The equation of the movement is a mathematical equation describing the movement of a physical object.

In general, the equation of the movement includes/understands the Accélération object according to its position, of its Speed, its Masse and all variables affecting one of this one. This equation is especially used in traditional Mécanique and is normally represented in the spherical form of Coordonnées, cylindrical Coordonnées or Cartesian Coordonnées and respects the Lois of the movement of Newton.

Equations of the movement in the space of a particle charged in an electromagnetic field

That is to say a specific particle of mass $m$ and load $q$ subjected to a Electric field $\backslash \; vec\; \{E\}$ and a Magnetic field $\backslash \; vec\; \{B\}$.

One takes for assumptions:

• Référentiel galiléen
• one neglects the forces of Pesanteur and Frottement

The force $\backslash \; vec\; \{F\}\; \_\; \{(M)\}$ which applies to this particle at the point $M$ is described not the relation: $\backslash \; vec\; \{F\}\; \_\; \{(M)\}=q.\; \backslash \; vec\; \{E\}\; +q.\; (\backslash \; vec\; \{v\}\; \backslash \; wedge\; \backslash \; vec\; \{B\})$

One finds the equation of the movement using the basic principle of dynamics (PFD).

$\backslash \; vec\; \{F\}\; \_\; \{(M)\}=m.\; \backslash \; vec\; \{has\}\; =q.\; \backslash \; vec\; \{E\}\; +q.\; (\backslash \; vec\; \{v\}\; \backslash \; wedge\; \backslash \; vec\; \{B\})$

with $\backslash \; vec\; \{has\}\; =\; \backslash \; frac\; \{D\; \backslash \; vec\; \{v\}\}\; \{dt\}$, the vector Accélération.

Three equations are found:
$m.\backslash frac\{d^2x\}\{dt^2\}=q.\{E\_x\}+q.(v\_y.\; B\_z-v\_z.B\_y)$
$m.\backslash frac\{d^2y\}\{dt^2\}=q.\{E\_y\}+q.(-\; (v\_x.\; B\_z-v\_z.B\_x)$
$m.\backslash frac\{d^2z\}\{dt^2\}=q.\{E\_z\}+q.(v\_x.\; B\_y-v\_y.B\_x)$

With $E\_x,\; E\_y,\; E\_z$, $B\_x,\; B\_y,\; B\_z$ and $v\_x,\; v\_y,\; v\_z$ Cartesian Coordinated space of the fields $\backslash \; vec\; \{E\}$, $\backslash \; vec\; \{B\}$ and $\backslash \; vec\; \{v\}$.

Equation of the movement of a particle in space in a field of gravity

One considers a specific particle of Masse Mr.

One takes for assumptions:

• Référentiel galiléen
• the forces of Frottement are neglected

The force $\backslash \; vec\; \{F\}\; \_\; \{(M)\}$ which applies to this particle at the point $M$ is described not the relation: $\backslash \; vec\; \{F\}\; \_\; \{(M)\}=\; \backslash \; vec\; \{P\}$.
P = Mg (G acceleration of the Gravity) corresponds to the Poids. One finds the equation of the movement using the basic principle of dynamics (PFD).

$\backslash \; vec\; \{F\}\; \_\; \{(M)\}=m.\; \backslash \; vec\; \{has\}\; =m.\; \backslash \; vec\; \{G\}$

with $\backslash \; vec\; \{has\}\; =\; \backslash \; frac\; \{D\; \backslash \; vec\; \{v\}\}\; \{dt\}\; =\; \backslash \; frac\; \{d^2\; \backslash \; vec\; \{R\}\}\; \{dt^2\}$, the vector Accélération. In a system of Coordinated Cartesian, the vector $\backslash \; vec\; \{G\}$ is directed according to $-\; \backslash \; vec\; \{u\_z\}$.
One thus has three equations:

• $m.\backslash frac\{d^2x\}\{dt^2\}=0$
  
• $m.\backslash frac\{d^2y\}\{dt^2\}=0$
  
• $m.\backslash frac\{d^2z\}\{dt^2\}=-g$

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