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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?

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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 $$[Q_j][q^j]~=~\text{dim. of work}~=~\text{dim. of energy}.$$

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  • $\begingroup$ does it have a physical meaning? $\endgroup$
    – yaser heba
    Mar 10 at 8:57
  • $\begingroup$ I updated the answer. $\endgroup$
    – Qmechanic
    Mar 10 at 9:08
  • $\begingroup$ still not clear why the dim. of work does not change either? $\endgroup$
    – yaser heba
    Mar 10 at 9:20

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