Newtonian dynamics
Encyclopedia
In physics, the Newtonian dynamics is understood as the dynamics
of a particle or a small body according to Newton's laws of motion
.
, 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
.
particles with masses 
in the regular three-dimensional Euclidean space
. Let
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
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 a flat multidimensional Euclidean space
, 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
of the dynamical system .
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 
:
can be constrained. Typical constraints
look like scalar equations of the form
Constraints of the form are called holonomic
and stationary
. 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
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
. 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
.
:
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.
of the constrained Newtonian dynamical system into the 
-dimensional flat comfiguration 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
of this induced metric are given by the formula
where
is the scalar product associated with the Euclidean structure .
particles is entroduced 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 .
. Such a maintaining force is perpendicular to 
. It is called the normal force
. 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
.
Like the velocity vector , the tangent force
has its internal presentation
The quantities
in are called the internal components of the force vector.
by the constraint equations is described by the differential equations
where
are Christoffel symbols
of the metric connection
produced by the Riemannian metric .
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
Dynamics (mechanics)
In the field of physics, the study of the causes of motion and changes in motion is dynamics. In other words the study of forces and why objects are in motion. Dynamics includes the study of the effect of torques on motion...
of a particle or a small body according to Newton's laws of motion
Newton's laws of motion
Newton's laws of motion are three physical laws that form the basis for classical mechanics. They describe the relationship between the forces acting on a body and its motion due to those forces...
.
Mathematical generalizations
Typically, the Newtonian dynamics occurs in a three-dimensional Euclidean spaceEuclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
, which is flat. However, in mathematics Newton's laws of motion
Newton's laws of motion
Newton's laws of motion are three physical laws that form the basis for classical mechanics. They describe the relationship between the forces acting on a body and its motion due to those forces...
can be generalized to multidimensional and curved
Curved space
Curved space often refers to a spatial geometry which is not “flat” where a flat space is described by Euclidean geometry. Curved spaces can generally be described by Riemannian geometry though some simple cases can be described in other ways. Curved spaces play an essential role in General...
spaces. Often the term Newtonian dynamics is narrowed to Newton's second law
.Newton's second law in a multidimensional space
Let's consider particles with masses in the regular three-dimensional Euclidean spaceEuclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
. Let
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 themThe 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 a flat multidimensional Euclidean space
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
, which is called the configuration space
Configuration space
- Configuration space in physics :In classical mechanics, the configuration space is the space of possible positions that a physical system may attain, possibly subject to external constraints...
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 spacePhase space
In mathematics and physics, a phase space, introduced by Willard Gibbs in 1901, is a space in which all possible states of a system are represented, with each possible state of the system corresponding to one unique point in 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. TheEuclidean structure of them is defined so that the kinetic energy
Kinetic energy
The kinetic energy of an object is the energy which 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 acceleration, the body maintains this kinetic energy unless its speed changes...
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 constraintsConstraint algorithm
In mechanics, a constraint algorithm is a method for satisfying constraints for bodies that obey Newton's equations of motion. There are three basic approaches to satisfying such constraints: choosing novel unconstrained coordinates , introducing explicit constraint forces, and minimizing...
look like scalar equations of the form
Constraints of the form are called holonomic
Holonomic
In mathematics and physics, the term holonomic may occur with several different meanings.-Holonomic basis:A holonomic basis for a manifold is a set of basis vectors ek for which all Lie derivatives vanish:[e_j,e_k]=0 \,...
and stationary
Stationary
Stationary can mean:* In statistics and probability: a stationary process.* In mathematics: a stationary point.* In mathematics: a stationary set.* In physics: a time-invariant quantity, such as a constant position or temperature....
. In terms of the radius-vector
of the Newtonian dynamical system they are written asEach 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 manifoldManifold
In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....
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 mechanicsLagrangian mechanics
Lagrangian mechanics is a re-formulation of classical mechanics that combines conservation of momentum with conservation of energy. It was introduced by the Italian-French mathematician Joseph-Louis Lagrange in 1788....
. 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 symboland 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 comfiguration space of the constrained Newtonian dynamical system into the -dimensional flat comfiguration space of the unconstrainedNewtonian 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 tensorMetric tensor
In the mathematical field of differential geometry, a metric tensor is a type of function defined on a manifold which takes as input a pair of tangent vectors v and w and produces a real number g in a way that generalizes many of the familiar properties of the dot product of vectors in Euclidean...
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 entroduced 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 forceNormal force
In mechanics, the normal force F_n\ is the component, perpendicular to the surface of contact, of the contact force exerted on an object by, for example, the surface of a floor or wall, preventing the object from penetrating the surface.The normal force is one of the components of the ground...
. The force
from is subdivided into two componentsThe first component in is tangent to the configuration manifold
. The second component is perpendicular to . In coincides with the normal forceNormal force
In mechanics, the normal force F_n\ is the component, perpendicular to the surface of contact, of the contact force exerted on an object by, for example, the surface of a floor or wall, preventing the object from penetrating the surface.The normal force is one of the components of the ground...
.Like the velocity vector , the tangent force
has its internal presentationThe 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 equationswhere
are Christoffel symbolsChristoffel symbols
In mathematics and physics, the Christoffel symbols, named for Elwin Bruno Christoffel , are numerical arrays of real numbers that describe, in coordinates, the effects of parallel transport in curved surfaces and, more generally, manifolds. As such, they are coordinate-space expressions for the...
of the metric connection
Metric connection
In mathematics, a metric connection is a connection in a vector bundle E equipped with a metric for which the inner product of any two vectors will remain the same when those vectors are parallel transported along any curve...
produced by the Riemannian metric .
Relation to Lagrange equations
Mechanical systems with constraints are usually described by Lagrange equations:where
is the kinetic energy the constrained dynamical system given by the formula . The quantities inare 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 procedureRaising and lowering indices
In mathematics and mathematical physics, given a tensor on a manifold M, in the presence of a nonsingular form on M , one can raise or lower indices: change a type tensor to a tensor or to a tensor...
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