Hi, let's get into the theorem of virtual powers. Once upon a time, there was the principle of equivalence discovered by Newton himself. This means that the force is equivalent somewhere to an acceleration and this formula is a victorial formula. And that's the problem. For numerical applications, modern applications, we need scalar expression of this theorem and Mr Dalembert, a French guy, found a way to make it scalar by the scalar product. He demonstrated very simply that it was equivalent to a scalar product by multiplying this formula with any, any vectorial field and that's the most, most important thing that it's valid for any vectorial field. We have F minus m multiplied by the acceleration scalar b and this time it is zero scalar. So it is strictly equivalent to the basic principle of dynamics because we can choose the vectorial field as we want. For instance, if we choose b identically equal to the derivate of the position, which means the velocity we find back the theorem of kinetic energy. Let's see F minus m O P dot dot scalar O P dot being zero scalar. Then we find here the power of F and we find here the derivate of the Kinetic energy. That's it. We have demonstrated the strict equivalence of the scalar formula for any vectorial field with the basic principle of dynamics. And it is Mr Lagrange will use it now to establish his formalism. Let's get into this formalism. Let's get into the parameters of Lagrange. Lagrange was a real genius, he had the idea to represent the position of any system by a finite number of geometrical parameters that he called, Qj. j being one to n. Then let's give an example. For instance, an aircraft in flight with its center, the position of the center can be represented by X, Y, Z coordinates, three translations. And the position around the center can be represented by three angles around the X, Y, Z axis, the three angles are the roll, the pitch, and the yaw. You see that with these six parameters, free translations, free rotations, we define the position, the configuration of the aircraft in flight. Well it's a photo of the aircraft. And now let's see how it works to make it move. And to make it move we don't need only scalars, we need function of times and that will be the role of the Lagrange equation to see how these parameters evolve and to see how the equations can be written to see the evolutions of these parameters. Let's get into the state variables of Lagrange. Lagrange observed the tree of Newton and he discovered that the unique difference between the apple and the moon was the initial condition. Both were falling around the center of Earth, but the apple was falling vertically because its velocity, its initial velocity was zero, and the moon was in orbit around the Earth because its initial velocity is not zero. So, as he was a genius, he discovered that the evolution, the dynamic evolution of the system did not depend only on the position, but also on the velocity. And then he defined the state variables as follows. The n geometrical parameters, the n derivate of the parameters versus time, and the time. And we don't forget that each parameter is depending on the time. Which means that in total we have two n plus one independent Lagrange variables. Let's give an example. The apple. The position of the apple, which is a material point, is given by its coordinates X, Y, Z. And the velocity is given by x dot, y dot, z dot. And then we, at the time. If we consider these seven variables, we have the basic variables of Lagrange to define the dynamics of any material point. If we come back now to the aircraft in flight with the six parameters of positions, this will require thirteen variables of Lagrange to define the dynamics. So with the state variables, we define the dynamics with the Lagrange equation.