Newtonian dynamics

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

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

Newton's second law in a multidimensional space

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

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

(1)

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

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

(2)

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

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

(3)

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

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

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

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

(4)

Constraints and internal coordinates

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

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

(5)

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

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

(6)

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

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

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

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

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

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):

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

(8)

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

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

(9)

and then treated as independent variables. The quantities

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

(10)

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

Embedding and the induced Riemannian metric

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

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

(11)

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

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

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

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

(13)

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

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

(14)

The quantities $F^{1},\,\ldots ,\,F^{n}$ 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 $\displaystyle M$ by the constraint equations (6) is described by the differential equations

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

(15)

where $\Gamma _{{ij}}^{s}$ 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:

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

(16)

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

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

(17)

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

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

(18)

Isaac Newton

Publications	De analysi per aequationes numero terminorum infinitas (1669, published 1711) Method of Fluxions (1671) De motu corporum in gyrum (1684) Philosophiæ Naturalis Principia Mathematica (1687) General Scholium (1713) Opticks (1704) The Queries (1704) Arithmetica Universalis (1707)

Other writings	Notes on the Jewish Temple Quaestiones quaedam philosophicae The Chronology of Ancient Kingdoms Amended (1728) An Historical Account of Two Notable Corruptions of Scripture (1754)

Newtonianism	Bucket argument Newton's inequalities Newton's law of cooling Newton's law of universal gravitation Post-Newtonian expansion Parameterized post-Newtonian formalism Newton–Cartan theory Schrödinger–Newton equation Gravitational constant Newton's laws of motion Newtonian dynamics Newton's method in optimization Gauss–Newton algorithm Truncated Newton method Newton's rings Newton's theorem about ovals Newton–Pepys problem Newtonian potential Newtonian fluid Classical mechanics Newtonian fluid Corpuscular theory of light Leibniz–Newton calculus controversy Rotating spheres Newton's cannonball Newton–Cotes formulas Newton's method Newton fractal Generalized Gauss–Newton method Newton's identities Newton polynomial Newton's theorem of revolving orbits Newton–Euler equations Newton number Kissing number problem Power number Solar mass Dynamics Absolute time and space Finite difference Table of Newtonian series Impact depth Structural coloration Inertia

Life	Cranbury Park Woolsthorpe Manor Early life of Isaac Newton Later life of Isaac Newton Religious views of Isaac Newton Isaac Newton's occult studies The Mysteryes of Nature and Art Scientific revolution Copernican Revolution

Friends and family	Catherine Barton John Conduitt William Clarke Benjamin Pulleyn William Stukeley William Jones Isaac Barrow Abraham de Moivre John Keill

Discoveries and inventions	Calculus Newton disc Newton polygon Newton–Okounkov body Newton's reflector Newtonian telescope Newton scale Newton's metal Newton's cradle Sextant

Phrases	"Hypotheses non fingo" "Standing on the shoulders of giants"

Theory expansions	Kepler's laws of planetary motion Problem of Apollonius

Related	Writing of Principia Mathematica Newton (Blake) List of things named after Isaac Newton Newton (unit) Isaac Newton in popular culture Elements of the Philosophy of Newton Isaac Newton S/O Philipose

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