Newtonian dynamics

In physics, Newtonian dynamics (also known as Newtonian mechanics) is the study of the dynamics of a particle or a small body according to Newton's laws of motion. Typically, the Newtonian dynamics occurs in a three-dimensional Euclidean space, which is flat. However, in mathematics Newton's laws of motion can be generalized to multidimensional and curved spaces. Often the term Newtonian dynamics is narrowed to Newton's second law . Consider particles with masses in the regular three-dimensional Euclidean space. Let be their radius-vectors in some inertial coordinate system. Then the motion of these particles is governed by Newton's second law applied to each of them The three-dimensional radius-vectors can be built into a single -dimensional radius-vector. Similarly, three-dimensional velocity vectors can be built into a single -dimensional velocity vector: In terms of the multidimensional vectors () the equations () are written as i.e. they take the form of Newton's second law applied to a single particle with the unit mass . Definition. The equations () are called the equations of a Newtonian dynamical system in a flat multidimensional Euclidean space, which is called the configuration space of this system. Its points are marked by the radius-vector The space whose points are marked by the pair of vectors is called the phase space of the dynamical system (). The configuration space and the phase space of the dynamical system () both are Euclidean spaces, i. e. they are equipped with a Euclidean structure. The Euclidean structure of them is defined so that the kinetic energy of the single multidimensional particle with the unit mass is equal to the sum of kinetic energies of the three-dimensional particles with the masses : In some cases the motion of the particles with the masses can be constrained. Typical constraints look like scalar equations of the form Constraints of the form () are called holonomic and scleronomic.