Understanding the Coordinates in the Lagrangian for a Pendulum

In summary, the conversation is about a doubt regarding the coordinates for a particle in a classical mechanics problem from Landau's book. The person asks for help in understanding the solution and mentions it has something to do with the harmonic oscillator. Another person suggests looking at a figure that shows the coordinates for the particle and explains that they are determined by the vector sum of two coordinates. The original person understands and thanks them for their help.
  • #1
p1ndol
7
3
So I've been studying classical mechanics and have come across a small doubt with the solution provided to the problem in question from Landau's book. My question is: why are the coordinates for the particle given as they are in the solution? I imagine it has something to do with the harmonic oscillator, but I'd like to properly understand. I appreciate any kind of help, and I'm sorry if this post is somehow incorrect, it is my first one regarding questions.
 

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  • #2
Did you look at the figure? For instance in (a), the support point ##p## has coordinates ##\mathbf{r}_p = a(\cos{\gamma t}, -\sin{\gamma t})## and the radius vector from ##p## to ##m## has coordinates ##\mathbf{R} = l(\sin{\phi}, \cos{\phi})## then the coordinates of ##m## are nothing but those of the vector ##\mathbf{r}_p + \mathbf{R}##.
 
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  • #3
I understand it now, thanks!
 

Related to Understanding the Coordinates in the Lagrangian for a Pendulum

1. What is the Lagrangian for a pendulum?

The Lagrangian for a pendulum is a mathematical expression that describes the energy and motion of a pendulum system. It takes into account the kinetic and potential energy of the pendulum and is derived from the principle of least action.

2. How is the Lagrangian for a pendulum derived?

The Lagrangian for a pendulum is derived using the principle of least action, which states that the motion of a system will follow the path that minimizes the action. This involves calculating the kinetic and potential energy of the pendulum and using the Euler-Lagrange equation to find the path that minimizes the action.

3. What is the significance of the Lagrangian for a pendulum?

The Lagrangian for a pendulum is significant because it allows us to accurately model and predict the behavior of a pendulum system. It also provides a more elegant and efficient way to describe the motion of a pendulum compared to traditional methods such as using Newton's laws of motion.

4. Can the Lagrangian for a pendulum be used for any type of pendulum?

Yes, the Lagrangian for a pendulum can be used for any type of pendulum, as long as the system can be described using the principles of classical mechanics. This includes simple pendulums, compound pendulums, and even double pendulums.

5. How does the Lagrangian for a pendulum differ from the Hamiltonian for a pendulum?

The Lagrangian and Hamiltonian for a pendulum are two different mathematical expressions that describe the same system. The Lagrangian takes into account the kinetic and potential energy of the pendulum, while the Hamiltonian takes into account the total energy of the system. They are related through the Legendre transformation and can be used interchangeably to describe the motion of a pendulum.

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