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Lagrangian

Lagrangian. Using the Lagrangian. Identify the degrees of freedom. One generalized coordinate for each Velocities as functions of generalized coordinates and velocities Find the Lagrangian Kinetic energy in terms of velocity components Potential energy in terms of generalized coordinates

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Lagrangian

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  1. Lagrangian

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

  3. 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 Simple Harmonic Oscillator

  4. Plane Pendulum • The plane pendulum is a 2-D system. • Two degrees of freedom • One constraint r = R • Angle q as generalized coordinate y x R q m

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

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

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