How does Generalized Work have dimensions of Work?

[patric44]

Homework Statement

how the generalized work has dimensions of work ?

Relevant Equations

W = sum(Qi.dqi) ; where Qi is the generalized force

hi guys
my analytical mechanics professor asked a question the other day about, how come the generalized forces$Q_{\alpha}$ doesn't need to have a dimension of force, and the generalized coordinated $q_{\alpha}$as well doesn't need to have a dimension of length, but the generalized work $\sum\;Q_{\alpha}\;q_{\alpha}$ must have the dimension of work ?

 

[stevendaryl]

It's true sort of by definition.

In regular Cartesian coordinates, the work due to a force $\vec{F}$ applied over a displacement $d\vec{r}$ is given by:

$dW = \sum_i F_i dx^i$

Under a generalized coordinate transformation, $dx^i = \sum_\alpha \Lambda^i_\alpha dq^\alpha$, where $\Lambda^i_\alpha$ is the matrix of partial derivatives: $\Lambda^i_\alpha = \dfrac{\partial x^i}{\partial q^\alpha}$. So in terms of the $q^\alpha$, we have:

$dW = \sum_i F_i dx^i = \sum_i F_i (\sum_\alpha \Lambda^i_\alpha dq^\alpha)$

Rearranging the terms gives:
$dW = \sum_i F_i dx^i = \sum_\alpha dq^\alpha (\sum_i F_i \Lambda^i_\alpha)$

Then you just define $Q_\alpha = \sum_i F_i \Lambda^i_\alpha$.

So the generalized forces were specifically defined so that $dW$ has the same value as in Cartesian coordinates.

 

[Steve4Physics]

To get an intuition, it helps to consider a specific example.

For simplicity, consider a system with only one degree of freedom - simple rotation. The generalised force, Q, will be a torque and the generalised coordinate, q, will be an angle.

dW = Qdq

Angles are dimensionless. It is no coincidence that the dimensions of torque are the same as those of work!

We choose Q and q for convenience; there is no reason they must have the dimensions of force and length. But their product must have the dimensions of work.

 

[patric44]

how come We choose Q and q for convenience, isn't that related to the system i am working on, how the system itself adjusts so that Q.q is the same in every coordinate?
or the system doesn't know about Q in the first place ,and it is only my definition?

so what i understood from you guys is that the generalized force is mathematically manufactured to get W upon doting it with q , why did we defined it in such a way ?
i can't get my head around this

 

[Steve4Physics]

Not sure I fully understand the questions (which are very different from the question in your first post!). But see if this helps.

The system does not ‘know’ about the existence of ‘Q’s and ‘q’s. Forces and coordinates are (of course) purely human choices/concepts to help us analyse a system’s behaviour.

We typically identify the degrees of freedom and assign a generalised coordinate $q_i$ to each. If $q_i$ changes by some amount $\delta q_i$ then some work $\delta W_i$ is done. The generalised force associated with this is $Q_i$ and, from an abstract point of view, it is simply the constant of proportionality between $\delta q_i$ and $\delta W_i$ giving $\delta W_i = Q_i \delta q_i$.

But $Q_i$ isn’t ‘mathematically manufactured’ this way without reason. Read the section entitled ‘Generalized forces’ here: https://en.wikipedia.org/wiki/Generalized_forces#Generalized_forces

Note that each generalized force, $Q_i$ is taken as having the same direction as $\delta q_i$. So dot products are not required between $Q_i$ and $\delta q_i$ (check the above link).

A different choice of coordinates would of course give a different set of $\delta W_i$s but the same total work.

I suspect that may not answer your questions! Someone more knowledgeable than I might be able to help. You could try posting a new, more specific, question.

 

[vela]

how come We choose Q and q for convenience, isn't that related to the system i am working on, how the system itself adjusts so that Q.q is the same in every coordinate?

Once you choose the $q$'s, they determine what the $Q$'s are. You don't have the freedom to choose both arbitrarily. That's what @stevendaryl is saying in his post above.

so what i understood from you guys is that the generalized force is mathematically manufactured to get W upon dotting it with q. Why did we defined it in such a way?

Because it's convenient. If you have a coordinate transformation, you can express the work in terms of displacements $\delta q_j$ where each $\delta q_j$ is multiplied by some complicated expression, and leave it at that. But we choose to give that complicated expression the name generalized force because it makes sense conceptually and it turns out to have a straightforward physical interpretation in some cases.