Lecture 5 - Classifications of Constraints in Mechanical Systems

追加: 2026-05-09

[00:00:06][music] Uh welcome again uh to the next lecture and uh we uh were discussing uh the classification of the constraint. So we have gone through the concept of what is constraint and how it uh uh restate the motion of the system. So now let's discuss what are the types of the classification of the constraints. So the topic today we are going to discuss is uh is uh classifications of constraints.

[00:01:01]Okay. So the first constraint uh is is holomic.

[00:01:09]So holomic constraints are those constraint which can be expressed as in terms of an equation which is say r1 r2 r3 and so on and time is equal to zero. So what are these r1 r2 r3 and time? These are the the coordinates of the system through which we can define the system. For example, like if if you say a simple pendolum.

[00:01:39]So if you say a simple pendolum which is uh uh is is the angle theta and if I I need an x and y coordinate uh to define the position of this simple pendulum. So but I have a restriction over here that is this x² + y² = l² and then I can write x + 2² + y² - l² is going to be zero. So this is one equation between this x and y. So this these x and y are the coordinates through which I can specify the certain configuration of the system.

[00:02:16]And if using these configuration coordinates we can define the constraint imposed on the system in like like in in this form then simple equation then such type of constraint are said to be holomic constant. For example, this is this simple pendulum is case and in case of the rigid body uh rigid body case or if you if you remember rigid body case then then we have this r i - r and this is uh minus cig is going to be zero. So this is again this is again is a form of uh this simple equation. So this is also going to be holomic constraint. And then we discuss about uh this uh beat problem beats motions we discuss about that that was y - x tan theta =0 and also y - x - v_t tan theta is equal to uh 0. If you remember like uh uh this case no here yeah this this one this case and then here uh this one. So this these are all this this one this these are all the cases of the holomic constraint. But if if if I don't have a provision to uh express my uh constraint in terms of this coordinates which are going to be a simple equation. So then this is going to be uh the second one this is going to be non holomic okay this is going to be nonomic constraints so that means uh no simple equation in this case no simple equation and what I have is going to equalities equalities for example For example, if if so this is r² - a² is greater than equal to 0. So this is one equation that is like in terms of the inequalities where you see that we have discussed uh the motion of like this one the motion of this ball this motion of this ball on the surface of this constraint to move on the surface of the sphere. So that uh the constraint equation is this r² - a² is greater than equal to zero. And in the same way this one this this equation is also nonomi because I don't have an equation but there's a constraint to the molecules has to be move inside the surface constraint surface under a b c. So these are these are so the examples of uh [snorts] non-holomic uh constants.

[00:05:14]Similarly, if I say that if I say that a particle has to move inside inside surface of a sphere. So what you are going to do is uh uh and that is that is going to be r² - a² is going to be less than equal to zero because that is outside the surface and then where particle is constant to move inside the surface then r² - a square is going to be less than equal to zero or this r² is basically if you write in terms of this x² + y² + z² And this is going to be always less than the radius of uh this uh so is not going to be equal to because it's always going to be less than the radius of the system because uh if you have a sphere and this this particle has to move inside the sphere here somewhere. So or this position vector of this particle this say this is and say this is a or so this is defined by xyz c. So this is always x² + y² + z² is always going to be less than uh this radius of this sphere has to be inside.

[00:06:32]So when when when I'm able to define my constraint equation in a simple equation this is going to be the holomic constraint. when I'm not able to do this, this is in terms of the inequalities. Then there is going to be the non-honomic constraint. There is another classification of uh constraint that could be depending upon time.

[00:06:54]So uh constraint uh we can have a constraint classification uh that depending upon time. So uh when uh the one is uh this is known as reanomomas.

[00:07:23]So this one is like when I have explicit explicit dependence on time.

[00:07:36]So whatever the constraint is whether uh this is holomic and non-holomic. So we normally talk about this holomic constraint that is uh that is physible to us. So when this is depending upon time is constraint equation depending upon time that is also uh known as the rionomous uh constant for example. So when when when when you see that this uh this bead this bead beads is bead is sliding over this length and then uh this this wire is also moving along in certain direction then this the time dependence is there. So this is like y - x - v_t then theta equal to zero is a is a kind of uh reonomous uh constant and uh when when it is not depending upon time so then this is uh this is said to be escarous esclar uh this is known as the esclaron nomous nomous constant. Okay. So I have aromas constant. So uh so time independent say time independent constraint equations.

[00:09:11]Okay.

