Editing Newtonian dynamics

{{Use American English|date = March 2019}}
{{Short description|Formulation of physics}}
{{mcn|date=October 2022}}
{{See introduction|Classical mechanics}}
In physics, '''Newtonian dynamics''' (also known as '''Newtonian mechanics''') is the study of the [[dynamics (mechanics)|dynamics]] of a particle or a small body according to [[Newton's laws of motion]].<ref>{{Cite book |last=Fitzpatrick |first=Richard |url=https://books.google.com/books?id=9gBSEAAAQBAJ&pg=PT13 |title=Newtonian Dynamics: An Introduction |date=2021-12-22 |publisher=[[CRC Press]] |isbn=978-1-000-50957-1 |language=en |at=Preface}}</ref><ref>{{Cite book |last1=Kasdin |first1=N. Jeremy |url=https://books.google.com/books?id=VWA44JiABm8C&pg=PA11 |title=Engineering Dynamics: A Comprehensive Introduction |last2=Paley |first2=Derek A. |date=2011-02-22 |publisher=[[Princeton University Press]] |page=11 |isbn=978-1-4008-3907-0 |language=en}}</ref><ref>{{Cite book |last=Barbour |first=Julian B. |url=https://books.google.com/books?id=pXc8DwAAQBAJ&pg=PA19 |title=The Discovery of Dynamics: A Study from a Machian Point of View of the Discovery and the Structure of Dynamical Theories |date=2001 |publisher=[[Oxford University Press]] |page=19 |isbn=978-0-19-513202-1 |language=en}}</ref>

==Mathematical generalizations==
Typically, the '''Newtonian dynamics''' occurs in a [[Three-dimensional space|three-dimensional]] [[Euclidean space]], which is flat. However, in mathematics [[Newton's laws of motion]] can be generalized to multidimensional and [[curved space|curved]] spaces. Often the term '''Newtonian dynamics''' is narrowed to [[Newton's second law]] <math>\displaystyle m\,\mathbf a=\mathbf F</math>.

==Newton's second law in a multidimensional space==
Consider <math>\displaystyle N</math> particles with masses <math>\displaystyle m_1,\,\ldots,\,m_N</math> in the regular three-dimensional [[Euclidean space]]. Let <math>\displaystyle \mathbf r_1,\,\ldots,\,\mathbf r_N</math> 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
{{NumBlk|:|<math>
\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. 
</math>|{{EquationRef|1}}}}
The three-dimensional radius-vectors <math>\displaystyle\mathbf r_1,\,\ldots,\,\mathbf r_N</math> can be built into a single <math>\displaystyle n=3N</math>-dimensional radius-vector. Similarly, three-dimensional velocity vectors <math>\displaystyle\mathbf v_1,\,\ldots,\,\mathbf v_N</math> can be built into a single <math>\displaystyle n=3N</math>-dimensional velocity vector:
{{NumBlk|:|<math>
\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}.
</math>|{{EquationRef|2}}}}
In terms of the multidimensional vectors ({{EquationNote|2}}) the equations ({{EquationNote|1}}) are written as 
{{NumBlk|:|<math>
\frac{d\mathbf r}{dt}=\mathbf v,\qquad\frac{d\mathbf v}{dt}=\mathbf F(\mathbf r,\mathbf v,t), 
</math>|{{EquationRef|3}}}}
i.e. they take the form of Newton's second law applied to a single particle with the unit mass <math>\displaystyle m=1</math>.

'''Definition'''. The equations ({{EquationNote|3}}) are called the
equations of a '''Newtonian [[dynamical system]]''' in a flat multidimensional [[Euclidean space]], which is called the [[Configuration space (physics)|configuration space]] of this system. Its points are marked by the radius-vector 
<math>\displaystyle\mathbf r</math>. The space whose points are marked by the pair of vectors <math>\displaystyle(\mathbf r,\mathbf v)</math> is called the [[phase space]] of the dynamical system ({{EquationNote|3}}).

