Form of mechanics

Kinematic

In Coordinated Cartesian

$\backslash overrightarrow\{OM\}=x\backslash overrightarrow\{u\_x\}+y\backslash overrightarrow\{u\_y\}+z\backslash overrightarrow\{u\_z\}$

$\backslash \; overrightarrow\; \{v\}\; (M)\; =\; \backslash \; frac\; \{\backslash \; text\; \{D\}\; \backslash \; overrightarrow\; \{OM\}\}\; \{\backslash \; text\; \{D\}\; T\}\; =\; \backslash \; frac\; \{\backslash \; text\; \{D\}\; X\}\; \{\backslash \; text\; \{D\}\; T\}\; \backslash \; overrightarrow\; \{u\_x\}\; +\; \backslash \; frac\; \{\backslash \; text\; \{D\}\; there\}\; \{\backslash \; text\; \{D\}\; T\}\; \backslash \; overrightarrow\; \{u\_y\}\; +\; \backslash \; frac\; \{\backslash \; text\; \{D\}\; Z\}\; \{\backslash \; text\; \{D\}\; T\}\; \backslash \; overrightarrow\; \{u\_z\}$

$\backslash \; overrightarrow\; \{has\}\; (M)\; =\; \backslash \; frac\; \{\backslash \; text\; \{D\}\; \backslash \; overrightarrow\; \{v\}\; (M)\}\{\backslash \; text\; \{D\}\; T\}\; =\; \backslash \; frac\; \{\backslash \; text\; \{D\}\; ^2\; \backslash \; overrightarrow\; \{OM\}\}\; \{\backslash \; text\; \{D\}\; t^2\}\; =\; \backslash \; frac\; \{\backslash \; text\; \{D\}\; ^2\; X\}\; \{\backslash \; text\; \{D\}\; t^2\}\; \backslash \; overrightarrow\; \{u\_x\}\; +\; \backslash \; frac\; \{\backslash \; text\; \{D\}\; ^2\; there\}\; \{\backslash \; text\; \{D\}\; t^2\}\; \backslash \; overrightarrow\; \{u\_y\}\; +\; \backslash \; frac\; \{\backslash \; text\; \{D\}\; ^2\; Z\}\; \{\backslash \; text\; \{D\}\; t^2\}\; \backslash \; overrightarrow\; \{u\_z\}$

In Coordinated cylindrical

$\backslash \; overrightarrow\; \{OM\}\; =\; \backslash \; rho\; \backslash \; overrightarrow\; \{u\_\; \backslash \; rho\}\; +z\; \backslash \; overrightarrow\; \{u\_z\}$

$\backslash \; overrightarrow\; \{v\}\; (M)\; =\; \backslash \; frac\; \{\backslash \; text\; \{D\}\; \backslash \; overrightarrow\; \{OM\}\}\; \{\backslash \; text\; \{D\}\; T\}\; =\; \backslash \; frac\; \{\backslash \; text\; \{D\}\; \backslash \; rho\}\; \{\backslash \; text\; \{D\}\; T\}\; \backslash \; overrightarrow\; \{u\_\; \backslash \; rho\}\; +\; \backslash \; rho\; \backslash \; frac\; \{\backslash \; text\; \{D\}\; \backslash \; phi\}\; \{\backslash \; text\; \{D\}\; T\}\; \backslash \; overrightarrow\; \{u\_\; \{\backslash \; phi\}\}\; +\; \backslash \; frac\; \{\backslash \; text\; \{D\}\; Z\}\; \{\backslash \; text\; \{D\}\; T\}\; \backslash \; overrightarrow\; \{u\_z\}$

