So here we're looking at a particle in a plane, this particle has mass M. And it's kind of spinning around in the plane and the rope is being pulled down by some force and the force is applied such that we have this constraint. In this form What do we call this form of a constraint? >> [INAUDIBLE] >> Yeah. Holonomic would not have rates in there. So this is a Is this an integrable constraint? >> [INAUDIBLE] >> Yeah, are not equal to a constant. >> I hope you can integrate that differential equation. Right, otherwise I have to go back to Freshman calculus. So this is definitely you could have written it as a holonomic constraint. But we have it here in a uniform because you see we tend to always implement things in a rate based poverty and form that we have there. Okay, good. So that's a constraint. So we know to enforce this constraint, it's going to have a constraint force that won't do any work, right? Not with admissible constraints. So where is that constraint force going to act in this system? Julian, what do you think? >> [INAUDIBLE] >> There's going to be in a constraint force? Like if a particle is rolling on the table and it has to be at 1 m height, it's a constraint force that's pushing up, holding that particle at 1 m. This is a constraint. It's a and a form of a holonomic constraint. How is that constraint force going to apply on the particle? >> [INAUDIBLE]. >> Yeah, so it's going to be in the Direction, right? We've shown here the forces pointing down, but that's because the rope did a 90° bend. And so ultimately it's going to come longer. It's not always obvious as you will find, is it going to be a working or nonworking force? I have these weird kinematic constraints where exactly is this force going to be? But when you do these definitions for generalized forces and do the stock products with partial velocities, the math will tell you. And so this is an example where we're going to do the math and I could have argued earlier this must be that force and therefore that wouldn't do any work and I don't even have to consider it, we will consider it and you will see it drops out anyway. So, here are position, we're using a rotating frame is little rer, and then therefore the inertial velocity proper transport theorem. Right, We get now our velocity expression, cool. Now I have to have the particles kinetic energy, mass over two velocities are dotted with itself. Right, that's the velocity magnitude squared and dotted with itself will give you r dot squared. And then we have r squared theta dot squared and r, we're now applying this constraint directly because it's not an independent variable. R is really dictated to this constraint are has to be whatever initial are you have minus C times time, that's the solution to integrating this and including initial conditions. So this is nothing but r squared and r dot is minus c, so squared just gives you c squared. Right, so we've taken now mass over 2, velocity dotted with itself and applied the constraint. This is now a minimal coordinates set. We have one degree of freedom. What's the angle of the particle going to be doing? So here we have generalized coordinates. Q is basically feta in this case and Q dot appears. Does Q appear in my kinetic energy in this example? Does theta appear in kinetic energy? No, right, so that'll be important in a moment. So this one only depends on my rates. Everything else is there. Good, now Lagrange in dynamics as we said, instead of Q and just, my Q is theta dot but I'm doing an explicit in terms of theta. Left hand side, there is no theta in this t, so it's partial is zero. I have to take the partial of t with respect to theta dot. Well, this is just a constant, that's all going to drop away. This Will give you 2 times a half times mass times this term times data dot which is here. And on the right hand side, I need my generalized force. So I'm not sure that's what I'm just assuming right now, I just want to include it just in case and I know this force on the rope will be in the direction. So I can write it out times the partial velocity of the place where this force is applied which is r and I'm doing the dot, the cancelation of dot property. So I can do the partial of our dot with respect to theta dot. That's again easier because then I don't have to do the partial of with respect to theta and then figure out what does that mean. So here you're just going to get re theta. And as you can see are the dotted with e theta has to be zero. Right, those are two orthogonal basis vectors. So the generalized force acting on the theta coordinate Is zero in this case. Now, why is this an interesting differential equation? I have the derivative of the square bracket zero equals to zero. Well that means whatever I'm differentiating, I could differentiate it and use chain rule. And this would give you a theta double dot differential equation and that's your equations of motion. But we can do even better. I can integrate that differential equation because I know its derivative must be zero. So therefore this bracketed term must be what? A constant, not zero. Right, if x start is equal to zero, that means x is a constant. It maybe zero, I don't know. But it also could be 15. Right, so this is simply going to be a constant. So you can apply initial conditions at time t zero. I only have r, not t is zero. And theta dot, that's your initial rotation rate and this is a constant. I can solve for theta dot. So I didn't have to solve a second order differential equation. In essence, what you have found is one term that is going to be preserved. The kinetic energy, is that going to be constant in this problem? No, something is doing work on it. It's actually pulling it in and doing weird stuff and his angular momentum going to be constant. Is it h dot equal to l? R across the force is going to be zero. So L is zero. So here momentum is constant and in essence what we've rediscovered is conservation of momentum, but this is a single degree of freedom system. What we'll find other cases is all of a sudden, momentum is not constant and energy is not constant. But maybe momentum around some body axis happens to be constant or any, or there's a sub result. So any time you find these kinetic energies that do not depend on the states, because that means this partial is going to go to zero. And there's no work in general, there's no working forces acting on the system. All these generalized coordinates rates are zero. Anytime this happens, then you're going to end up with this partial derivative of this partial has to be zero. So therefore this partial of energy with respect to the generalized coordinate rate is a constant and in fact they call this term generalized momentum. P theta is generalized momentum. You will see, you're laughing Julian. Any particular reason. >> I just like the [INAUDIBLE] generalized. >> Generalized or they really love generalized, that's true. So this is the generalized momenta and you will see it when we get to hamiltonian and stuff that appears in those places as well formulation. ? But this is nice because you may have xy theta pitch mrps, who knows what in there? And if this condition holds for one of the co-ordonates there's no sigma 1, there's no q sigma 1, then immediately you've got variant of motion. All right And that's very helpful, one I don't have to solve a second order differential equation. I can enforce that constraint and put it in directly because you notice in this differential equation, I don't have to worry about what was the initial rate and how does that go? You can just plug it in and off you go. So that's kind of nice if you're doing integration checks like we've been doing with the SEMG. We used energy and energy rate and momentum and momentum rate to kind of go or we, is everything acting the way it should. If you have these kinds of invariants of motion, this is really nice when you didn't actually solve the differential equations or you have another form is dissatisfied at all times? Or sometimes you need it analytically. I just need to know the energy rate and I have a direct analytic answer here, in terms of time. You can probably integrate this actually, I haven't tried it. This one is not a complicated equation, so it probably has an integral. But that's so if you find invariants of motion, that's a good thing. Think of it like all of a sudden energy is preserved and here it isn't. But it is actually a form of mental that is happening here. Any questions on this example? So takeaways, we only had energy right now. There was some force acting on it but it turned out to be non zero. And then we have these invariants of motion we can find in this one. And if in doubt put the force in there, the math thought that force with the partial velocities. And it'll tell you right away it's going to be zero or not.
