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 generalizationsEdit

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  .

Newton's second law in a multidimensional spaceEdit

Consider   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

 

 

 

 

 

(1)

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:

 

 

 

 

 

(2)

In terms of the multidimensional vectors (2) the equations (1) are written as

 

 

 

 

 

(3)

i.e. they take the form of Newton's second law applied to a single particle with the unit mass  .

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  . The space whose points are marked by the pair of vectors   is called the phase space of the dynamical system (3).

Euclidean structureEdit

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   is equal to the sum of kinetic energies of the three-dimensional particles with the masses  :

 .

 

 

 

 

(4)

Constraints and internal coordinatesEdit

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

 .

 

 

 

 

(5)

Constraints of the form (5) are called holonomic and scleronomic. In terms of the radius-vector   of the Newtonian dynamical system (3) they are written as

 .

 

 

 

 

(6)

Each such constraint reduces by one the number of degrees of freedom of the Newtonian dynamical system (3). Therefore, the constrained system has   degrees of freedom.

Definition. The constraint equations (6) define an  -dimensional manifold   within the configuration space of the Newtonian dynamical system (3). 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  :

 .

 

 

 

 

(7)

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  .

Internal presentation of the velocity vectorEdit

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

 .

 

 

 

 

(8)

The quantities   are called internal components of the velocity vector. Sometimes they are denoted with the use of a separate symbol

 

 

 

 

 

(9)

and then treated as independent variables. The quantities

 

 

 

 

 

(10)

are used as internal coordinates of a point of the phase space   of the constrained Newtonian dynamical system.

Embedding and the induced Riemannian metricEdit

Geometrically, the vector-function (7) 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 (3). 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

 ,

 

 

 

 

(11)

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

Kinetic energy of a constrained Newtonian dynamical systemEdit

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:

 .

 

 

 

 

(12)

The formula (12) is derived by substituting (8) into (4) and taking into account (11).

Constraint forcesEdit

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  . Such a maintaining force is perpendicular to  . It is called the normal force. The force   from (6) is subdivided into two components

 .

 

 

 

 

(13)

The first component in (13) is tangent to the configuration manifold  . The second component is perpendicular to  . In coincides with the normal force  .
Like the velocity vector (8), the tangent force   has its internal presentation

 .

 

 

 

 

(14)

The quantities   in (14) are called the internal components of the force vector.

Newton's second law in a curved spaceEdit

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

 ,

 

 

 

 

(15)

where   are Christoffel symbols of the metric connection produced by the Riemannian metric (11).

Relation to Lagrange equationsEdit

Mechanical systems with constraints are usually described by Lagrange equations:

 ,

 

 

 

 

(16)

where   is the kinetic energy the constrained dynamical system given by the formula (12). The quantities   in (16) are the inner covariant components of the tangent force vector   (see (13) and (14)). They are produced from the inner contravariant components   of the vector   by means of the standard index lowering procedure using the metric (11):

 ,

 

 

 

 

(17)

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

 .

 

 

 

 

(18)

See alsoEdit