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
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 (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 .
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).
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 :
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 (5) are called holonomic and scleronomic. In terms of the radius-vector 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 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 :
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 .
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
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:
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 . Such a maintaining force is perpendicular to . It is called the normal force. The force from (6) is subdivided into two components
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
The quantities in (14) are called the internal components of the force vector.
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