Newtonian dynamics

In physics, the Newtonian dynamics is understood as 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 m a = F {displaystyle displaystyle m,mathbf {a} =mathbf {F} } . Consider N {displaystyle displaystyle N} particles with masses m 1 , … , m N {displaystyle displaystyle m_{1},,ldots ,,m_{N}} in the regular three-dimensional Euclidean space. Let r 1 , … , r N {displaystyle displaystyle mathbf {r} _{1},,ldots ,,mathbf {r} _{N}} 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 r 1 , … , r N {displaystyle displaystyle mathbf {r} _{1},,ldots ,,mathbf {r} _{N}} can be built into a single n = 3 N {displaystyle displaystyle n=3N} -dimensional radius-vector. Similarly, three-dimensional velocity vectors v 1 , … , v N {displaystyle displaystyle mathbf {v} _{1},,ldots ,,mathbf {v} _{N}} can be built into a single n = 3 N {displaystyle displaystyle n=3N} -dimensional velocity vector: In terms of the multidimensional vectors (2) the equations (1) are written as i.e. they take the form of Newton's second law applied to a single particle with the unit mass m = 1 {displaystyle displaystyle m=1} . Definition. The equations (3) are called theequations 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 r {displaystyle displaystyle mathbf {r} } . The space whose points are marked by the pair of vectors ( r , v ) {displaystyle displaystyle (mathbf {r} ,mathbf {v} )} is called the phase space of the dynamical system (3). The configuration space and the phase space of the dynamical system (3) both are Euclidean spaces, i. e. they are equipped with a Euclidean structure. TheEuclidean structure of them is defined so that the kinetic energy of the single multidimensional particle with the unit mass m = 1 {displaystyle displaystyle m=1} is equal to the sum of kinetic energies of the three-dimensional particles with the masses m 1 , … , m N {displaystyle displaystyle m_{1},,ldots ,,m_{N}} :