[00:09:13][snorts] So uh you see that when your uh constraint equation is depending upon time so this is realomomas when it's not depending upon time this is going to be scalomous there are conditions that like when when when the condition of bead so if I have if I have uh if I have a bead uh coming down this this wire so this is y this is x and so I have I'm going to have a certain uh angle theta. So I have this y and this x component. So this have y = xin theta. So there's no time dependency here. So this is going to be escalenormous but it it there are there may be cases that while this this uh bead is coming down towards the thread and if thread is not rigid. So that the thread is while it is coming this if this is not rigid uh so if uh the thread is not rigid.

[00:10:17]So uh it can move it can move while uh bead is going along going along it.

[00:10:32]So then then when it's moving uh then then there there comes a time. So so like when be come from this position to this position and while coming this this this this this had certain motion this thread has certain motion then the configuration of this this this thread is also coming into play. So the time dependency itself is going to be included inside uh the coordinates which are going to specify the motion of this thread. So this is not explicitly dependent on this. This is coming across uh inside the coordinate system. So that is not going to be the rionomomas. This is going to be the escaronomomas. So this is not explicit dependence. So we have u uh classification of the constraint that is one is holomic that is uh when when I am able to define my equations uh based on uh the coordinate of the system in a simple equation that is going to be holomic constraint and when it's not the conditions that we are going to have the equalities then this is going to be uh the non-holomic constant and further division is is based on uh uh the time dependence that if it depends upon time explicitly then it's going to be realomous constraint but uh if it is not then it is going to be the clear and normous constraint.

[00:11:54]Okay. So uh I hope that uh the concept of the constraint and its its uh uh classification is clear to you. There are many there could be many many examples but these are the standard example through which through which you can understand you can imagine what do you mean by the constraint and how to identify whe the constraint is holomic non-holomic rionomous or escalon one uh one more uh way uh to identify the uh constraint is whether the constraint which is going to be holomic is integraable or non- integraable. So uh let's try to understand this that is that is known as uh uh say let me have another page that is known as the velocity dependent constraints. Okay. So what do you mean by velocity dependent uh constraint? So uh for example the best example to understand the velocity dependent constraint is that uh we consider the example of uh pure rotation that is pure rotation means I have the rotation uh uh without slipping. Okay.

[00:13:29]So if I if I consider a a disc which is rotating along this surface and uh rotating this is going to have this angular velocity omega and this is uh so what we need to specify this rotation first first thing uh we need is uh the coordinates of the center of mass center of mass that is this thing and this is going to be say this is x and y whatever it is and then uh the angle of rotation angle of rotation say this is going to be theta in this case and this is the point of contact it is also going to have uh so this this when this is rotating and this is moving forward this this center of mass is not rotating because it is fixed pointed it is all it is it is going to have a translation motion that the velocity of the center of mass is uh this is uh if this is r is the radius of this uh desk and this is the omega so this is this is going to have a translational motion and this is this is the omega and this point of contact which which could have x and y but this point of contact is always a stationary because is a pure rotation. So whatever the point of contact is it is uh going to be always stationary. So the condition is that here is that v cm is going to be r omega and we can write it as like is moving in like in say this is x direction. So this is dx by dt is going to the r and omega is written in terms of this is if the angle of rotation of theta. So this is r d theta by dt. Okay. So this this we have a differential equations. Now the restriction is that uh this center of mass is not moving because we are talking about the pure rotation. The center of mass is going a translational motions at vcm r omega. So what I'm going to have is a differential equation.

[00:15:53]So when I'm going to have a differential equation sort of thing and if I integrate it, let try to integrate it.

[00:15:58]If I integrate it, what I'm going to get is dx = say r d theta and from here I get x = r theta and if uh so this is what I'm going to get. So x r is for x and theta. So what I'm going to get is like when uh the constraint equation which is in this case is going to be holomic. So vcm minus r omega equal to zero. So what I have is a differential equation sort of thing because there is a velocity velocity coming into play and that is this is uh in terms of the differential equation. So when your constraint are in terms of the differential equation and then if you integrate it and you get a relation between the coordinates then that south that that indicates that the condition or the constraint is uh kind of oromic.

[00:16:53]So if if the differential equation form is given and when you are able to integrate it and then you're going to have a relation like this between the coordinates which satisfy the system then that that is going to be the holomic constant. There's a second case which is where you see that we have the differential form but we we cannot fully integrate it then that is going to be the case of non-holomic constant for example.

[00:17:20]So the condition is that uh a desk uh uh uh rolling down rolling down a long horizontal surface.

[00:17:42]The condition is uh with uh always vertical in position.

