Newtonian dynamics explained
See also: Classical mechanics. In physics, Newtonian dynamics (also known as Newtonian mechanics) is the study of the dynamics of a particle or a small body according to Newton's laws of motion.[1] [2] [3]
Mathematical generalizations
.
Newton's second law in a multidimensional space
Consider
particles with masses
\displaystylem1,\ldots,mN
in the regular three-dimensional
Euclidean space. Let
\displaystyler1,\ldots,rN
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
\displaystyler1,\ldots,rN
can be built into a single
-dimensional radius-vector. Similarly, three-dimensional velocity vectors
\displaystylev1,\ldots,vN
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 theequations 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
The configuration space and the phase space of the dynamical system 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
\displaystylem1,\ldots,mN
:
Constraints and internal coordinates
In some cases the motion of the particles with the masses
\displaystylem1,\ldots,mN
can be constrained. Typical
constraints look like scalar equations of the form Constraints of the form are called
holonomic and
scleronomic. 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
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
\displaystyleq1,\ldots,qn
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
\displaystyleq1,\ldots,qn
:The vector-function resolves the constraint equations in the sense that upon substituting into the equations are fulfilled identically in
\displaystyleq1,\ldots,qn
.
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 quantitiesare 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 configuration space
of the constrained Newtonian dynamical system into the
-dimensional flat configuration 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 formulawhere
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 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 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 force. 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 force
.
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 symbols of the
metric connection produced by the Riemannian metric .
Relation to Lagrange equations
Mechanical systems with constraints are usually described by Lagrange equations:where
T=T(q1,\ldots,qn,w1,\ldots,wn)
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 andother geometric features of the configuration manifold
are not explicit in . The metric can be recovered from the kinetic energy
by means of the formula
See also
Notes and References
- Book: Fitzpatrick, Richard . Newtonian Dynamics: An Introduction . 2021-12-22 . . 978-1-000-50957-1 . en . Preface.
- Book: Kasdin . N. Jeremy . Engineering Dynamics: A Comprehensive Introduction . Paley . Derek A. . 2011-02-22 . . 11 . 978-1-4008-3907-0 . en.
- Book: Barbour, Julian B. . The Discovery of Dynamics: A Study from a Machian Point of View of the Discovery and the Structure of Dynamical Theories . 2001 . . 19 . 978-0-19-513202-1 . en.
