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.
[1][2][3] 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
Then the motion of these particles is governed by Newton's second law applied to each of them The three-dimensional radius-vectors
-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
The equations (3) are called the equations of a Newtonian dynamical system in a flat multidimensional Euclidean space, which is called the configuration space of this system.
The space whose points are marked by the pair of vectors
is called the phase space of the dynamical system (3).
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
of the Newtonian dynamical system (3) they are written as Each such constraint reduces by one the number of degrees of freedom of the Newtonian dynamical system (3).
is called the configuration space of the constrained system.
is called the phase space of the constrained system.
The velocity vector of the constrained Newtonian dynamical system is expressed in terms of the partial derivatives of the vector-function (7): The quantities
are called internal components of the velocity vector.
Sometimes they are denoted with the use of a separate symbol and then treated as independent variables.
The quantities are used as internal coordinates of a point of the phase space
Geometrically, the vector-function (7) implements an embedding of the configuration space
-dimensional flat configuration space of the unconstrained Newtonian dynamical system (3).
Due to this embedding the Euclidean structure of the ambient space induces the Riemannian metric onto the manifold
particles is introduced through their kinetic energy, the induced Riemannian structure on the configuration space
of a constrained system preserves this relation to the kinetic energy: The formula (12) is derived by substituting (8) into (4) and taking into account (11).
For a constrained Newtonian dynamical system the constraints described by the equations (6) are usually implemented by some mechanical framework.
in (14) are called the internal components of the force vector.
The Newtonian dynamical system (3) constrained to the configuration manifold
Mechanical systems with constraints are usually described by Lagrange equations: where
is the kinetic energy the constrained dynamical system given by the formula (12).
in (16) are the inner covariant components of the tangent force vector
However, the metric (11) and other geometric features of the configuration manifold