$\backslash \; overrightarrow\; \{has\}\; (M)\; =\; \backslash \; frac\; \{\backslash \; text\; \{D\}\; \backslash \; overrightarrow\; \{v\}\; (M)\}\{\backslash \; text\; \{D\}\; T\}\; =\; \backslash \; frac\; \{\backslash \; text\; \{D\}\; ^2\; \backslash \; overrightarrow\; \{OM\}\}\; \{\backslash \; text\; \{D\}\; t^2\}\; =\; \backslash \; left\; (\backslash \; frac\; \{\backslash \; text\; \{D\}\; ^2\; \backslash \; rho\}\; \{\backslash \; text\; \{D\}\; t^2\}\; -\; \backslash \; rho\; \backslash \; left\; (\backslash \; frac\; \{\backslash \; text\; \{D\}\; \backslash \; phi\}\; \{\backslash \; text\; \{D\}\; T\}\; \backslash \; right)\; ^2\; \backslash \; right)\; \backslash \; overrightarrow\; \{u\_\; \backslash \; rho\}\; +\; \backslash \; left\; (2\; \backslash \; frac\; \{\backslash \; text\; \{D\}\; \backslash \; rho\}\; \{\backslash \; text\; \{D\}\; T\}\; \backslash frac\; \{\backslash \; text\; \{D\}\; \backslash \; phi\}\; \{\backslash \; text\; \{D\}\; T\}\; +\; \backslash \; rho\; \backslash \; frac\; \{\backslash \; text\; \{D\}\; ^2\; \backslash \; phi\}\; \{\backslash \; text\; \{D\}\; t^2\}\; \backslash \; right)\; \backslash \; overrightarrow\; \{u\_\; \{\backslash \; phi\}\}\; +\; \backslash \; frac\; \{\backslash \; text\; \{D\}\; ^2\; Z\}\; \{\backslash \; text\; \{D\}\; t^2\}\; \backslash \; overrightarrow\; \{u\_z\}$

While using: $\backslash \; displaystyle\; \backslash \; frac\; \{\backslash \; text\; \{D\}\; \backslash \; overrightarrow\; \{u\_\; \backslash \; rho\}\}\; \{\backslash \; text\; \{D\}\; T\}\; =\; \backslash \; frac\; \{\backslash \; text\; \{D\}\; \backslash \; phi\}\; \{\backslash \; text\; \{D\}\; T\}\; \backslash \; overrightarrow\; \{u\_\; \{\backslash \; phi\}\}$ and $\backslash \; frac\; \{\backslash \; text\; \{D\}\; \backslash \; overrightarrow\; \{u\_\; \{\backslash \; phi\}\}\}\; \{\backslash \; text\; \{D\}\; T\}\; =\; \backslash \; frac\; \{\backslash \; text\; \{D\}\; \backslash \; phi\}\; \{\backslash \; text\; \{D\}\; T\}\; \backslash \; overrightarrow\; \{u\_\; \backslash \; rho\}$

In Coordinated spherical

$\backslash overrightarrow\{OM\}=r\backslash overrightarrow\{u\_r\}$

$\backslash \; overrightarrow\; \{v\}\; (M)\; =\; \backslash \; frac\; \{\backslash \; text\; \{D\}\; \backslash \; overrightarrow\; \{OM\}\}\; \{\backslash \; text\; \{D\}\; T\}\; =\; \backslash \; frac\; \{\backslash \; text\; \{D\}\; R\}\; \{\backslash \; text\; \{D\}\; T\}\; \backslash \; overrightarrow\; \{u\_r\}\; +r\; \backslash \; frac\; \{\backslash \; text\; \{D\}\; \backslash \; theta\}\; \{\backslash \; text\; \{D\}\; T\}\; \backslash \; overrightarrow\; \{u\_\; \{\backslash \; theta\}\}\; +r\; \backslash \; frac\; \{\backslash \; text\; \{D\}\; \backslash \; varphi\}\; \{\backslash \; text\; \{D\}\; T\}\; \backslash \; sin\; \backslash \; theta\; \backslash \; overrightarrow\; \{u\_\; \backslash \; varphi\}$

$\backslash \; overrightarrow\; \{has\}\; (M)\; =\; \backslash \; frac\; \{\backslash \; text\; \{D\}\; \backslash \; overrightarrow\; \{v\}\; (M)\}\{\backslash \; text\; \{D\}\; T\}\; =\; \backslash \; frac\; \{\backslash \; text\; \{D\}\; ^2\; \backslash \; overrightarrow\; \{OM\}\}\; \{\backslash \; text\; \{D\}\; t^2\}\; =a\_r\; \backslash \; overrightarrow\; \{u\_r\}\; +a\_\; \backslash \; theta\; \backslash \; overrightarrow\; \{u\_\; \backslash \; theta\}\; +a\_\; \backslash \; varphi\; \backslash \; overrightarrow\; \{u\_\; \backslash \; varphi\}$

with:

