what conditions are necessary for mechanical work to be done
Lagrangian mechanics
∂ L ∂ q j − d d t ∂ L ∂ q  ̇ j = 0 . ( \ displaystyle ( \ frac ( \ partial L ) ( \ partial q_ ( j ) ) ) - ( \ frac ( \ mathrm ( d ) ) ( \ mathrm ( d ) t ) ) ( \ frac ( \ partial L ) ( \ partial ( \ dot ( q ) ) _ ( j ) ) ) = 0 \ , . )
∂ L ∂ q j − d d t ∂ L ∂ q  ̇ j = 0 . ( \ displaystyle ( \ frac ( \ partial L ) ( \ partial q_ ( j ) ) ) - ( \ frac ( \ mathrm ( d ) ) ( \ mathrm ( d ) t ) ) ( \ frac ( \ partial L ) ( \ partial ( \ dot ( q ) ) _ ( j ) ) ) = 0 \ , . )