Deriving Kinetic Energy in a Double Pendulum System

In summary, the conversation discusses a problem in Lagrangian mechanics involving a double pendulum without a gravitational field. The goal is to compute the time derivatives of the Cartesian velocity components in terms of the angles to calculate the kinetic energy. The solution involves using the chain rule for derivatives to find the anticipated solutions for the kinetic energies.
  • #1
Andreas C
197
20
Ok, I'm reading up on Lagrangian mechanics, and there is a problem that I don't really understand: the double pendulum (in this case, without a gravitational field). So, I want to take it step by step to make sure I understand all of it.

We've got a pendulum (1) with a weight mass m=1kg attached to a rod of length r=1m making an angle of θ with the vertical, and another identical pendulum (2) attached on the weight of pendulum (1), making an angle α with it. Transforming these coordinates to Cartesian ones, we get x(1)=sinθ, y(1)=cosθ and x(2)=sinθ+sin(α+θ) and y(2) = cosθ+cos(α+θ).

Ok. So far so good. Now I am supposed to compute the time derivatives of the Cartesian velocity components in terms of the angles to compute the kinetic energy. How exactly am I supposed to do that? Since there is no t in these, I can't directly find their time derivatives, unless I use the equations that describe the motions of pendulums that are already known, but aren't these wrong in the case of a double pendulum?

I know that the results I am supposed to get for the kinetic energies are these:

T1=(dθdt)^2/2

T2=((dθdt)^2+(dθdt+dαdt)^2)/2+(dθdt)*(dθdt+dαdt)*cosα

I just have no idea how to get them.
 
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  • #2
Andreas C said:
Since there is no t in these
actually there is ##\alpha=\alpha(t),\quad \theta=\theta(t)##
Andreas C said:
with a weight mass m=1kg
this is not a good manner: you deprive yourself from opportunity of using the dimensions of physics quantities to check your formulas
 
Last edited:
  • #3
wrobel said:
actually there is α=α(t),θ=θ(t)

What I meant was that it's not explicitly shown. Of course there is, but I don't know what the value of the function α(t) is.

wrobel said:
this is not a good manner: you deprive yourself from opportunity of using the dimensions of physics quantities to check your formulas

Well, I'm not the one who made the problem :)
 
  • #4
Anyway, the question still stands, I still can't figure out what to do here.
 
  • #5
Ok, I found the answer, here it is for anyone who might be interested:

Using the chain rule, we write say dx1/dt=sinθ as dθ/dt ⋅ d(sinθ)/dθ = dθ/dt ⋅ cosθ. If we plug this (and y1) into the equation for kinetic energy in this case (which is T1=(x^2+y^2)/2), and do some algebra, we eventually get the anticipated solution of T1=(dθ/dt)^2/2. It's the same thing for T2, only a bit more complicated. The point is that you're meant to apply the chain rule for derivatives.
 

1. What is the Lagrangian of a double pendulum?

The Lagrangian of a double pendulum is a mathematical function used to describe the motion of a double pendulum system. It takes into account the kinetic and potential energy of the two masses as well as the constraints of the system.

2. How is the Lagrangian of a double pendulum calculated?

The Lagrangian of a double pendulum is calculated using the Lagrangian mechanics approach. This involves finding the kinetic and potential energy of each mass, and then using the Lagrangian equation L = T - V, where T is the total kinetic energy and V is the total potential energy.

3. What are the equations of motion for a double pendulum?

The equations of motion for a double pendulum can be derived from the Lagrangian using the Euler-Lagrange equations. These equations describe the position, velocity, and acceleration of each mass in terms of the Lagrangian, and can be used to solve for the motion of the system.

4. What are the applications of the Lagrangian of a double pendulum?

The Lagrangian of a double pendulum has applications in various fields such as robotics, physics, and engineering. It can be used to model and analyze the motion of pendulums, gyroscopes, and other mechanical systems.

5. How does the Lagrangian of a double pendulum differ from a single pendulum?

The Lagrangian of a double pendulum is more complex than that of a single pendulum due to the addition of another mass and the constraints of the system. It also results in a more complex motion, including chaotic behavior, compared to the simple harmonic motion of a single pendulum.

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