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- 1 Mathematical generalizations
- 2 Newton's second law in a multidimensional space
- 3 Euclidean structure
- 4 Constraints and internal coordinates
- 5 Internal presentation of the velocity vector
- 6 Embedding and the induced Riemannian metric
- 7 Kinetic energy of a constrained Newtonian dynamical system
- 8 Constraint forces
- 9 Newton's second law in a curved space
- 10 Relation to Lagrange equations
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 space
Let's 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:
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 :
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
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 :
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 (7):
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 (7) implements an embedding of the configuration space of the constrained Newtonian dynamical system into the -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 . 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 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:
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.
Newton's second law in a curved space
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 (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):
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