[00:17:58]So that means the orientation is always has to be uh vertical. So if I say this is my uh this is my say x-axis along this direction and this is my z-axis and this is my say this is my yaxis and if some desk is disc has to rotate along okay so if the disc has to rotate along this some some path it is coming towards certain path while rotating along this uh xy plane and this this disc has to be vertical always. So what I need is uh the first thing is uh the coordinates of coordinates of center of mass of this and then the second thing is the angle of rotation to specify let's this is going to be phi and then orientation angle angle so this is theta so what I have This is is this is this is my phi and uh this is my angle theta. So and the velocity is in this direction say this is my v. So and this orientation angle is with this is this is with respect to say this with respect to x-axis that it has to the disc has to be always vertical vertically and rolling down along the xy plane and so these are the things we need to specify uh the rotation of this along this line. So first thing first equation what I'm going to have is like uh this uh the velocity is going to be a fi dot this ai dot and then that is basically a d5 by dt and then second thing I'm going to have is this velocity is going to have uh two component that is vxi and uh vyj and this this I'm not going to calculate the angle sort of thing. This is going to be uh this v uh sin theta i plus uh v cos theta j. Okay.

[00:20:28][snorts] So this [clears throat] this v is basically uh uh basically is going to be dx or say like if I say like this. So this is vx component is v sin theta and uh so I have dx by dt is equal to uh say this is v is v is a this is d5 by dt and I have sin theta here. Okay. So if I integrate it, what I'm going to get is dx is equal to a sin theta d5 or I'm going to have an equation is a sin theta d5 =0. Similarly from uh this uh uh vy component this is going to be minus cos theta this is going to be have dy - a uh + cos theta d5 equal to z. So I have these two differential equation and uh one one is this. So here you have a differential equation which we can integrate and we get a relation which relates completely to the coordinates. But uh here in this case uh if I if I try to solve these two equation solving these two differential equation means that what uh what I'm going to get is x uh y because there is no z component and this is theta and phi. So I have these four coordinates.

[00:22:09]So if I solve these differential equation and I get a relation which relates completely this x y theta and phi then this is going to be uh integraable uh and going to be holomic.

[00:22:22]But th this is this is no position because uh if if if my if my say if my uh uh if the disc is going from one position to some other position what I'm going to get is a different different uh uh x1 y1 if I have x1 y1 theta 1 and 51 at one position and then from there it goes to some x2 y2 2 theta_2 and uh 52 position.

[00:22:54]So whether I'm able to connect it or not because uh if if this is if I fix x1 y1 and theta1 so this is so this is going to have uh uh say it's like rotating along this direction.

[00:23:18]So if I have this is my 51 and then it go from here to that position then I go to some another so 52 it's okay and say some theta angle is there this is depends upon theta and theta_2 but for this if I fix this uh so I can going to from 51 to I'm going to get 52 but the 52 is not going to be unique because while it is rotating if I get 51 to 52 to from here I can get another value of phi like going from this direction to this direction. So five phi is not unique in this case. So I cannot have a a good relation which can relate this uh x y and theta and I cannot have this x y theta and equal to zero kind of this is not possible. So this is is going to be is going to be non-integraable non-integraable non-integraable.

[00:24:22]So this is uh this is uh non-holomic nonomic constraint. Okay. So uh two points we can uh we can uh remember here is uh one is that uh is that if the constant if uh if constraints are integraable so we have holomic constraint and if [snorts] constraints or non-integraable then this is nonomic.

[00:25:14]Okay. So first thing is if if integraable and non-integraable concept comes when we have a con the constraint is given in terms of the differential form and when we have in the differential form that is going to have uh the velocity dependent okay so when we have velocity dependence I'm going to have in differential form of that and when I'm going to integrate it and got a good relation so this is going to be holomic and if I'm not able to integrate it completely which can give me the relation uh of uh f of this sort in terms of the coordinate with which I specify the system is going to be the non-holomic uh case. So this is all about this for the constraint system that what is the constraint holomic non-olonomic time dependent is going to be in terms of the reonomic and escalomic and then the another kind of uh uh dependence that is in terms of uh the velocity which is integraable and non-tegrable then this is going to be holomic and nonomic uh classification of this.

[00:26:24]Okay. So this is all about uh a constraint system and uh we move to uh uh the second thing which is uh I think is a degree of freedom but uh uh I think I I I'll stop this uh at this position this uh next from the next onward lecture next lecture we will cover try to cover the degree of freedoms and the generalized coordinate and in that generalized coordinate we had to discuss about the configuration space. So degree of freedom, generalized coordinate in the configuration space. This is is uh a very uh very conceptual point which we need to learn about while we are able to set up the lang for this system. So I hope uh this is all good. If you have any queries, try to uh mail your queries and questions and don't hesitate if you have and I'll try to answer your questions as soon as possible. Thank you very much.

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