How Does the Principle of Virtual Work Apply to Particle Equilibrium?

[cosmicraj]

I could not understand the Principle of virtual work.
Suppose we have two (x,y,t) [at two different points].By principle of least action we will get a trajectory such that it minimize lagrangian.
Does the principle of virtual work say that it will vary the path a little but having the same end points?

 

[torquil]

http://en.wikipedia.org/wiki/Lagrangian_mechanics

Quote: "Start with D'Alembert's principle for the virtual work of applied forces, \mathbf{F}_i, and inertial forces on a three dimensional accelerating system of n particles, i, whose motion is consistent with its constraints"

 

[cosmicraj]

For understanding D alembert's pinciple. one should know the principle of virtual work.

 

[Raheel Riaz]

i think It is not difficult as people have made it...
The simple thing is that the use of this principle is to find the equilibrium position of anybody or set of particles.

 

[Raheel Riaz]

Thanx to all and please tell me more if required...

 

[PhDorBust]

I don't understand well enough to put this in the context of what you are talking about. But...

A virtual displacement is when you change the coordinates of the particles by an infinitesimal distance. This is different from a normal displacement in that this displacement does not take place over an time interval dt.

Now, if you have a system of particles in equilibrium, then $$\sum_i F_i \cdot \delta r_i = 0$$, because each F_i = 0.

The principle of virtual work says that forces of constraint do no work. So $$\sum_i F^{constraint}_i \cdot \delta r_i = 0$$ and therefore $$\sum_i F^{applied}_i \cdot \delta r_i = 0$$ as $$F_i = F^{constraint}_i + F^{applied}_i$$.

A colloquial way to see the validity of this is to imagine a particle constrained to travel on a sphere. The force of constraint will be perpendicular to the surface while the virtual displacement will be tangent to the surface, so their dot product is 0.