$a\_r=\; \backslash \; left\; (\backslash \; frac\; \{\backslash \; text\; \{D\}\; ^\; 2r\}\; \{\backslash \; text\; \{D\}\; t^2\}\; -\; R\; \backslash \; left\; (\backslash \; frac\; \{\backslash \; text\; \{D\}\; \backslash \; theta\}\; \{\backslash \; text\; \{D\}\; T\}\; \backslash \; right)\; ^2+r\; \backslash \; left\; (\backslash \; frac\; \{\backslash \; text\; \{D\}\; \backslash \; varphi\}\; \{\backslash \; text\; \{D\}\; T\}\; \backslash \; right)\; ^2\; \backslash \; sin^2\; \backslash \; theta\; \backslash \; right)$

$a\_\; \backslash \; theta=\; \backslash \; left\; (R\; \backslash \; frac\; \{\backslash \; text\; \{D\}\; ^2\; \backslash \; theta\}\; \{\backslash \; text\; \{D\}\; t^2\}\; +2\; \backslash \; frac\; \{\backslash \; text\; \{D\}\; R\}\; \{\backslash \; text\; \{D\}\; T\}\; \backslash \; frac\; \{\backslash \; text\; \{D\}\; \backslash \; theta\}\; \{\backslash \; text\; \{D\}\; T\}\; -\; R\; \backslash \; left\; (\backslash \; frac\; \{\backslash \; text\; \{D\}\; \backslash \; varphi\}\; \{\backslash \; text\; \{D\}\; T\}\; \backslash \; right)\; ^2\; \backslash \; sin\; \backslash \; theta\; \backslash \; cos\; \backslash \; theta\; \backslash \; right)$

$a\_\; \backslash \; varphi=\; \backslash \; left\; (R\; \backslash \; frac\; \{\backslash \; text\; \{D\}\; ^2\; \backslash \; varphi\}\; \{\backslash \; text\; \{D\}\; t^2\}\; \backslash \; sin\; \backslash \; theta\; +2\; \backslash \; frac\; \{\backslash \; text\; \{D\}\; R\}\; \{\backslash \; text\; \{D\}\; T\}\; \backslash \; frac\; \{\backslash \; text\; \{D\}\; \backslash \; varphi\}\; \{\backslash \; text\; \{D\}\; T\}\; \backslash \; sin\; \backslash \; theta\; +\; 2r\; \backslash \; frac\; \{\backslash \; text\; \{D\}\; \backslash \; varphi\}\; \{\backslash \; text\; \{D\}\; T\}\; \backslash \; frac\; \{\backslash \; text\; \{D\}\; \backslash \; theta\}\; \{\backslash \; text\; \{D\}\; T\}\; \backslash \; cos\; \backslash \; theta\; \backslash \; right)$

Change of reference frame

Speed of drive: $\backslash \; overrightarrow\; \{v\_e\}\; (M)\; =\; \backslash \; displaystyle\; \backslash \; left\; (\backslash \; frac\; \{\backslash \; text\; \{D\}\; \backslash \; overrightarrow\; \{OO\text{'}\}\}\; \{\backslash \; text\; \{D\}\; T\}\; \backslash \; right)\; \_R+\; \backslash \; overrightarrow\; \{\backslash \; Omega\}\; \backslash \; wedge\; \backslash \; overrightarrow\; \{O\text{'}\; M\}$

Law of composition speeds: $\backslash \; overrightarrow\; \{v\_R\}\; (M)\; =\; \backslash \; overrightarrow\; \{v\_\; \{R\text{'}\}\}\; (M)\; +\; \backslash \; overrightarrow\; \{v\_e\}\; (M)$

