = 0 In between K > 0 Motion forbidden outside range V x E x1x1 x2x2 x0x0 E0E0

Potential Well  An arbitrary potential near equilibrium can be approximated with a spring potential. Second order series expansion First derivative is zero V x x0x0 E0E0

Stability  For positive k, the motion is like a spring. Stable oscillations about a pointStable oscillations about a point  For negative k, the motion is unstable. V x xSxS E0E0 xUxU unstable

Complex Solutions  The differential equation at stable equilibrium has a complex solution. Euler’s formulaEuler’s formula Real part is physicalReal part is physical  r ir sin  r cos  Re Im Complex conjugate for real solution

Damping Force  Small damping forces are velocity dependent. Not from a potential Generalized force on right side  The differential equation can be solved with an exponential. Possibly complex Quadratic expression must vanish

Three Cases  The quadratic equation in has three forms depending on the constants.  If  ,  is real. Overdamped solutionOverdamped solution  If  ,  is zero. Critically damped solutionCritically damped solution  If  ,  is imaginary. Underdamped solutionUnderdamped solution

Quality Factor  The energy in a damped oscillator is dissipated. Work done by friction  Lightly damped systems have periods close to undamped. Damping   0  Quality factor Q measures energy loss per radian. next