Lagrangian

Using the Lagrangian  Identify the degrees of freedom. One generalized coordinate for eachOne generalized coordinate for each Velocities as functions of generalized coordinates and velocitiesVelocities as functions of generalized coordinates and velocities  Find the Lagrangian Kinetic energy in terms of velocity componentsKinetic energy in terms of velocity components Potential energy in terms of generalized coordinatesPotential energy in terms of generalized coordinates  Write Lagrange’s equations of motion.

Simple Harmonic Oscillator  The 1-D simple harmonic oscillator has one force. F =  kx Conservative force  Select x as the generalized coordinate. T, V in terms of generalized coordinate and velocity  Use Lagrange’s EOM. Usual Newtonian equation

Plane Pendulum  The plane pendulum is a 2-D system. Two degrees of freedomTwo degrees of freedom One constraint r = ROne constraint r = R Angle  as generalized coordinateAngle  as generalized coordinate x y R  m

Oscillating Support  The moving support depends only on time. Not a new degree of freedom – add to xNot a new degree of freedom – add to x Angle  still the generalized coordinateAngle  still the generalized coordinate x y R  m

Forced Oscillator  The support term is time dependent. Must take derivatives when needed Provides a driving force  The Lagrangian method gives the equation of motion.