==Euclidean structure==
The configuration space and the phase space of the dynamical system ({{EquationNote|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 <math>\displaystyle m=1</math> is equal to the sum of kinetic energies of the three-dimensional particles with the masses <math>\displaystyle m_1,\,\ldots,\,m_N</math>:
{{NumBlk|:|<math>
T=\frac{\Vert\mathbf v\Vert^2}{2}=\sum^N_{i=1}m_i\,\frac{\Vert\mathbf v_i\Vert^2}{2}</math>.|{{EquationRef|4}}}}

==Constraints and internal coordinates==
In some cases the motion of the particles with the masses <math>\displaystyle m_1,\,\ldots,\,m_N</math> can be constrained. Typical [[constraint algorithm|constraints]] look like scalar equations of the form 
{{NumBlk|:|<math>\displaystyle\varphi_i(\mathbf r_1,\ldots,\mathbf r_N)=0,\quad i=1,\,\ldots,\,K</math>.|{{EquationRef|5}}}}
Constraints of the form ({{EquationNote|5}}) are called [[Holonomic constraints|holonomic]] and [[Scleronomous|scleronomic]]. In terms of the radius-vector <math>\displaystyle\mathbf r</math> of the Newtonian dynamical system ({{EquationNote|3}}) they are written as
{{NumBlk|:|<math>\displaystyle\varphi_i(\mathbf r)=0,\quad i=1,\,\ldots,\,K</math>.|{{EquationRef|6}}}}
Each such constraint reduces by one the number of degrees of freedom of the Newtonian dynamical system ({{EquationNote|3}}). Therefore, the constrained system has <math>\displaystyle n=3\,N-K</math> degrees of freedom.

'''Definition'''. The constraint equations ({{EquationNote|6}}) define an <math>\displaystyle n</math>-dimensional [[manifold]] <math>\displaystyle M</math> within the configuration space of the Newtonian dynamical system ({{EquationNote|3}}). This manifold <math>\displaystyle M</math> is called the configuration space of the constrained system. Its tangent bundle <math>\displaystyle TM</math> is called the phase space of the constrained system.

Let <math>\displaystyle q^1,\,\ldots,\,q^n</math> be the internal coordinates of a point of <math>\displaystyle M</math>. Their usage is typical for the [[Lagrangian mechanics]]. The radius-vector <math>\displaystyle\mathbf r</math> is expressed as some definite function of <math>\displaystyle q^1,\,\ldots,\,q^n</math>:
{{NumBlk|:|<math>\displaystyle\mathbf r=\mathbf r(q^1,\,\ldots,\,q^n)
</math>.|{{EquationRef|7}}}}
The vector-function ({{EquationNote|7}}) resolves the constraint equations ({{EquationNote|6}}) in the sense that upon substituting ({{EquationNote|7}}) into ({{EquationNote|6}}) the equations ({{EquationNote|6}}) are fulfilled identically in <math>\displaystyle q^1,\,\ldots,\,q^n</math>.

==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
({{EquationNote|7}}):
{{NumBlk|:|<math>\displaystyle\mathbf v=\sum^n_{i=1}\frac{\partial\mathbf r}{\partial q^i}\,\dot q^i
</math>.|{{EquationRef|8}}}}
The quantities <math>\displaystyle\dot q^1,\,\ldots,\,\dot q^n</math> are called internal components of the velocity vector. Sometimes they are denoted with the use of a separate symbol
{{NumBlk|:|<math>\displaystyle\dot q^i=w^i,\qquad i=1,\,\ldots,\,n
</math>|{{EquationRef|9}}}}
and then treated as independent variables. The quantities
{{NumBlk|:|<math>\displaystyle q^1,\,\ldots,\,q^n,\,w^1,\,\ldots,\,w^n
</math>|{{EquationRef|10}}}}
are used as internal coordinates of a point of the phase space <math>\displaystyle TM</math> of the constrained Newtonian dynamical system.

==Embedding and the induced Riemannian metric==
Geometrically, the vector-function ({{EquationNote|7}}) implements an embedding of the configuration space <math>\displaystyle M</math> of the constrained Newtonian dynamical system into the <math>\displaystyle 3\,N</math>-dimensional flat configuration space of the unconstrained 
Newtonian dynamical system ({{EquationNote|3}}). Due to this embedding the Euclidean structure of the ambient space induces the Riemannian metric onto the manifold <math>\displaystyle M</math>. The components of the [[metric tensor]] of this induced metric are given by the formula
{{NumBlk|:|<math>\displaystyle g_{ij}=\left(\frac{\partial\mathbf r}{\partial q^i},\frac{\partial\mathbf r}{\partial q^j}\right)
</math>,|{{EquationRef|11}}}}
where <math>\displaystyle(\ ,\ )</math> is the scalar product associated with the Euclidean structure ({{EquationNote|4}}).

