[MUSIC] Hi and welcome again to our class simulation and modeling of natural processes. The next module is about Newton's laws of motion. And I will introduce some concepts of forces and potentials. Newton's law of motions consists of three laws. The first law state that if there is no force which acts on a body, the body just stays at a rest. Or travels through space along a straight path, with a constant velocity. If there is a force acting on a body, then the second lowest state that while the sum of all forces acting on the body is proportional to the mass of the body times its acceleration. And the third law is the law of action reaction. If a body act on a certain other body it became force, then the other body act back on the first body with the same force and magnitude but with enough opposite direction. In all the models we are gonna talk about in this module, we will focus on forces which act on an object. And which come from the interaction with the other objects here present in the system. There might be other types of forces in practice, but we will not consider them here. We restrict our self to models of Newtonian mechanics, which is we are only gonna consider classical potentials and forces. In some cases, the actual physics is non classical. For example, when we are gonna consider molecular dynamics, the actual physics is the physics of quantum mechanics. For the models we consider here we approximate quantum mechanics with Newtonian potentials and simulate these approximated classical systems. One of the implications of Newtonian mechanics is that the interaction between the particles or the objects in the system can't be decomposed into pair wise particles. Which means we can calculate individually the forces acting between pairs of particles or pairs of objects and in the end sum up all these contributions. Let's start with the first example, which is gravity. As I have said before, we consider first only pair wise interactions, and then we sum up all these pair wise interactions. The laws of gravity state that the force acting on an object coming from the gravitational interaction from another object is a proportional to the mass of both objects. In this equation and in all the coming equations we use a boldface notation to represent vectors. So F is the vector force Fij is the vector force, acting on body I, coming from body J. It is proportional to G, the gravitational constant, proportional to the mass of body I, proportional to the mass of body J. The interaction goes like the inverse query of the distance between the two bodies. On the numerator of this expression we have to vector which points from body I to body J and on the denominator we have the norm of this vector which is the distance between the two bodies. To the power 3. If we divide the length of the vector in the numerator by the distance between the two bodies, we are left with a system which goes like 1 over the distance is clear. If we wanna know the total force which acts on body I, we sum over all the other bodies. So we take a sum going from 1 to n, where n is the total amount of bodies in the system, excluding the body I itself because there is no gravitational force of the body onto itself. And sum up the force contribution from each individual body. There exists a second way to describe this forces mathematically in case the system is conservative. Which means the forces conserve the energy in the system. In this case the forces can be considered as derivatives of a scale of potential. Scale potential is an interesting object by it self. And we will see that, when we investigate the potential. We gain potential insights in to physics of the interaction between the objects. The force acting on body I is nothing else than the partial delivery of the potential. With respect to the three coordinates, the three coordinates of the position of body I. And the scalar potential in that gravitational system is the sum over all pairs of objects or in the numerator, the gravitational constant times the mass of the first object in the pair times the mass of the second object in the pair divided by the norm of the distance between the two objects. As we execute the sum over all pairs we must be careful to not take the pairs twice because you remember from Newton's Law of Mechanics that the force of object A acting on object B, it's the same in normal off the active force of object P acting on object A. So the pair needs to be taken into account only once when you calculate this K potential. Let me now introduce a second type of law in physics, which you an use to simulate a system of many point like particles. It is Coulomb's law. It is again a inverse square law like a law of gravity, which in this case, describes the electrostatic interaction between electrically charged particles. In this case, the force acting on object I coming from object J is again proportional to the inverse square of the distance between these two objects. And it is directed along the vector which points from object I to its object J. In this case, the force is proportional to the electric charge of object I. And directing charge of object J. And there is a proportionality constant, which evolves the constant epsilon 0 which is the permitivity of empty space. This force has the exact same space as the force which we have in the case of gravity. Except that there are different constants in front of it. Additionally, this force can be both repulsive or attractive. It is attractive if the two charges have a different sign. And repulsive if they have the same sign. Whereas the forces of