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
.

Let's consider
N
particles with masses
m
1
,
…
,
m
N
in the regular three-dimensional Euclidean space. Let
r
1
,
…
,
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
can be built into a single
n
=
3
N
-dimensional radius-vector. Similarly, three-dimensional velocity vectors
v
1
,
…
,
v
N
can be built into a single
n
=
3
N
-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
.

**Definition**. 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. Its points are marked by the radius-vector
r
. The space whose points are marked by the pair of vectors
(
r
,
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. The Euclidean structure of them is defined so that the kinetic energy of the single multidimensional particle with the unit mass
m
=
1
is equal to the sum of kinetic energies of the three-dimensional particles with the masses
m
1
,
…
,
m
N
:

In some cases the motion of the particles with the masses
m
1
,
…
,
m
N
can be constrained. Typical constraints look like scalar equations of the form

Constraints of the form (**5**) are called holonomic and scleronomic. In terms of the radius-vector
r
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**). Therefore the constrained system has
n
=
3
N
−
K
degrees of freedom.

**Definition**. The constraint equations (**6**) define an
n
-dimensional manifold
M
within the configuration space of the Newtonian dynamical system (**3**). This manifold
M
is called the configuration space of the constrained system. Its tangent bundle
T
M
is called the phase space of the constrained system.

Let
q
1
,
…
,
q
n
be the internal coordinates of a point of
M
. Their usage is typical for the Lagrangian mechanics. The radius-vector
r
is expressed as some definite function of
q
1
,
…
,
q
n
:

The vector-function (**7**) resolves the constraint equations (**6**) in the sense that upon substituting (**7**) into (**6**) the equations (**6**) are fulfilled identically in
q
1
,
…
,
q
n
.

The velocity vector of the constrained Newtonian dynamical system is expressed in terms of the partial derivatives of the vector-function (**7**):

The quantities
q
˙
1
,
…
,
q
˙
n
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
T
M
of the constrained Newtonian dynamical system.

Geometrically, the vector-function (**7**) implements an embedding of the configuration space
M
of the constrained Newtonian dynamical system into the
3
N
-dimensional flat comfiguration 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
M
. The components of the metric tensor of this induced metric are given by the formula

where
(
,
)
is the scalar product associated with the Euclidean structure (**4**).

Since the Euclidean structure of an unconstrained system of
N
particles is entroduced through their kinetic energy, the induced Riemannian structure on the configuration space
N
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. This framework produces some auxiliary forces including the force that maintains the system within its configuration manifold
M
. Such a maintaining force is perpendicular to
M
. It is called the normal force. The force
F
from (**6**) is subdivided into two components

The first component in (**13**) is tangent to the configuration manifold
M
. The second component is perpendicular to
M
. In coincides with the normal force
N
.

Like the velocity vector (**8**), the tangent force
F
∥
has its internal presentation

The quantities
F
1
,
…
,
F
n
in (**14**) are called the internal components of the force vector.

The Newtonian dynamical system (**3**) constrained to the configuration manifold
M
by the constraint equations (**6**) is described by the differential equations

where
Γ
i
j
s
are Christoffel symbols of the metric connection produced by the Riemannian metric (**11**).

Mechanical systems with constraints are usually described by Lagrange equations:

where
T
=
T
(
q
1
,
…
,
q
n
,
w
1
,
…
,
w
n
)
is the kinetic energy the constrained dynamical system given by the formula (**12**). The quantities
Q
1
,
…
,
Q
n
in (**16**) are the inner covariant components of the tangent force vector
F
∥
(see (**13**) and (**14**)). They are produced from the inner contravariant components
F
1
,
…
,
F
n
of the vector
F
∥
by means of the standard index lowering procedure using the metric (**11**):

The equations (**16**) are equivalent to the equations (**15**). However, the metric (**11**) and other geometric features of the configuration manifold
M
are not explicit in (**16**). The metric (**11**) can be recovered from the kinetic energy
T
by means of the formula