==Kinetic energy of a constrained Newtonian dynamical system==
Since the Euclidean structure of an unconstrained system of <math>\displaystyle N</math> particles is introduced through their kinetic energy, the induced Riemannian structure on the configuration space <math>\displaystyle N</math> of a constrained system preserves this relation to the kinetic energy:
{{NumBlk|:|<math>
T=\frac{1}{2}\sum^n_{i=1}\sum^n_{j=1}g_{ij}\,w^i\,w^j</math>.|{{EquationRef|12}}}}
The formula ({{EquationNote|12}}) is derived by substituting ({{EquationNote|8}}) into ({{EquationNote|4}}) and taking into account ({{EquationNote|11}}).

==Constraint forces==

For a constrained Newtonian dynamical system the constraints described by the equations ({{EquationNote|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 <math>\displaystyle M</math>. Such a maintaining force is perpendicular to <math>\displaystyle M</math>. It is called the [[normal force]]. The force <math>\displaystyle\mathbf F</math> from ({{EquationNote|6}}) is subdivided into two components
{{NumBlk|:|<math>
\mathbf F=\mathbf F_\parallel+\mathbf F_\perp</math>.|{{EquationRef|13}}}}
The first component in ({{EquationNote|13}}) is tangent to the configuration manifold <math>\displaystyle M</math>. The second component is perpendicular to <math>\displaystyle M</math>. In coincides with the [[normal force]] <math>\displaystyle\mathbf N</math>.<br>
Like the velocity vector ({{EquationNote|8}}), the tangent force 
<math>\displaystyle\mathbf F_\parallel</math> has its internal presentation
{{NumBlk|:|<math>\displaystyle\mathbf F_\parallel=\sum^n_{i=1}\frac{\partial\mathbf r}{\partial q^i}\,F^i</math>.|{{EquationRef|14}}}}
The quantities <math>F^1,\,\ldots,\,F^n</math> in ({{EquationNote|14}}) are called the internal components of the force vector.

==Newton's second law in a curved space==
The Newtonian dynamical system ({{EquationNote|3}}) constrained to the configuration manifold <math>\displaystyle M</math> by the constraint equations ({{EquationNote|6}}) is described by the differential equations
{{NumBlk|:|<math>
\frac{dq^s}{dt}=w^s,\qquad\frac{d w^s}{dt}+\sum^n_{i=1}\sum^n_{j=1}\Gamma^s_{ij}\,w^i\,w^j=F^s,\qquad s=1,\,\ldots,\,n</math>,|{{EquationRef|15}}}}
where <math>\Gamma^s_{ij}</math> are [[Christoffel symbols]] of the [[metric connection]] produced by the Riemannian metric ({{EquationNote|11}}).

==Relation to Lagrange equations==
Mechanical systems with constraints are usually described by [[Lagrangian mechanics#Lagrange equations of the second kind|Lagrange equations]]:
{{NumBlk|:|<math>
\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</math>,|{{EquationRef|16}}}}
where <math>T=T(q^1,\ldots,q^n,w^1,\ldots,w^n)</math> is the kinetic energy the constrained dynamical system given by the formula ({{EquationNote|12}}). The quantities <math>Q_1,\,\ldots,\,Q_n</math> in 
({{EquationNote|16}}) are the inner [[tensor#Tensor valence|covariant components]] of the tangent force vector <math>\mathbf F_\parallel</math> (see ({{EquationNote|13}}) and ({{EquationNote|14}})). They are produced from the inner [[tensor#Tensor valence|contravariant components]] <math>F^1,\,\ldots,\,F^n</math> of the vector <math>\mathbf F_\parallel</math> by means of the standard [[raising and lowering indices|index lowering procedure]] using the metric ({{EquationNote|11}}):
{{NumBlk|:|<math>
Q_s=\sum^n_{r=1}g_{sr}\,F^r,\qquad s=1,\,\ldots,\,n</math>,|{{EquationRef|17}}}}
The equations ({{EquationNote|16}}) are equivalent to the equations ({{EquationNote|15}}). However, the metric ({{EquationNote|11}}) and
other geometric features of the configuration manifold <math>\displaystyle M</math> are not explicit in ({{EquationNote|16}}). The metric ({{EquationNote|11}}) can be recovered from the kinetic energy <math>\displaystyle T</math> by means of the formula
{{NumBlk|:|<math>
g_{ij}=\frac{\partial^2T}{\partial w^i\,\partial w^j}</math>.|{{EquationRef|18}}}}

==See also==
*[[Modified Newtonian dynamics]]

==References==
{{reflist}}

{{Isaac Newton}}

{{DEFAULTSORT:Newtonian Dynamics}}
[[Category:Classical mechanics]]
[[Category:Isaac Newton]]