gravity are always attractive. The third example I would like to show you is a type of interaction which has a very different shape. It is the Lennard Jones potential. It is a simple model which we use to approximately model the interaction between pairs of neutral atoms or molecules. This potential, Vij, between two neutral objects consists of two terms. The first term goes like the inverse distance between the two object to the power 12. And the second terms goes the reverse distance between the two objects to the power of six, and it has negative sign. Let's look at the graph which shows this potential. On the x axis we have r which we defined to be the distance between the two charged objects and on the y axis we have the value of the potential. You see that as the distance goes to zero which means as these two objects for example two molecules getting very close to each other the potential gets very high, very positive. In this case it is repulsive. The potential is there to avoid that the two objects penetrate into each other. At a little bit of a longer range the potential becomes negative, it is attractive. The long range behavior comes from that inverse power 6 term. It is the Van der Waals force which is attractive which makes molecules stick to each other when they can get close to each other. That inverse r to the power 12 term is the Pauli repulsion. It stems from the overlap of the orbitals of the two molecules or two atoms as they get close to each other. The Van der Waals force is actually physically motivated. It is the appropriate term to use in these circumstances. The Pauli repulsion is a little bit more ad hoc. Actually, as molecules get very close to each other, their interaction, as Pauli described by Newtonian mechanics must be fully resolved if you wanna know the details by quantum mechanics. The Pauli repulsion term which goes to the power 12 is just put there for numerical reasons to make this object behave like classical solid objects which cannot penetrate into each other as they get too close. The power 12 has just been chosen for numerical reasons because it is very cheap to calculate once you have calculated R to the power of minus six to get Van der Waals term. You just take the square of this previously calculated term to get the Pauli repulsion term, which goes to the power of 12. If could have been the power 13 or the power of 11 or whatever but we chose it to be 12 for computational efficiency. In the Lennard Jones potential we have two constants. One constant, let's go back to the potential, one constant epsilon is the depths of the potential well which is formed as we are in between the two readings of Pauli repulsion and Van der Waals attraction. As molecules or atoms get trapped into this well, they stick to each other and they need additional energy to free each other again from this bond. The second constant, is in some sense, the size of the atom or of the molecule. Of course, atoms that molecules don't have an actual size, because they are no solid object in the classical sense. Again, their interaction is fully determined by quantum mechanics which does not work with the concept of solid objects. But if you approximate these objects, these atoms or these molecules by a Newtonian model, you can imagine them as a hard ball sphere which interacts with other spheres. And this sphere would have a radius which is approximately sigma. On this graph of the Lennard Jones potential, the x-axis is formulated as multiples of these molecular radius and the Y-axis is formulated as multiples of the depth of the potential well epsilon. You see that this is a force which is quite different than the force of attraction or repulsion, which you have in Coulomb's Law or in the laws of gravity. The force in the Lennard Jones potential goes to zero quite fast as the two objects, the two molecules get far from each other. As you see on this graph, if you just go to a distance of three radius's of the molecules, the fourth is very close to zero. This is why in Agard most of the time the molecules don't interact with each other, they just follow a straight path. And only when they get very close to each other they collide. In summary, I have introduced three potentials which we are gonna investigate further in this lesson. The first potential represents, describes the force of gravity. At a long range, and actually also at a short range, it follows an inverse square row, which means it decreases like the square of the distance between two bodies. We call these a long range force, because the inverse square of distance is a quantity which decreases quite slowly. You could see this when you saw the simulation of two colliding galaxies. In this case, all stars were feeling the attraction of all of other stars and as a matter of fact the two galaxies were revolving around a common center of mass. Coulomb's law is exactly the same. However it is used at the much smaller scale describing the electrostatic interaction between charged particles. In these again the long range force, because the force as well, decreases like in the square. The Lennard Jones potential has two terms, but its long range behavior decreases like the power six. Six of the distance between the two objects. It is a short range force because the power six decreases quite fast. This means that with this type of potential, two objects, two particle don't feel the presence of each other unless they are very close to each other This ends our module on the Newton's law of motion and on the discussion of forces and potentials. Stay tuned for the next module. [MUSIC]