Double Pendulum Lagrange's Equation Problem

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SUMMARY

The discussion focuses on deriving Lagrange's equations of motion for a double pendulum system, where both pendula have equal lengths and masses, and are confined to move in the same plane without assuming small angles. The key equation used is L = T - U, where T represents kinetic energy and U represents potential energy. The main challenge identified is formulating the correct equations of constraint due to the interdependence of the pendula's motions. Participants emphasize the need to express the lengths of both pendulums as functions of their respective coordinates over time.

PREREQUISITES
  • Understanding of Lagrangian mechanics
  • Familiarity with kinetic and potential energy concepts
  • Knowledge of constraint equations in mechanical systems
  • Basic proficiency in calculus and differential equations
NEXT STEPS
  • Study the derivation of Lagrange's equations for multi-body systems
  • Explore the concept of constraint forces in mechanics
  • Learn about the mathematical formulation of the double pendulum
  • Investigate numerical methods for simulating non-linear dynamics
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Homework Statement


A double pendulum consists of two simple pendula, with one pendulum suspended from the bob of the other. If the two pendula have equal lengths and have bobs of equal mass and if both pendula are confined to move in the same plane, find Lagrange's equations of motion for the system. Do no assume small angles.


Homework Equations


L=T-U
partial L/partial q-(d/dt)(partial L/partial q')+lambda(partial f/partial q)


The Attempt at a Solution


Main problem is finding an equation of constraint to use. I know the motion of the pendula must be dependent on each other since they are connected, but not sure how to correctly state that. Also, do I need to set things up piece by piece? Meaning, have an equation of motion for x1,x2,y1,and y2? Or should it be combined?
 
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Well, suppose that the upper pendulum hangs from the origin, and its bob has coordinates (x_1(t),y_1(t)) at time t...can you find an expression for the length of that pendulum as a function of x_1(t) and y_1(t)?

How about the length of the second pendulum if its bob has coordinates (x_2(t),y_2(t)) at time t? (keep in mind that the second pendulum is not hanging from the origin :wink:)

That should give you two constraint equations.
 

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