Acceleration of drive: $\backslash \; overrightarrow\; \{a\_e\}\; (M)\; =\; \backslash \; displaystyle\; \backslash \; left\; (\backslash \; frac\; \{\backslash \; text\; \{D\}\; ^2\; \backslash \; overrightarrow\; \{OO\text{'}\}\}\; \{\backslash \; text\; \{D\}\; t^2\}\; \backslash \; right)\; \_R+\; \backslash \; frac\; \{\backslash \; text\; \{D\}\; \backslash \; overrightarrow\; \{\backslash \; Omega\}\}\; \{\backslash \; text\; \{D\}\; T\}\; \backslash \; wedge\; \backslash \; overrightarrow\; \{O\text{'}\; M\}\; +\; \backslash \; overrightarrow\; \{\backslash \; Omega\}\; \backslash \; wedge\; \backslash \; left\; (\; \backslash \; overrightarrow\; \{\backslash \; Omega\}\; \backslash \; wedge\; \backslash \; overrightarrow\; \{O\text{'}\; M\}\; \backslash \; right)$

Acceleration of Coriolis: $\backslash \; overrightarrow\; \{a\_c\}\; (M)\; =2\; \backslash \; overrightarrow\; \{\backslash \; Omega\}\; \backslash \; wedge\; \backslash \; overrightarrow\; \{v\_\; \{R\text{'}\}\}\; (M)$

Law of composition of accelerations: $\backslash \; overrightarrow\; \{a\_R\}\; (M)\; =\; \backslash \; overrightarrow\; \{a\_\; \{R\text{'}\}\}\; (M)\; +\; \backslash \; overrightarrow\; \{a\_e\}\; (M)\; +\; \backslash \; overrightarrow\; \{a\_c\}\; (M)$

Dynamics

Some forces

Weight: $\backslash \; overrightarrow\; \{P\}\; =m\; \backslash \; overrightarrow\; \{G\}$

Electromagnetic interaction: $\backslash \; overrightarrow\; \{F\_\; \{1\; \backslash \; rightarrow\; 2\}\}\; =\; \backslash \; displaystyle\; \backslash \; frac\; \{1\}\; \{4\; \backslash \; pi\; \backslash \; varepsilon\_0\}\; \backslash \; frac\; \{q\_1q\_2\}\; \{(M\_1\; M\_2)\; ^3\}\; \backslash \; overrightarrow\; \{M\_1\; M\_2\}$

Gravitational interaction: $\backslash \; overrightarrow\; \{G\_\; \{1\; \backslash \; rightarrow\; 2\}\}\; =\; \backslash \; mathcal\; \{G\}\; \backslash \; displaystyle\; \backslash \; frac\; \{m\_1m\_2\}\; \{(M\_1\; M\_2)\; ^3\}\; \backslash \; overrightarrow\; \{M\_1\; M\_2\}$

Tension of a spring: $\backslash \; overrightarrow\; \{T\}\; =-k\; (l-l\_0)\; \backslash \; overrightarrow\; \{U\}$ Fluid friction: $\backslash \; overrightarrow\; \{F\}\; =\; \backslash \; lambda\; \backslash \; overrightarrow\; \{v\}$

Inertia of drive: $\backslash overrightarrow\{f\_\{i\_e\}\}=-m\backslash overrightarrow\{a\_e\}$

Inertia of Coriolis: $\backslash overrightarrow\{f\_\{i\_c\}\}=-m\backslash overrightarrow\{a\_c\}$

Principle fundamental of dynamics

Vector momentum: $\backslash \; overrightarrow\; \{p\}\; =m\; \backslash \; overrightarrow\; \{v\}\; (M)$

Principle fundamental of dynamics: $\backslash \; displaystyle\; \backslash \; frac\; \{\backslash \; text\; \{D\}\; \backslash \; overrightarrow\; \{p\}\; (M)\}\{\backslash \; text\; \{D\}\; T\}\; =\; \backslash \; sum\; \backslash \; overrightarrow\; \{F\}\; +\; \backslash \; overrightarrow\; \{f\_\; \{i\_e\}\}\; +\; \backslash \; overrightarrow\; \{f\_\; \{i\_c\}\}$

