Now everyone's itching to get to conservative forces, right? This is a lagrange's most popular form, because often we have just stuff happening, bouncing balls, gravity problems. There's just conservative forces acting on it, that's it. And then we will see, we have a very nice simple form. But let's own it first, let's derive it. So the generalized force was the summation of all the forces acting on it, so not one. And we've excluded constrained compliant forces all ready, they didn't do work. Times these partial velocities which with the cancelation of dots, I'm going to use, Which version? I'm going to use this one here, I think down below. I could do either but this will help you with my chain rule that I need. Now if the force acting on the system is a conservative force, then by definition there exists a potential function V, capital V, right? Where if you take the negative gradient of V with respect to the position vector, you will get the force vector out of it, right? That's how conservative forces are defined. So, if we use that now and plug that in here, right, we're going to say generalized force is the force dotted times this partial velocity. And we have to sum up over all the particles and all the forces acting on each particle. That was the classic for definition. Now, I plug in this definition. So you get minus the summation of partial V, partial R times partial R partial q, Freshman calculus again, right, this is equivalent to the partial of V with respect to q, right? If V depends on R and R depends on q. Doing this and going backwards, you'd have to take the partial of V with respect to R and then R depends on q, so times partial R with respect to q. We're just going the other way. So all of this in the end, the generalized coordinates acting on the system are nothing but minus the partial of the potential function with respect to the generalized coordinates, Right? And they go through. >> What happens to the sum there? >> Actually, I was thinking the same thing. The question is, what happens to the sum? It drops because the i's are dropped, [SOUND] because you have to do this for all stuff. If you did this backwards, you'd have to do it for every particle. because particle one may depend on this coordinate particle two position may depend on the coordinate. [LAUGH] I was thinking the same thing, what happens with my summation? So that's why that's done through the chain rule essentially. So going from here to here is almost easier and then you go, so therefore from there and go to the right. But it really simplifies it, no more summation. A single scaler and the rest is just algebra. Okay, so what does that mean now? If we plug that into our earlier equations that we have, we're going to compare to this and we can start putting that in. And that gives you a partial of V with respect to q, here we have a partial of T with respect to q. Which motivates now this definition of lagrangian. And the lagrangian ends up being T minus the potential kinetic energy minus the potential energy of the system. because if we do that now, again, this lagrangian T can depend on states and rates, most generally, definitely rates but it could have states. V can it depend on states? Yes. Can V depend on rates? No, that's impossible, potential functions are always just state dependent, not rate, otherwise that wouldn't work. So, we can say, okay, that the partial of L with respect to q dot is equivalent to the partial of T with respect to q dots, right? So earlier when I had partial T I can just replace it with partial like L, the script L, lagrangian. Whereas partial L with respect to q, well, T can depend on q and V can depend on q and you get this. So if I just go back a slide, here we have T minus this, this was minus V. So if you bring this over to the left hand side, you would have minus T partial plus V partial. So since script L is T minus V, you end up with just minus L with respect to q. This is the classic form. If you only have conservative forces. This is where people love lagrangian because all they have to do is get kinetic energy, know the potential energy, get mathematical, let it rip, it's just going to do it all. Where things always get fuzzy as now, I'm giving you damping, that bastard, he gave me a non conservative force. And then you have to figure out what's this generalized force. But you guys and I hope you're starting to get comfortable with this. It's just forces dotted with partial velocities acting on every one of those coordinates. All right, so this is the conservative system, is there. And you notice, this is the classic way to write it, partial L, partial q dot. You could also write partial T partial q dot, Right, because we know this. But this is how people tend to write it. Just but that's end ever, because the partial of V with respect to q should always be q dots, the rates should always be zero. So that's how you write it. If you have other forces, if not all the F's are due to springs and gravity potentials, you may have damping, you may have thrusters, you may have weird ropes pulling and twirling on this thing, who knows? Then you have what we call non conservative jth coordinate force, so ncj. So we just have to consider those on the right hand side to compute the generalized force acting on that coordinate. We don't have to consider that, this is a modification, it's not the same as the earlier generalized force, that would include gravity. This is basically just a non conservative components of the generalized force acting on the jth coordinate. So you do this for QX, Q theta, for everything. And that definition remains the same, right, we've simply absorbed the conservative forces because they can be written as partial into the lagrangian. And we don't have to think about all these partial velocities and to get them. So this is the more general version of lagrangian dynamics subject to general forces. Still talking about minimal coordinates set. But we now have potential of general forces, everything covered. Any questions on this?
