Generalized coordinates: Understanding Kinetic Energy

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Homework Help Overview

The discussion revolves around the use of generalized coordinates in calculating kinetic energy, particularly in systems like a double pendulum. The original poster questions how to ensure that kinetic contributions from different coordinates do not lead to double counting, especially when considering the motion of angles that may be moving in opposite directions.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants explore the nature of kinetic energy as a scalar quantity and discuss the implications of adding contributions from different objects. There are questions about the validity of simply adding kinetic energies associated with different angles and concerns about potential cancellation of kinetic energy when angles move in opposite directions.

Discussion Status

Some participants provide insights into the scalar nature of kinetic energy and suggest that contributions from different masses in a system can be added. However, there is also skepticism about whether kinetic energies from different angles can be directly summed without further consideration of their relationships. The discussion reflects a mix of interpretations and approaches to the problem.

Contextual Notes

There is an emphasis on starting from an inertial frame to calculate kinetic energies, and participants note the importance of expressing positions in terms of generalized coordinates to avoid double counting. The discussion highlights the need for clarity regarding the definitions of angles and their relation to the physical objects involved.

ehrenfest
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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

 
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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.
 
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.
 
ehrenfest said:
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.
 

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