Welcome to the course of advanced classical Mechanics. My name is Mikhail Frolov and today we're going to discuss Kelly's problem. So what is Kelly's problem? So let's say we have an age of the dog on the edge of this dog. We have a real with a rope and free part of this rope hands down and falls down according to the 2nd law in Newton, because gravity forces of acting. So now the question, so what is the acceleration of this arm of robe? So it should be equal gravity acceleration or should be less of gravity acceleration or should be washed than gravity acceleration. So, the Kelly's problem answer on that questions. So how we can calculate or how we can solve the problem. So, first, we will solve this problem qualitatively. So, according to the 2nd Law of Newton, changing of momentum of our system, a caused by forces. So in that situation, our forces, gravity force in our momentum is just a product of mass multiplied by velocity. But because of robe or the length of this, pre robe is increasing, then mass is also increasing. So now it's a mess is of some function of time t. So in general we have such a simple differential equation and we just try to expand. There is a total derivative which we have in the left side of that equation. So if we expand that equation is the second term proportional acceleration of the rope. So we can express acceleration in terms of gravity acceleration and some additional term which is a proportional changing of mass and velocity. For that case, changing of mass is a positive because if we increase time t so more mass will be involved in our dynamics. So velocity here direct in the same direction as the acceleration of gravity, then the additional term is a positive. So our acceleration is just the difference between acceleration of gravity minus some positive term means that acceleration of rope here is less than acceleration of rabbits. So how we can solve this problem by using Lagrange formalism or how we can solve by using Lagrange equation of 2nd type. So for this case we have to introduce linear density. So our mass is given by product of linear density, gamma multiplied by Z and Z is a length of this rope or free robe, which we discussed. So let's consider a small part delta z and that part of the rope which has lang delta. See the mass is a delta M here. And that mass is given by product of gamma multiplied by dz. So we can easily calculate elementary kinetic energy of this small piece of our rope, which is is just a product al mass, elementary mass, delta M. Multiplied by speed of this piece. So we can easily calculate the total kinetic energy by taking integral from left and right side of that equation. And they can arrive to the final expression which is quite intuitively understandable. So it is just a regular kinetic energy of our system. So it's just a product of total mass which involved in economics multiplied by the square of velocity. So let's calculate now potential energy of our system. So let's consider a small piece with the length of delta Z. And for that small piece. So we can calculate potential energy in the gravity forces. So that is a elementary potential energy is given by product of three terms delta M, G and Z. G here is a gravity acceleration. So we can easily can calculate total potential energy by taking integral from left and right sides of that equation. So in the final, so we obtain an expression for potential energy which is represented by regular potential energy which is given by product mass, gravity, acceleration and z divide by 2. So the total Lagrange function here is just a difference between kinetic energy and potential energy and given by that expression. So we know that our Lagrange function, you find it up to some constant. So we can cancel out common factor gamma. So we can cancel out factors too, so and we can now construct a Lagrange equation. Lagrange equation can be constructed by calculating partial derivative in generalized velocity and partial derivative in z. So finally we have seen such a result which can be presented in such a form. So, and now the problem, how we can solve z differential equations. So z differential is not so simple. So because there is not linear differential equation, but we know that a time T equals 0, so the length of rope on the age equals 0. So let's consider a simple case. So we assume that our solution can be presented in a very simple form. So that is an acceleration a multiplied by the square of time, divide by 2. And in that expression a is acceleration. So let's substitute all that expression to our differential equations. Make some simple calculation and we obtain just simple algebraic equation for a. So and we can conclude that our acceleration is just half of acceleration of gravity. So the robe accelerates with the half of acceleration of gravity. So there is a Kelly's problem. And we see that if we have a system which was best maybe change during of dynamics of our system. So we will have an extra forces here and that extra forces may accelerate or or deaccelerate our system. So in that case which we consider here so that extra forces here they accelerate our system. So on the next lecture, we will discuss more tough problem in mechanics. So it's a scattering problem. So we will consider two cases scattering on the shortage potential and scattering on the cooling potential. So, and they consider what's the difference between these two cases?