# I'm studying analytical mechanics and it states that it always true that generalized coordinates times generalized forces have the dimension of energy

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?

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