List of equations in classical mechanics

;Nomenclature

: "a" = acceleration (m/s²): "g" = gravitational field strength/acceleration in free-fall (m/s²): "F" = force (N = kg m/s²): "E"_k = kinetic energy (J = kg m²/s²): "E"_p = potential energy (J = kg m²/s²): "m" = mass (kg): "p" = momentum (kg m/s): "s" = displacement (m): "R" = radius (m): "t" = time (s): "v" = velocity (m/s): "v"₀ = velocity at time t=0: "W" = work (J = kg m²/s²): "τ" = torque (m N, not J) (torque is the rotational form of force): "s"(t) = position at time t: "s"₀ = position at time t=0: "r"_unit = unit vector pointing from the origin in polar coordinates: "θ"_unit = unit vector pointing in the direction of increasing values of theta in polar coordinates

Note: All quantities in bold represent vectors.

Classical mechanics is the branch of physics used to describe the motion of macroscopic objects. [Harvnb|Mayer|Sussman|Wisdom|2001|p=xiii] It is the most familiar of the theories of physics. The concepts it covers, such as mass, acceleration, and force, are commonly used and known. [Harvnb|Berkshire|Kibble|2004|p=1] The subject is based upon a three-dimensional Euclidean space with fixed axes, called a frame of reference. The point of concurrency of the three axes is known as the origin of the particular space. [Harvnb|Berkshire|Kibble|2004|p=2]

Classical mechanics utilises many equation—as well as other mathematical concepts—which relate various physical quantities to one another. These include differential equationss, manifolds, Lie groups, and ergodic theory. [Harvnb|Arnold|1989|p=v] This page gives a summary of the most important of these.

Equations

Velocity

: $mathbf\{v\}\_\{mbox\{average\; =\; \{Delta\; mathbf\{d\}\; over\; Delta\; t\}$: $mathbf\{v\}\; =\; \{dmathbf\{s\}\; over\; dt\}$

Acceleration

: $mathbf\{a\}\_\{mbox\{average\; =\; frac\{Deltamathbf\{v\{Delta\; t\}$ : $mathbf\{a\}\; =\; frac\{dmathbf\{v\{dt\}\; =\; frac\{d^2mathbf\{s\{dt^2\}$

*Centripetal Acceleration

: $|mathbf\{a\}\_c\; |\; =\; omega^2\; R\; =\; v^2\; /\; R$("R" = radius of the circle, ω = "v/R" angular velocity)

Momentum

: $mathbf\{p\}\; =\; mmathbf\{v\}$

Force

:$sum\; mathbf\{F\}\; =\; frac\{dmathbf\{p\{dt\}\; =\; frac\{d(mmathbf\{v\})\}\{dt\}$

:$sum\; mathbf\{F\}\; =\; mmathbf\{a\}\; quad$ (Constant Mass)

Impulse

:$mathbf\{J\}\; =\; Delta\; mathbf\{p\}\; =\; int\; mathbf\{F\}\; dt$:

$mathbf\{J\}\; =\; mathbf\{F\}\; Delta\; t\; quad$

if F is constant

Moment of inertia

For a single axis of rotation:The moment of inertia for an object is the sum of the products of the mass element and the square of their distances from the axis of rotation:

$I\; =\; sum\; r\_i^2\; m\_i\; =int\_M\; r^2\; mathrm\{d\}\; m\; =\; iiint\_V\; r^2\; ho(x,y,z)\; mathrm\{d\}\; V$

Angular momentum

:$|L|\; =\; mvr\; quad$ if v is perpendicular to r

Vector form:

:$mathbf\{L\}\; =\; mathbf\{r\}\; imes\; mathbf\{p\}\; =\; mathbf\{I\},\; omega$

(Note: I can be treated like a vector if it is diagonalized first, but it is actually a 3×3 matrix - a tensor of rank-2)

r is the radius vector.

Torque

::if |r| and the sine of the angle between r and p remains constant.:This one is very limited, more added later. α = dω/dt

Precession

Omega is called the precession angular speed, and is defined:

:

(Note: w is the weight of the spinning flywheel)

Energy

for "m" as a constant:

:

:$Delta\; E\_p\; =\; mgDelta\; h\; quad\; ,!$ in field of gravity

Central force motion

: $frac\{d^2\}\{d\; heta^2\}left(frac\{1\}\{mathbf\{r\; ight)\; +\; frac\{1\}\{mathbf\{r\; =\; -frac\{mumathbf\{r\}^2\}\{mathbf\{l\}^2\}mathbf\{F\}(mathbf\{r\})$

Equations of motion (constant acceleration)

These equations can be used only when acceleration is constant. If acceleration is not constant then calculus must be used.

:$v\; =\; v\_0+at\; ,$

:$s\; =\; frac\; \{1\}\; \{2\}(v\_0+v)\; t$

:$s\; =\; v\_0\; t\; +\; frac\; \{1\}\; \{2\}\; a\; t^2$

:$v^2\; =\; v\_0^2\; +\; 2\; a\; s\; ,$

:$s\; =\; vt\; -\; frac\; \{1\}\; \{2\}\; a\; t^2$

Derivation of these equation in vector format and without having is shown here These equations can be adapted for angular motion, where angular acceleration is constant:

: $omega\; \_1\; =\; omega\; \_0\; +\; alpha\; t\; ,$

: $heta\; =\; frac\{1\}\{2\}(omega\; \_0\; +\; omega\; \_1)t$

: $heta\; =\; omega\; \_0\; t\; +\; frac\{1\}\{2\}\; alpha\; t^2$

: $omega\; \_1^2\; =\; omega\; \_0^2\; +\; 2alpha\; heta$

: $heta\; =\; omega\; \_1\; t\; -\; frac\{1\}\{2\}\; alpha\; t^2$

ee also

*Acoustics
*Classical mechanics
*Electromagnetism
*Mechanics
*Optics
*Thermodynamics

Notes

References

*citation|title=Mathematical Methods of Classical Mechanics|last=Arnold|first=Vladimir I.|publisher=Springer|year=1989|isbn=978-0-387-96890-2|edition=2nd
*citation|title=Classical Mechanics|last1=Berkshire|last2=Kibble|first1=Frank H.|first2=T. W. B.|edition=5th|publisher=Imperial College Press|year=2004|isbn=978-1860944352
*citation|title=Structure and Interpretation of Classical Mechanics|last1=Mayer|last2=Sussman|last3=Wisdom|first1=Meinhard E.|first2=Gerard J.|first3=Jack|publisher=MIT Press|year=2001|isbn=978-0262194556

External links

* [http://www.astro.uvic.ca/~tatum/classmechs.html Lectures on classical mechanics]
* [http://scienceworld.wolfram.com/biography/Newton.html Biography of Isaac Newton, a key contributor to classical mechanics]