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.
Mathematical generalizations.
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 formula_1.
Newton's second law in a multidimensional space.
Consider formula_2 particles with masses formula_3 in the regular three-dimensional Euclidean space. Let formula_4 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 formula_5 can be built into a single formula_6-dimensional radius-vector. Similarly, three-dimensional velocity vectors formula_7 can be built into a single formula_6-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 formula_9.
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 
formula_10. The space whose points are marked by the pair of vectors formula_11 is called the phase space of the dynamical system ().
Euclidean structure.
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 formula_9 is equal to the sum of kinetic energies of the three-dimensional particles with the masses formula_3:
Constraints and internal coordinates.
In some cases the motion of the particles with the masses formula_3 can be constrained. Typical constraints look like scalar equations of the form 
Constraints of the form () are called holonomic and scleronomic. In terms of the radius-vector formula_10 of the Newtonian dynamical system () they are written as
Each such constraint reduces by one the number of degrees of freedom of the Newtonian dynamical system (). Therefore, the constrained system has formula_16 degrees of freedom.
Definition. The constraint equations () define an formula_17-dimensional manifold formula_18 within the configuration space of the Newtonian dynamical system (). This manifold formula_18 is called the configuration space of the constrained system. Its tangent bundle formula_20 is called the phase space of the constrained system.
Let formula_21 be the internal coordinates of a point of formula_18. Their usage is typical for the Lagrangian mechanics. The radius-vector formula_10 is expressed as some definite function of formula_21:
The vector-function () resolves the constraint equations () in the sense that upon substituting () into () the equations () are fulfilled identically in formula_21.
Internal presentation of the velocity vector.
The velocity vector of the constrained Newtonian dynamical system is expressed in terms of the partial derivatives of the vector-function
The quantities formula_26 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 formula_20 of the constrained Newtonian dynamical system.
Embedding and the induced Riemannian metric.
Geometrically, the vector-function () implements an embedding of the configuration space formula_18 of the constrained Newtonian dynamical system into the formula_29-dimensional flat configuration space of the unconstrained 
Newtonian dynamical system (). Due to this embedding the Euclidean structure of the ambient space induces the Riemannian metric onto the manifold formula_18. The components of the metric tensor of this induced metric are given by the formula
where formula_31 is the scalar product associated with the Euclidean structure ().
Kinetic energy of a constrained Newtonian dynamical system.
Since the Euclidean structure of an unconstrained system of formula_2 particles is introduced through their kinetic energy, the induced Riemannian structure on the configuration space formula_2 of a constrained system preserves this relation to the kinetic energy:
The formula () is derived by substituting () into () and taking into account ().
Constraint forces.
For a constrained Newtonian dynamical system the constraints described by the equations () are usually implemented by some mechanical framework. This framework produces some auxiliary forces including the force that maintains the system within its configuration manifold formula_18. Such a maintaining force is perpendicular to formula_18. It is called the normal force. The force formula_36 from () is subdivided into two components
The first component in () is tangent to the configuration manifold formula_18. The second component is perpendicular to formula_18. In coincides with the normal force formula_39.<br>
Like the velocity vector (), the tangent force 
formula_40 has its internal presentation
The quantities formula_41 in () are called the internal components of the force vector.
Newton's second law in a curved space.
The Newtonian dynamical system () constrained to the configuration manifold formula_18 by the constraint equations () is described by the differential equations
where formula_43 are Christoffel symbols of the metric connection produced by the Riemannian metric ().
Relation to Lagrange equations.
Mechanical systems with constraints are usually described by Lagrange equations:
where formula_44 is the kinetic energy the constrained dynamical system given by the formula (). The quantities formula_45 in 
() are the inner covariant components of the tangent force vector formula_46 (see () and ()). They are produced from the inner contravariant components formula_41 of the vector formula_46 by means of the standard index lowering procedure using the metric ():
The equations () are equivalent to the equations (). However, the metric () and
other geometric features of the configuration manifold formula_18 are not explicit in (). The metric () can be recovered from the kinetic energy formula_50 by means of the formula