Principle of the reciprocal actions: $\backslash \; overrightarrow\; \{f\_\; \{has\; \backslash \; rightarrow\; B\}\}\; =\; \backslash \; overrightarrow\; \{f\_\; \{B\; \backslash \; rightarrow\; has\}\}$

Energy aspect

Elementary work of a force: $\backslash \; delta\; W\; (M)\; =\; \backslash \; overrightarrow\; \{F\}\; \backslash \; cdot\; \backslash \; text\; \{D\}\; \backslash \; overrightarrow\; \{OM\}$

Work along a way $\backslash \; Gamma\_\; \{AB\}$: $\backslash \; displaystyle\; W\_\; \{has\; \backslash \; rightarrow\; B\}\; =\; \backslash \; int\_\; \{M\; \backslash \; in\; \backslash \; Gamma\_\; \{AB\}\}\; \backslash \; delta\; W\; (M)\; =\; \backslash \; int\_\; \{M\; \backslash \; in\; \backslash \; Gamma\_\; \{AB\}\}\; \backslash \; overrightarrow\; \{F\}\; \backslash \; cdot\; \backslash \; text\; \{D\}\; \backslash \; overrightarrow\; \{OM\}$

Power: $\backslash \; mathcal\; \{P\}\; =\; \backslash \; displaystyle\; \backslash \; frac\; \{\backslash \; delta\; W\}\; \{\backslash \; text\; \{D\}\; T\}$

kinetic Energy: $E\_c\; (M)\; =\; \backslash \; displaystyle\; \backslash \; frac\; \{1\}\; \{2\}\; mv^2\; (M)$

Theorem of the kinetic energy: $\backslash \; displaystyle\; \backslash \; Delta\; E\_c=\; \backslash \; sum\; W\; (F)\; +W\; (f\_\; \{i\_e\})\; +W\; (f\_\; \{i\_c\})$

Mechanical energy: $E\_m=E\_c+E\_p$

Potential energy for some conservative forces

$E\_p\; (\backslash \; overrightarrow\; \{P\})\; =mgz+C$

$E\_p\; (\backslash \; overrightarrow\; \{T\})\; =\; \backslash \; displaystyle\; \backslash \; frac\; \{1\}\; \{2\}\; K\; (l-l\_0)\; ^2+C$

$E\_p\; (\backslash \; overrightarrow\; \{F\})\; =\; \backslash \; displaystyle\; \backslash \; frac\; \{1\}\; \{4\; \backslash \; pi\; \backslash \; varepsilon\_0\}\; \backslash \; frac\; \{q\_1q\_2\}\; \{R\}$

$E\_p\; (\backslash \; overrightarrow\; \{G\})\; =\; \backslash \; displaystyle\; -\; \backslash \; mathcal\; \{G\}\; \backslash \; frac\; \{m\_1m\_2\}\; \{R\}$

Concept of Moment

kinetic Moment of a point $M$: $\backslash \; overrightarrow\; \{L\_O\}\; (M)\; =\; \backslash \; overrightarrow\; \{OM\}\; \backslash \; wedge\; m\; \backslash \; overrightarrow\; \{v\}\; (M)$

$\backslash \; overrightarrow\; \{L\_\; \{O\text{'}\}\}\; (M)\; =\; \backslash \; overrightarrow\; \{L\_O\}\; (M)\; +\; \backslash \; overrightarrow\; \{O\text{'}\; O\}\; \backslash \; wedge\; m\; \backslash \; overrightarrow\; \{v\}\; (M)$

Moment of a force $\backslash \; overrightarrow\; \{F\}$ compared to $O$: $\backslash \; overrightarrow\; \{\backslash \; mathcal\; \{M\}\; \_O\}\; (\backslash \; overrightarrow\; \{F\})\; =\; \backslash \; overrightarrow\; \{OM\}\; \backslash \; wedge\; \backslash \; overrightarrow\; \{F\}$

$\backslash \; overrightarrow\; \{\backslash \; mathcal\; \{M\}\; \_\; \{O\text{'}\}\}\; (\backslash \; overrightarrow\; \{F\})\; =\; \backslash \; overrightarrow\; \{\backslash \; mathcal\; \{M\}\; \_O\}\; (\backslash \; overrightarrow\; \{F\})\; +\; \backslash \; overrightarrow\; \{O\text{'}\; O\}\; \backslash \; wedge\; \backslash \; overrightarrow\; \{F\}$

