When do I need to use virtual work in writing the equations of motion?

[jhosamelly]

I'm studying for our comprehensive exam . I just need to clarify something. So the equation of motion for lagrangian dynamics is $\frac{d}{dt}\frac{\partial L}{\partial\dot{q}_{i}}$ = $\frac{\partial L}{\partial {q}_{i}}$

However, in my notes there are example which uses the principle of virtual work wherein $\frac{d}{dt}\frac{\partial L}{\partial\dot{q}_{i}}$ - $\frac{\partial L}{\partial {q}_{i}}$ = $F_{q}$

Then we look for $F_{q}$ using virtual work.

However isn't $\frac{d}{dt}\frac{\partial L}{\partial\dot{q}_{i}}$ - $\frac{\partial L}{\partial {q}_{i}}$ = 0 ?

 

[Joey21]

I think I can help you, but I don't want to answer without knowing what F sub q stands for just in case I make things worse. Can you specify what it represents please?

 

[jhosamelly]

q is the generalized coordinate.

For example if I have r (radial distance) as generalized coordinate I'll have


$\frac{d}{dt}\frac{\partial L}{\partial\dot{r}}$ - $\frac{\partial L}{\partial {r}}$ = $F_{r}$

 

[WannabeNewton]

The second equation is for when non-conservative forces are present; one can then use the virtual work principle to find the $F_{q}$ (the non-conservative generalized forces) and incorporate them into the modified Euler-Lagrange equations written above. See here: http://ocw.mit.edu/courses/mechanic...control-i-spring-2007/lecture-notes/lec14.pdf

 

[PJC01]

Virtual work is a powerful tool that is often used in the study of mechanics and dynamics. It allows us to analyze systems by considering the work done by virtual displacements, which are hypothetical displacements that do not actually occur in the system. This allows us to simplify complex systems and derive equations of motion without having to consider all possible real displacements.

In the context of writing equations of motion, virtual work is commonly used in Lagrangian dynamics. The equation you mentioned, \frac{d}{dt}\frac{\partial L}{\partial\dot{q}_{i}} = \frac{\partial L}{\partial {q}_{i}}, is known as the Lagrange's equation and is derived using the principle of virtual work. This principle states that the work done by the virtual displacements is equal to zero, which is why you see \frac{d}{dt}\frac{\partial L}{\partial\dot{q}_{i}} - \frac{\partial L}{\partial {q}_{i}} = 0 in your notes.

However, there are cases where we need to consider external forces acting on the system, which cannot be expressed solely in terms of the generalized coordinates and velocities. This is where the modified form of Lagrange's equation, \frac{d}{dt}\frac{\partial L}{\partial\dot{q}_{i}} - \frac{\partial L}{\partial {q}_{i}} = F_{q}, comes into play. Here, F_{q} represents the generalized forces acting on the system, and it is found using the principle of virtual work.

In summary, virtual work is a useful tool in writing equations of motion, especially in Lagrangian dynamics. It allows us to simplify complex systems and consider external forces in our analysis. It is important to understand when and how to use virtual work in order to accurately describe the dynamics of a system.