This articleneeds additional citations forverification. Please helpimprove this article byadding citations to reliable sources. Unsourced material may be challenged and removed. Find sources: "Newtonian dynamics" – news ·newspapers ·books ·scholar ·JSTOR(October 2022) (Learn how and when to remove this message)

In physics,Newtonian dynamics (also known asNewtonian mechanics) is the study of thedynamics of a particle or a small body according toNewton's laws of motion.^[1][2][3]

Typically, theNewtonian dynamics occurs in athree-dimensionalEuclidean space, which is flat. However, in mathematicsNewton's laws of motion can be generalized to multidimensional andcurved spaces. Often the termNewtonian dynamics is narrowed toNewton's second law${\displaystyle {\displaystyle m\mathbf{a}=\mathbf{F}}}$.

Consider${\displaystyle {\displaystyle N}}$ particles with masses${\displaystyle {\displaystyle m_1,\dots ,m_N}}$ in the regular three-dimensionalEuclidean space. Let${\displaystyle {\displaystyle {\mathbf{r}}_1,\dots ,{\mathbf{r}}_N}}$ be their radius-vectors in someinertial coordinate system. Then the motion of these particles is governed by Newton's second law applied to each of them

${\displaystyle \frac{d{\mathbf{r}}_i}{dt}={\mathbf{v}}_i,\frac{d{\mathbf{v}}_i}{dt}=\frac{{\mathbf{F}}_i({\mathbf{r}}_1,\dots ,{\mathbf{r}}_N,{\mathbf{v}}_1,\dots ,{\mathbf{v}}_N,t)}{m_i},i=1,\dots ,N.}$

1

The three-dimensional radius-vectors${\displaystyle {\displaystyle {\mathbf{r}}_1,\dots ,{\mathbf{r}}_N}}$ can be built into a single${\displaystyle {\displaystyle n=3N}}$-dimensional radius-vector. Similarly, three-dimensional velocity vectors${\displaystyle {\displaystyle {\mathbf{v}}_1,\dots ,{\mathbf{v}}_N}}$ can be built into a single${\displaystyle {\displaystyle n=3N}}$-dimensional velocity vector:

${\displaystyle \mathbf{r}=\Vert \begin{array}{c}{\mathbf{r}}_1\\ ⋮\\ {\mathbf{r}}_N\end{array}\Vert ,\mathbf{v}=\Vert \begin{array}{c}{\mathbf{v}}_1\\ ⋮\\ {\mathbf{v}}_N\end{array}\Vert .}$

2

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

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

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 theequations of aNewtoniandynamical system in a flat multidimensionalEuclidean space, which is called theconfiguration 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 thephase 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 thekinetic 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${\displaystyle {\displaystyle m_1,\dots ,m_N}}$:

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

4

In some cases the motion of the particles with the masses${\displaystyle {\displaystyle m_1,\dots ,m_N}}$ can be constrained. Typicalconstraints look like scalar equations of the form

${\displaystyle {\displaystyle {\phi }_i({\mathbf{r}}_1,\dots ,{\mathbf{r}}_N)=0,i=1,\dots ,K}}$.

5

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

${\displaystyle {\displaystyle {\phi }_i(\mathbf{r})=0,i=1,\dots ,K}}$.

6

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

Definition. The constraint equations (6) define an${\displaystyle {\displaystyle n}}$-dimensionalmanifold${\displaystyle {\displaystyle M}}$ 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${\displaystyle {\displaystyle q^1,\dots ,q^n}}$ be the internal coordinates of a point of${\displaystyle {\displaystyle M}}$. Their usage is typical for theLagrangian mechanics. The radius-vector${\displaystyle {\displaystyle \mathbf{r}}}$ is expressed as some definite function of${\displaystyle {\displaystyle q^1,\dots ,q^n}}$:

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

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${\displaystyle {\displaystyle q^1,\dots ,q^n}}$.

The velocity vector of the constrained Newtonian dynamical system is expressed in terms of the partial derivatives of the vector-function(7):

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

8

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

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

9

and then treated as independent variables. The quantities

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

10

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

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

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

11

where${\displaystyle {\displaystyle (,)}}$ is the scalar product associated with the Euclidean structure (4).

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).

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 thenormal 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 thenormal force${\displaystyle {\displaystyle \mathbf{N}}}$.
Like the velocity vector (8), the tangent force${\displaystyle {\displaystyle {\mathbf{F}}_{\parallel }}}$ has its internal presentation

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

14

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

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

${\displaystyle \frac{d{q}^s}{dt}=w^s,\frac{d{w}^s}{dt}+\sum _{i=1}^n\sum _{j=1}^n{\mathrm{\Gamma }}_{ij}^s{w}^i{w}^j=F^s,s=1,\dots ,n}$,

15

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

Mechanical systems with constraints are usually described byLagrange equations:

${\displaystyle \frac{d{q}^s}{dt}=w^s,\frac{d}{dt}\left(\frac{\mathrm{\partial }T}{\mathrm{\partial }{w}^s}\right)-\frac{\mathrm{\partial }T}{\mathrm{\partial }{q}^s}=Q_s,s=1,\dots ,n}$,

16

where${\displaystyle T=T(q^1,\dots ,q^n,w^1,\dots ,w^n)}$ is the kinetic energy the constrained dynamical system given by the formula (12). The quantities${\displaystyle Q_1,\dots ,Q_n}$ in (16) are the innercovariant components of the tangent force vector${\displaystyle {\mathbf{F}}_{\parallel }}$ (see (13) and (14)). They are produced from the innercontravariant components${\displaystyle F^1,\dots ,F^n}$ of the vector${\displaystyle {\mathbf{F}}_{\parallel }}$ by means of the standardindex lowering procedure using the metric (11):

${\displaystyle Q_s=\sum _{r=1}^n{g}_{sr}{F}^r,s=1,\dots ,n}$,

17

The equations (16) are equivalent to the equations (15). However, the metric (11) andother 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

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

18

• Modified Newtonian dynamics

• Classical mechanics
• Isaac Newton

• Use American English from March 2019
• All Wikipedia articles written in American English
• Articles with short description
• Short description matches Wikidata
• Articles needing additional references from October 2022
• All articles needing additional references
• Articles with separate introductions