Lagrange equation of motion

In summary, the Lagrange equation of motion is used to find the equation of motion for a double pendulum system with equal lengths and equal bob masses, moving in the same plane without assuming small angles. The chosen generalized coordinates are the two angles formed with the vertical, which are independent of each other. The constraints are taken into account by making use of a picture to visualize the system.
  • #1
yukawa
13
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Lagrange equation of motion



(from Marion 7-7)

A double pendulum consists of two simpe pendula, with one pendulum suspended from the bob of the other. If the two pendula have equal lenghts and have bobs of equal mass and if both pendula are confirned to move in the same plane, find Lagrange's equation of motion for the system. Do not assume small angles.

Which generalized coordinates should it choose? And how to made use of the constrains?
 
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  • #2
Make a picture. You can see that the 2 angles formed with the vertical are the 2 needed generalized coordinates.
 
  • #3
but are these two angles independent of each other? (in fact, i don't know how to determine whether two coordinates are independent of each other or not)
 
  • #4
Of course they are independent.
 

What is the Lagrange equation of motion?

The Lagrange equation of motion is a mathematical equation used to describe the motion of a system of particles. It is a form of the Euler-Lagrange equation and is based on the principle of least action, which states that the path taken by a system between two points in time is the one that minimizes the action, a quantity defined as the integral of the system's Lagrangian over time.

What does the Lagrange equation of motion tell us?

The Lagrange equation of motion tells us how a system will evolve over time based on the forces acting on it. It takes into account the system's kinetic and potential energies, as well as any external forces that may be acting on it. By solving the Lagrange equation, we can determine the equations of motion for each particle in the system.

How is the Lagrange equation of motion derived?

The Lagrange equation of motion is derived from the principle of least action. The first step is to define the system's Lagrangian, which is a function that combines the system's kinetic and potential energies. Then, the Euler-Lagrange equation is applied to this Lagrangian, resulting in the Lagrange equation of motion.

What are the advantages of using the Lagrange equation of motion?

One advantage of using the Lagrange equation of motion is that it provides a more elegant and concise way of describing a system's motion compared to using Newton's laws of motion. It also allows for a more systematic approach to solving problems, as the Lagrangian can be easily modified to account for different types of forces or constraints.

When is the Lagrange equation of motion most useful?

The Lagrange equation of motion is most useful when dealing with complex systems where the number of particles and forces involved makes it difficult to use traditional methods. It is also commonly used in theoretical physics and engineering, particularly in fields such as celestial mechanics, quantum mechanics, and control theory.

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