Lagrangian

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Lagrangian
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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 Lagranges equations of motion.

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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 Lagranges EOM.
• Usual Newtonian equation

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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
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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
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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.