Generalized coordinates: Understanding Kinetic Energy

[ehrenfest]

Homework Statement

When I use generalized coordinates how do I know that I can add the kinetic contributions from each to get the total kinetic energy? How do I know that you are not "counting the same KE twice"?

e.g. if you have a double pendulum how do you know that you can just add the KE due to one angle to the KE due to the other angle?

What if the angles are moving in opposite directions? Couldn't some KE cancel out then?

Homework Equations

The Attempt at a Solution

 

[javierR]

Kinetic energy is a *scalar* quantity, there is no inherent direction associated with it (i.e., it doesn't matter in what directions objects in a single system are moving with respect to each other or a static frame of reference, etc). There can only be *positive contributions* to kinetic energy fom each object; and, e.g., two masses m and M in a single system, the objects *individually contribute* to the kinetic energy, so that they simply add as
1/2 m s^2 + 1/2 M S^2,
where s and S are the speeds of the two masses m and M *with respect to a single reference frame*. This is therefore true in the special case of the double pendulum, where it's usually simpler to write the linear speeds "s" in terms of the rotational speed
d(Angle)/dt.

 

[ehrenfest]

Actually I do not even think it is true that you can just add the KE due to one to the KE due to the other angle. You need to express x and y in terms of the angles and then add the squares of their derivatives.

 

[nrqed]

Actually I do not even think it is true that you can just add the KE due to one to the KE due to the other angle. You need to express x and y in terms of the angles and then add the squares of their derivatives.

I guess that by "angles" you really mean the two *objects*! (It makes no sense to me to talk about the energy of angle! There is the energy of a mass whose position is described by an angle).

You are correct that we always start from an *inertial frame* to calculate the kinetic energies of the objects. And *then* we rexpress those energies in terms of the generalized coordinates. This way there is no double counting problems.