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
Newton's laws of motion are three basic laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows:
# A body remains at rest, or in moti ...
.
Mathematical generalizations
Typically, the Newtonian dynamics occurs in a three-dimensional
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
, which is flat. However, in mathematics
Newton's laws of motion
Newton's laws of motion are three basic laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows:
# A body remains at rest, or in moti ...
can be generalized to multidimensional and
curved
In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight.
Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that a ...
spaces. Often the term Newtonian dynamics is narrowed to
Newton's second law
Newton's laws of motion are three basic laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows:
# A body remains at rest, or in motion ...
.
Newton's second law in a multidimensional space
Consider
particles with masses
in the regular three-dimensional
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
. Let
be their radius-vectors in some
inertial
In classical physics and special relativity, an inertial frame of reference (also called inertial reference frame, inertial frame, inertial space, or Galilean reference frame) is a frame of reference that is not undergoing any acceleration. ...
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 mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space. Examples include the mathematical models that describe the swinging of a ...
in a flat multidimensional
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
, 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
In dynamical system theory, a phase space is a space in which all possible states of a system are represented, with each possible state corresponding to one unique point in the phase space. For mechanical systems, the phase space usually ...
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
In physics, the kinetic energy of an object is the energy that it possesses due to its motion.
It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its accele ...
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
:
Constraints and internal coordinates
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
A mechanical system is scleronomous if the equations of constraints do not contain the time as an explicit variable and the equation of constraints can be described by generalized coordinates. Such constraints are called scleronomic constraints. ...
. In terms of the radius-vector
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
degrees of freedom.
Definition. The constraint equations () define an
-dimensional
manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
within the configuration space of the Newtonian dynamical system (). This manifold
is called the configuration space of the constrained system. Its tangent bundle
is called the phase space of the constrained system.
Let
be the internal coordinates of a point of
. Their usage is typical for the
Lagrangian mechanics
In physics, Lagrangian mechanics is a formulation of classical mechanics founded on the stationary-action principle (also known as the principle of least action). It was introduced by the Italian-French mathematician and astronomer Joseph-Lou ...
. The radius-vector
is expressed as some definite function of
:
The vector-function () resolves the constraint equations () in the sense that upon substituting () into () the equations () are fulfilled identically in
.
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
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
of the constrained Newtonian dynamical system.
Embedding and the induced Riemannian metric
Geometrically, the vector-function () implements an embedding of the configuration space
of the constrained Newtonian dynamical system into the
-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
. The components of the
metric tensor
In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold (such as a surface) that allows defining distances and angles, just as the inner product on a Euclidean space allows ...
of this induced metric are given by the formula
where
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
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 () 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
. Such a maintaining force is perpendicular to
. It is called the
normal force
In mechanics, the normal force F_n is the component of a contact force that is perpendicular to the surface that an object contacts, as in Figure 1. In this instance ''normal'' is used in the geometric sense and means perpendicular, as oppose ...
. The force
from () is subdivided into two components
The first component in () is tangent to the configuration manifold
. The second component is perpendicular to
. In coincides with the
normal force
In mechanics, the normal force F_n is the component of a contact force that is perpendicular to the surface that an object contacts, as in Figure 1. In this instance ''normal'' is used in the geometric sense and means perpendicular, as oppose ...
.
Like the velocity vector (), the tangent force
has its internal presentation
The quantities
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
by the constraint equations () is described by the differential equations
where
are
Christoffel symbols
In mathematics and physics, the Christoffel symbols are an array of numbers describing a metric connection. The metric connection is a specialization of the affine connection to surfaces or other manifolds endowed with a metric, allowing distance ...
of the
metric connection
In mathematics, a metric connection is a connection in a vector bundle ''E'' equipped with a bundle metric; that is, a metric for which the inner product of any two vectors will remain the same when those vectors are parallel transported along ...
produced by the Riemannian metric ().
Relation to Lagrange equations
Mechanical systems with constraints are usually described by
Lagrange equations
In physics, Lagrangian mechanics is a formulation of classical mechanics founded on the stationary-action principle (also known as the principle of least action). It was introduced by the Italian-French mathematician and astronomer Joseph-Lo ...
:
where
is the kinetic energy the constrained dynamical system given by the formula (). The quantities
in
() are the inner
covariant components of the tangent force vector
(see () and ()). They are produced from the inner
contravariant components of the vector
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
are not explicit in (). The metric () can be recovered from the kinetic energy
by means of the formula
See also
*
Modified Newtonian dynamics
Modified Newtonian dynamics (MOND) is a hypothesis that proposes a modification of Newton's law of universal gravitation to account for observed properties of galaxies. It is an alternative to the hypothesis of dark matter in terms of explaining ...
References
{{DEFAULTSORT:Newtonian Dynamics
Classical mechanics
Isaac Newton