# Newtonian dynamics

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In physics, the Newtonian dynamics is understood as the dynamics of a particle or a small body according to Newton's laws of motion.

Dynamics is the branch of classical mechanics concerned with the study of forces and their effects on motion. Isaac Newton defined the fundamental physical laws which govern dynamics in physics, especially his second law of motion.

Newton's laws of motion are three physical laws that, together, laid the foundation for classical mechanics. They describe the relationship between a body and the forces acting upon it, and its motion in response to those forces. More precisely, the first law defines the force qualitatively, the second law offers a quantitative measure of the force, and the third asserts that a single isolated force doesn't exist. These three laws have been expressed in several ways, over nearly three centuries, and can be summarised as follows:

## 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 ${\displaystyle \displaystyle m\,\mathbf {a} =\mathbf {F} }$.

Euclidean space is the fundamental space of geometry. Originally, this was the three-dimensional space of Euclidean geometry, but, in modern mathematics, there are Euclidean spaces of any nonnegative integer dimension, including the three-dimensional space and the Euclidean plane. It has been introduced by the Ancient Greek mathematician Euclid of Alexandria, and the qualifier Euclidean has been added for distinguishing it from other spaces that are considered in physics and modern mathematics.

Curved space often refers to a spatial geometry which is not "flat", where a flat space is described by Euclidean geometry. Curved spaces can generally be described by Riemannian geometry though some simple cases can be described in other ways. Curved spaces play an essential role in general relativity, where gravity is often visualized as curved space. The Friedmann-Lemaître-Robertson-Walker metric is a curved metric which forms the current foundation for the description of the expansion of space and shape of the universe.

## Newton's second law in a multidimensional space

Consider ${\displaystyle \displaystyle N}$ particles with masses ${\displaystyle \displaystyle m_{1},\,\ldots ,\,m_{N}}$ in the regular three-dimensional Euclidean space. Let ${\displaystyle \displaystyle \mathbf {r} _{1},\,\ldots ,\,\mathbf {r} _{N}}$ 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

${\displaystyle {\frac {d\mathbf {r} _{i}}{dt}}=\mathbf {v} _{i},\qquad {\frac {d\mathbf {v} _{i}}{dt}}={\frac {\mathbf {F} _{i}(\mathbf {r} _{1},\ldots ,\mathbf {r} _{N},\mathbf {v} _{1},\ldots ,\mathbf {v} _{N},t)}{m_{i}}},\quad i=1,\ldots ,N.}$

(1)

The three-dimensional radius-vectors ${\displaystyle \displaystyle \mathbf {r} _{1},\,\ldots ,\,\mathbf {r} _{N}}$ can be built into a single ${\displaystyle \displaystyle n=3N}$-dimensional radius-vector. Similarly, three-dimensional velocity vectors ${\displaystyle \displaystyle \mathbf {v} _{1},\,\ldots ,\,\mathbf {v} _{N}}$ can be built into a single ${\displaystyle \displaystyle n=3N}$-dimensional velocity vector:

${\displaystyle \mathbf {r} ={\begin{Vmatrix}\mathbf {r} _{1}\\\vdots \\\mathbf {r} _{N}\end{Vmatrix}},\qquad \qquad \mathbf {v} ={\begin{Vmatrix}\mathbf {v} _{1}\\\vdots \\\mathbf {v} _{N}\end{Vmatrix}}.}$

(2)

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

${\displaystyle {\frac {d\mathbf {r} }{dt}}=\mathbf {v} ,\qquad {\frac {d\mathbf {v} }{dt}}=\mathbf {F} (\mathbf {r} ,\mathbf {v} ,t),}$

(3)

i.e. they take the form of Newton's second law applied to a single particle with the unit mass ${\displaystyle \displaystyle m=1}$.

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 ${\displaystyle \displaystyle \mathbf {r} }$. The space whose points are marked by the pair of vectors ${\displaystyle \displaystyle (\mathbf {r} ,\mathbf {v} )}$ is called the phase space of the dynamical system ( 3 ).

In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in a geometrical space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, and the number of fish each springtime in a lake.

In classical mechanics, the parameters that define the configuration of a system are called generalized coordinates, and the vector space defined by these coordinates is called the configuration space of the physical system. It is often the case that these parameters satisfy mathematical constraints, such that the set of actual configurations of the system is a manifold in the space of generalized coordinates. This manifold is called the configuration manifold of the system.

