In physics, the Newtonian dynamics is understood as the dynamics of a particle or a small body according to Newton's laws of motion.
Contents
- Mathematical generalizations
- Newtons second law in a multidimensional space
- Euclidean structure
- Constraints and internal coordinates
- Internal presentation of the velocity vector
- Embedding and the induced Riemannian metric
- Kinetic energy of a constrained Newtonian dynamical system
- Constraint forces
- Newtons second law in a curved space
- Relation to Lagrange equations
- References
Mathematical generalizations
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
The three-dimensional radius-vectors
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
Euclidean structure
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
Constraints and internal coordinates
In some cases the motion of the particles with the masses
Constraints of the form (5) are called holonomic and scleronomic. In terms of the radius-vector
Each such constraint reduces by one the number of degrees of freedom of the Newtonian dynamical system (3). Therefore the constrained system has
Definition. The constraint equations (6) define an
Let
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 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
and then treated as independent variables. The quantities
are used as internal coordinates of a point of the phase space
Embedding and the induced Riemannian metric
Geometrically, the vector-function (7) implements an embedding of the configuration space
where
Kinetic energy of a constrained Newtonian dynamical system
Since the Euclidean structure of an unconstrained system of
The formula (12) is derived by substituting (8) into (4) and taking into account (11).
Constraint forces
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
The first component in (13) is tangent to the configuration manifold
Like the velocity vector (8), the tangent force
The quantities
Newton's second law in a curved space
The Newtonian dynamical system (3) constrained to the configuration manifold
where
Relation to Lagrange equations
Mechanical systems with constraints are usually described by Lagrange equations:
where
The equations (16) are equivalent to the equations (15). However, the metric (11) and other geometric features of the configuration manifold