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## Main Question or Discussion Point

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:

T

T

I just have no idea how to get them.

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:

T

_{1}=(dθdt)^2/2T

_{2}=((dθdt)^2+(dθdt+dαdt)^2)/2+(dθdt)*(dθdt+dαdt)*cosαI just have no idea how to get them.