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| In physics, the '''Newtonian dynamics''' is understood as the [[dynamics (mechanics)|dynamics]] of a particle or a small body according to [[Newton's laws of motion]].
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| ==Mathematical generalizations==
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| 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 space|curved]] spaces. Often the term '''Newtonian dynamics''' is narrowed to [[Newton's second law]] <math>\displaystyle m\,\mathbf a=\mathbf F</math>.
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| ==Newton's second law in a multidimensional space==
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| Let's 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
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| {{NumBlk|:|<math>
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| \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.
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| </math>|{{EquationRef|1}}}}
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| 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:
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| {{NumBlk|:|<math>
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| \mathbf r=\begin{Vmatrix}
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| \mathbf r_1\\ \vdots\\ \mathbf r_N\end{Vmatrix},\qquad\qquad
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| \mathbf v=\begin{Vmatrix}
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| \mathbf v_1\\ \vdots\\ \mathbf v_N\end{Vmatrix}.
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| </math>|{{EquationRef|2}}}}
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| In terms of the multidimensional vectors ({{EquationNote|2}}) the equations ({{EquationNote|1}}) are written as
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| {{NumBlk|:|<math>
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| \frac{d\mathbf r}{dt}=\mathbf v,\qquad\frac{d\mathbf v}{dt}=\mathbf F(\mathbf r,\mathbf v,t),
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| </math>|{{EquationRef|3}}}}
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| 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>.
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| '''Definition'''. The equations ({{EquationNote|3}}) are called the
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| 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
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| <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}}).
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| ==Euclidean structure==
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| 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 space#Euclidean structure|Euclidean structure]]. The
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| 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>:
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| {{NumBlk|:|<math>
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| 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}}}}
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| ==Constraints and internal coordinates==
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| 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
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| {{NumBlk|:|<math>\displaystyle\varphi_i(\mathbf r_1,\ldots,\mathbf r_N)=0,\quad i=1,\,\ldots,\,K</math>.|{{EquationRef|5}}}}
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| 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
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| {{NumBlk|:|<math>\displaystyle\varphi_i(\mathbf r)=0,\quad i=1,\,\ldots,\,K</math>.|{{EquationRef|6}}}}
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| 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.
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| '''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.
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| 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>:
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| {{NumBlk|:|<math>\displaystyle\mathbf r=\mathbf r(q^1,\,\ldots,\,q^n)
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| </math>.|{{EquationRef|7}}}}
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| 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>.
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| ==Internal presentation of the velocity vector==
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| The velocity vector of the constrained Newtonian dynamical system is expressed in terms of the partial derivatives of the vector-function
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| ({{EquationNote|7}}):
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| {{NumBlk|:|<math>\displaystyle\mathbf v=\sum^n_{i=1}\frac{\partial\mathbf r}{\partial q^i}\,\dot q^i
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| </math>.|{{EquationRef|8}}}}
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| 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
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| {{NumBlk|:|<math>\displaystyle\dot q^i=w^i,\qquad i=1,\,\ldots,\,n
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| </math>|{{EquationRef|9}}}}
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| and then treated as independent variables. The quantities
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| {{NumBlk|:|<math>\displaystyle q^1,\,\ldots,\,q^n,\,w^1,\,\ldots,\,w^n
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| </math>|{{EquationRef|10}}}}
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| are used as internal coordinates of a point of the phase space <math>\displaystyle TM</math> of the constrained Newtonian dynamical system.
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| ==Embedding and the induced Riemannian metric==
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| 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 comfiguration space of the unconstrained
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| 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
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| {{NumBlk|:|<math>\displaystyle g_{ij}=\left(\frac{\partial\mathbf r}{\partial q^i},\frac{\partial\mathbf r}{\partial q^j}\right)
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| </math>,|{{EquationRef|11}}}}
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| where <math>\displaystyle(\ ,\ )</math> is the scalar product associated with the Euclidean structure ({{EquationNote|4}}). | |
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| ==Kinetic energy of a constrained Newtonian dynamical system==
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| Since the Euclidean structure of an unconstrained system of <math>\displaystyle N</math> particles is entroduced 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:
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| {{NumBlk|:|<math>
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| T=\frac{1}{2}\sum^n_{i=1}\sum^n_{j=1}g_{ij}\,w^i\,w^j</math>.|{{EquationRef|12}}}}
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| The formula ({{EquationNote|12}}) is derived by substituting ({{EquationNote|8}}) into ({{EquationNote|4}}) and taking into account ({{EquationNote|11}}).
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| ==Constraint forces==
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| 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
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| {{NumBlk|:|<math>
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| \mathbf F=\mathbf F_\parallel+\mathbf F_\perp</math>.|{{EquationRef|13}}}}
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| 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>
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| Like the velocity vector ({{EquationNote|8}}), the tangent force
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| <math>\displaystyle\mathbf F_\parallel</math> has its internal presentation
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| {{NumBlk|:|<math>\displaystyle\mathbf F_\parallel=\sum^n_{i=1}\frac{\partial\mathbf r}{\partial q^i}\,F^i</math>.|{{EquationRef|14}}}}
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| The quantities <math>F^1,\,\ldots,\,F^n</math> in ({{EquationNote|14}}) are called the internal components of the force vector.
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| ==Newton's second law in a curved space==
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| 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
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| {{NumBlk|:|<math>
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| \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}}}}
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| where <math>\Gamma^s_{ij}</math> are [[Christoffel symbols]] of the [[metric connection]] produced by the Riemannian metric ({{EquationNote|11}}).
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| ==Relation to Lagrange equations==
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| Mechanical systems with constraints are usually described by [[Lagrangian mechanics#Lagrange equations of the second kind|Lagrange equations]]:
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| {{NumBlk|:|<math>
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| \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}}}}
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| 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
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| ({{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}}):
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| {{NumBlk|:|<math>
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| Q_s=\sum^n_{r=1}g_{sr}\,F^r,\qquad s=1,\,\ldots,\,n</math>,|{{EquationRef|17}}}}
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| The equations ({{EquationNote|16}}) are equivalent to the equations ({{EquationNote|15}}). However, the metric ({{EquationNote|11}}) and
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| 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
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| {{NumBlk|:|<math>
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| g_{ij}=\frac{\partial^2T}{\partial w^i\,\partial w^j}</math>.|{{EquationRef|18}}}}
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| {{DEFAULTSORT:Newtonian Dynamics}}
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| [[Category:Classical mechanics]]
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| [[Category:Mechanics]]
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| [[Category:Isaac Newton]]
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| [[Category:History of physics]]
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| [[Category:Concepts in physics]]
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