In dynamical system theory, a phase space is a space in which all possible states of a system are represented, with each possible state corresponding to one unique point in the phase space. For mechanical systems, the phase space usually consists of all possible values of position and momentum variables. The concept of phase space was developed in the late 19th century by Ludwig Boltzmann, Henri Poincaré, and Josiah Willard Gibbs.

## 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 ${\displaystyle \displaystyle m=1}$ is equal to the sum of kinetic energies of the three-dimensional particles with the masses ${\displaystyle \displaystyle m_{1},\,\ldots ,\,m_{N}}$:

In physics, the kinetic energy (KE) of an object is the energy that it possesses due to its motion. It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acceleration, the body maintains this kinetic energy unless its speed changes. The same amount of work is done by the body when decelerating from its current speed to a state of rest.

${\displaystyle T={\frac {\Vert \mathbf {v} \Vert ^{2}}{2}}=\sum _{i=1}^{N}m_{i}\,{\frac {\Vert \mathbf {v} _{i}\Vert ^{2}}{2}}}$.

(4)

## Constraints and internal coordinates

In some cases the motion of the particles with the masses ${\displaystyle \displaystyle m_{1},\,\ldots ,\,m_{N}}$ can be constrained. Typical constraints look like scalar equations of the form

${\displaystyle \displaystyle \varphi _{i}(\mathbf {r} _{1},\ldots ,\mathbf {r} _{N})=0,\quad i=1,\,\ldots ,\,K}$.

(5)

Constraints of the form ( 5 ) are called holonomic and scleronomic. In terms of the radius-vector ${\displaystyle \displaystyle \mathbf {r} }$ of the Newtonian dynamical system ( 3 ) they are written as

${\displaystyle \displaystyle \varphi _{i}(\mathbf {r} )=0,\quad i=1,\,\ldots ,\,K}$.

(6)

Each such constraint reduces by one the number of degrees of freedom of the Newtonian dynamical system ( 3 ). Therefore, the constrained system has ${\displaystyle \displaystyle n=3\,N-K}$ degrees of freedom.

Definition. The constraint equations ( 6 ) define an ${\displaystyle \displaystyle n}$-dimensional manifold ${\displaystyle \displaystyle M}$ within the configuration space of the Newtonian dynamical system ( 3 ). This manifold ${\displaystyle \displaystyle M}$ is called the configuration space of the constrained system. Its tangent bundle ${\displaystyle \displaystyle TM}$ is called the phase space of the constrained system.

Let ${\displaystyle \displaystyle q^{1},\,\ldots ,\,q^{n}}$ be the internal coordinates of a point of ${\displaystyle \displaystyle M}$. Their usage is typical for the Lagrangian mechanics. The radius-vector ${\displaystyle \displaystyle \mathbf {r} }$ is expressed as some definite function of ${\displaystyle \displaystyle q^{1},\,\ldots ,\,q^{n}}$:

${\displaystyle \displaystyle \mathbf {r} =\mathbf {r} (q^{1},\,\ldots ,\,q^{n})}$.

(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 ${\displaystyle \displaystyle q^{1},\,\ldots ,\,q^{n}}$.

## 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 ):

${\displaystyle \displaystyle \mathbf {v} =\sum _{i=1}^{n}{\frac {\partial \mathbf {r} }{\partial q^{i}}}\,{\dot {q}}^{i}}$.

(8)

The quantities ${\displaystyle \displaystyle {\dot {q}}^{1},\,\ldots ,\,{\dot {q}}^{n}}$ are called internal components of the velocity vector. Sometimes they are denoted with the use of a separate symbol

${\displaystyle \displaystyle {\dot {q}}^{i}=w^{i},\qquad i=1,\,\ldots ,\,n}$

(9)

and then treated as independent variables. The quantities

${\displaystyle \displaystyle q^{1},\,\ldots ,\,q^{n},\,w^{1},\,\ldots ,\,w^{n}}$

(10)

are used as internal coordinates of a point of the phase space ${\displaystyle \displaystyle TM}$ of the constrained Newtonian dynamical system.

