Since the terms $q$ "generalized coordinates" are not necessarily ‘lengths’, the quantities $Q$ "generalized forces" also do not necessarily have the dimension of a ‘force’. However, it is always true that: $[Q]\cdot [q] =$ energy. Why the units of energy do not change?
$\begingroup$
$\endgroup$
Add a comment
|
1 Answer
$\begingroup$
$\endgroup$
3
Yes, it follows directly from the definition of generalized forces $$ Q_j~:=~\sum_{i=1}^N {\bf F}_i \cdot \frac{\partial {\bf r}_i}{\partial q^j}$$ that the the virtual work is $$ \delta W~=~\sum_{i=1}^N {\bf F}_i \cdot \delta {\bf r}_i~=~\sum_{j=1}^n Q_j\delta q^j, $$ and hence that the product $$[Q_j][q^j]~=~\text{dim. of work}~=~\text{dim. of energy}.$$
-
-
-
$\begingroup$ still not clear why the dim. of work does not change either? $\endgroup$ Mar 10 at 9:20