Finding equation of motion from lagrangian

The equation of motion from the Lagrangian is given by (\omega^2-\frac{k}{m})r = \frac{d^2r}{dt^2}. In summary, to solve the given problem, we need to use the Euler-Lagrange equation and the special value for \omega^2.
  • #1
adjklx
13
0
Hi, I am trying to solve this problem here:

http://img201.imageshack.us/img201/7006/springqo9.jpg

We're supposed to find the equation of motion from the lagrangian and not Newton's equations.

Attempted solution:

[tex] L = T - U = \frac{I\omega^2}{2} + \frac{mv^2}{2} - \frac{kr^2}{2} [/tex]
[tex]I = m(r^2 + l^2) [/tex]
[tex]v = \frac{dr}{dt}[/tex]

From the euler-lagrange equation [tex]\frac{\partial L}{\partial r} = \frac{d}{dt}\frac{\partial L}{\partial v}[/tex] I get:

[tex]m\omega^2r - kr = m \frac{d^2r}{dt^2}[/tex]
[tex](\omega^2-\frac{k}{m})r = \frac{d^2r}{dt^2}[/tex]

If anyone can see any mistakes i'd appreciate it if they could let me know. Thanks
 
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  • #2
Your equation

[tex](\omega^2-\frac{k}{m})r = \frac{d^2r}{dt^2}[/tex]

appears correct. Recall that

[tex]\frac{k}{m} = \omega^2_{spring}[/tex]

So, this shows the special value for

[tex]\omega^2[/tex]

Particularly,

[tex]\omega^2=\omega^2_{spring}[/tex]
 

Related to Finding equation of motion from lagrangian

1. What is the Lagrangian method for finding the equation of motion?

The Lagrangian method is a mathematical approach used to find the equations of motion for a physical system. It is based on the principle of least action, which states that the motion of a system follows the path that minimizes the action, a quantity that is related to the system's energy.

2. How is the Lagrangian method different from other methods for finding the equation of motion?

The Lagrangian method differs from other methods, such as Newton's laws or Hamilton's equations, in that it does not require the use of forces or accelerations. Instead, it uses the concept of generalized coordinates and the Lagrangian function to describe the system's dynamics.

3. What is the role of the Lagrangian function in finding the equation of motion?

The Lagrangian function is a mathematical expression that combines the kinetic and potential energies of a physical system. It is used in the Lagrangian method to derive the equations of motion for the system, by taking the derivative of the Lagrangian with respect to the generalized coordinates.

4. Can the Lagrangian method be applied to any physical system?

Yes, the Lagrangian method can be applied to any physical system, as long as the system can be described using generalized coordinates and the Lagrangian function can be defined. This method is particularly useful for systems with complex dynamics or constraints, making it a versatile tool in physics and engineering.

5. Are there any limitations to using the Lagrangian method for finding the equation of motion?

While the Lagrangian method is a powerful tool, it does have some limitations. It may not be suitable for systems with non-conservative forces or systems with highly nonlinear dynamics. In these cases, other methods may be more appropriate for finding the equations of motion.

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