## Embedding and the induced Riemannian metric

Geometrically, the vector-function ( 7 ) implements an embedding of the configuration space ${\displaystyle \displaystyle M}$ of the constrained Newtonian dynamical system into the ${\displaystyle \displaystyle 3\,N}$-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 ${\displaystyle \displaystyle M}$. The components of the metric tensor of this induced metric are given by the formula

${\displaystyle \displaystyle g_{ij}=\left({\frac {\partial \mathbf {r} }{\partial q^{i}}},{\frac {\partial \mathbf {r} }{\partial q^{j}}}\right)}$,

(11)

where ${\displaystyle \displaystyle (\ ,\ )}$ 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 ${\displaystyle \displaystyle N}$ particles is introduced through their kinetic energy, the induced Riemannian structure on the configuration space ${\displaystyle \displaystyle N}$ of a constrained system preserves this relation to the kinetic energy:

${\displaystyle T={\frac {1}{2}}\sum _{i=1}^{n}\sum _{j=1}^{n}g_{ij}\,w^{i}\,w^{j}}$.

(12)

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 ${\displaystyle \displaystyle M}$. Such a maintaining force is perpendicular to ${\displaystyle \displaystyle M}$. It is called the normal force. The force ${\displaystyle \displaystyle \mathbf {F} }$ from ( 6 ) is subdivided into two components

${\displaystyle \mathbf {F} =\mathbf {F} _{\parallel }+\mathbf {F} _{\perp }}$.

(13)

The first component in ( 13 ) is tangent to the configuration manifold ${\displaystyle \displaystyle M}$. The second component is perpendicular to ${\displaystyle \displaystyle M}$. In coincides with the normal force ${\displaystyle \displaystyle \mathbf {N} }$.
Like the velocity vector ( 8 ), the tangent force ${\displaystyle \displaystyle \mathbf {F} _{\parallel }}$ has its internal presentation

${\displaystyle \displaystyle \mathbf {F} _{\parallel }=\sum _{i=1}^{n}{\frac {\partial \mathbf {r} }{\partial q^{i}}}\,F^{i}}$.

(14)

The quantities ${\displaystyle F^{1},\,\ldots ,\,F^{n}}$ in ( 14 ) are called the internal components of the force vector.

## Newton's second law in a curved space

The Newtonian dynamical system ( 3 ) constrained to the configuration manifold ${\displaystyle \displaystyle M}$ by the constraint equations ( 6 ) is described by the differential equations

${\displaystyle {\frac {dq^{s}}{dt}}=w^{s},\qquad {\frac {dw^{s}}{dt}}+\sum _{i=1}^{n}\sum _{j=1}^{n}\Gamma _{ij}^{s}\,w^{i}\,w^{j}=F^{s},\qquad s=1,\,\ldots ,\,n}$,

(15)

where ${\displaystyle \Gamma _{ij}^{s}}$ are Christoffel symbols of the metric connection produced by the Riemannian metric ( 11 ).

## Relation to Lagrange equations

Mechanical systems with constraints are usually described by Lagrange equations:

${\displaystyle {\frac {dq^{s}}{dt}}=w^{s},\qquad {\frac {d}{dt}}\left({\frac {\partial T}{\partial w^{s}}}\right)-{\frac {\partial T}{\partial q^{s}}}=Q_{s},\qquad s=1,\,\ldots ,\,n}$,

(16)

where ${\displaystyle T=T(q^{1},\ldots ,q^{n},w^{1},\ldots ,w^{n})}$ is the kinetic energy the constrained dynamical system given by the formula ( 12 ). The quantities ${\displaystyle Q_{1},\,\ldots ,\,Q_{n}}$ in ( 16 ) are the inner covariant components of the tangent force vector ${\displaystyle \mathbf {F} _{\parallel }}$ (see ( 13 ) and ( 14 )). They are produced from the inner contravariant components ${\displaystyle F^{1},\,\ldots ,\,F^{n}}$ of the vector ${\displaystyle \mathbf {F} _{\parallel }}$ by means of the standard index lowering procedure using the metric ( 11 ):

${\displaystyle Q_{s}=\sum _{r=1}^{n}g_{sr}\,F^{r},\qquad s=1,\,\ldots ,\,n}$,

(17)

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

${\displaystyle g_{ij}={\frac {\partial ^{2}T}{\partial w^{i}\,\partial w^{j}}}}$.

(18)

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