Theorem of the kinetic moment: $\backslash \; frac\; \{\backslash \; text\; \{D\}\; \backslash \; overrightarrow\; \{L\_O\}\; (M)\}\{\backslash \; text\; \{D\}\; T\}\; =\; \backslash \; sum\; \backslash \; overrightarrow\; \{\backslash \; mathcal\; \{M\}\; \_O\}\; (\backslash \; overrightarrow\; \{F\})\; +\; \backslash \; overrightarrow\; \{\backslash \; mathcal\; \{M\}\; \_O\}\; (\backslash \; overrightarrow\; \{f\_\; \{i\_e\}\})\; +\; \backslash \; overrightarrow\; \{\backslash \; mathcal\; \{M\}\; \_O\}\; (\backslash \; overrightarrow\; \{f\_\; \{i\_c\}\})$

Oscillator

Harmonic oscillator (without damping):

Differential equation of the form: $\backslash \; frac\; \{\backslash \; text\; \{D\}\; ^2\; X\}\; \{\backslash \; text\; \{D\}\; t^2\}\; +\; \backslash \; omega\_0^2x=0$

own Pulsation: $\backslash \; omega\_0=$; clean Period: $T\_0=\; \backslash \; displaystyle\; \backslash \; frac\; \{2\; \backslash \; pi\}\; \{\backslash \; omega\_0\}$

Solution in the form: $x\; (T)\; =A\; \backslash \; cos\; (\backslash \; omega\_0\; T)\; +B\; \backslash \; sin\; (\backslash \; omega\_0\; T)$

The constant ones has and B are determined by the initial conditions.

Oscillator with factor of Damping $\backslash \; lambda$:

Differential equation of the form: $\backslash \; frac\; \{\backslash \; text\; \{D\}\; ^2\; X\}\; \{\backslash \; text\; \{D\}\; t^2\}\; +2\; \backslash \; lambda\; \backslash \; frac\; \{\backslash \; text\; \{D\}\; X\}\; \{\backslash \; text\; \{D\}\; T\}\; +\; \backslash \; omega\_0^2x=0$

Three cases according to the value of the Discriminant of the characteristic equation:

$\backslash \; Delta=4\; (\backslash \; lambda^2-\; \backslash \; omega\_0^2)$

$\backslash \; ast$ if is , then $x\; (T)\; =e^\; \{-\; \backslash \; lambda\; T\}\; \backslash \; left\; \backslash \; cos\; (\backslash \; Omega\; T)\; +B\; \backslash \; sin\; (\backslash \; Omega\; T)\; \backslash \; right$ (pseudoperiodic mode)

Pseudo-pulsation: $\backslash Omega=\backslash sqrt\{\backslash omega\_0^2-\backslash lambda^2\}$ ; Pseudo-period: $T=\; \backslash \; displaystyle\; \backslash \; frac\; \{2\; \backslash \; pi\}\; \{\backslash \; Omega\}$

$\backslash \; ast$ if $\backslash \; Delta=0$ is $\backslash \; lambda=\; \backslash \; omega\_0$, then $x\; (T)\; =\; (At+B)\; e^\; \{-\; \backslash \; lambda\; T\}$ (mode criticizes)

$\backslash \; ast$ if $\backslash \; Delta>0$ is $\backslash \; lambda>\; \backslash \; omega\_0$, then $x\; (T)\; =e^\; \{-\; \backslash \; lambda\; T\}\; (Ae^\; \{\backslash \; sqrt\; \{\backslash \; lambda^2-\; \backslash \; omega\_0^2\}\; .t\}\; +Be^\; \{-\; \backslash \; sqrt\; \{\backslash \; lambda^2-\; \backslash \; omega\_0^2\}\; .t\})$ (aperiodic mode)

In each case, the constant ones has and B are determined by the initial conditions.

Internal bonds

• Mechanical
• Mechanical Newtonian

Random links:

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