![p = mv](/media/stories/equations/classicalmechanics/pointdynamics01.gif)
![p = mv](/media/stories/equations/classicalmechanics/pointdynamics02.gif)
Newton's 3rd law is given by:
![F action = F reaction](/media/stories/equations/classicalmechanics/pointdynamics03.gif)
For the power P holds:
![P = W = F v](/media/stories/equations/classicalmechanics/pointdynamics04.gif)
![W = T + U; T = U](/media/stories/equations/classicalmechanics/pointdynamics05.gif)
![T = (1/2)mv2](/media/stories/equations/classicalmechanics/pointdynamics06.gif)
The kick
![S](/media/stories/equations/classicalmechanics/pointdynamics07.gif)
![](/media/stories/equations/classicalmechanics/pointdynamics08.gif)
The work A, delivered by a force, is
![](/media/stories/equations/classicalmechanics/pointdynamics09.gif)
The torque
![torque](/media/stories/equations/classicalmechanics/pointdynamics10.gif)
![angular momentum](/media/stories/equations/classicalmechanics/pointdynamics11.gif)
![angular momentum](/media/stories/equations/classicalmechanics/pointdynamics12.gif)
![](/media/stories/equations/classicalmechanics/pointdynamics13.gif)
Hence, the conditions for a mechanical equilibrium are:
![](/media/stories/equations/classicalmechanics/pointdynamics14.gif)
The force of friction is usually proportional to the force perpendicular to the surface, except when the motion starts, when a threshold has to be overcome:
![F(fric) = f . F(norm) . et](/media/stories/equations/classicalmechanics/pointdynamics15.gif)
Conservative force fields
A conservative force can be written as the gradient of a potential:![](/media/stories/equations/classicalmechanics/pointdynamics16.gif)
![](/media/stories/equations/classicalmechanics/pointdynamics17.gif)
![](/media/stories/equations/classicalmechanics/pointdynamics18.gif)
So the work delivered by a conservative force field depends not on the trajectory covered but only on the starting and ending points of the motion.
Gravitation
The Newtonian law of gravitation is (in GRT one also uses K instead of G):![](/media/stories/equations/classicalmechanics/pointdynamics19.gif)
The gravitational potential is then given by
![V = -Gm/r](/media/stories/equations/classicalmechanics/pointdynamics20.gif)
![](/media/stories/equations/classicalmechanics/pointdynamics20.gif)
Orbital equations
If![V = V (r)](/media/stories/equations/classicalmechanics/pointdynamics22.gif)
![](/media/stories/equations/classicalmechanics/pointdynamics22_2.gif)
![](/media/stories/equations/classicalmechanics/pointdynamics23.gif)
For the radial position as a function of time can be found that:
![](/media/stories/equations/classicalmechanics/pointdynamics24.gif)
The angular equation is then:
![](/media/stories/equations/classicalmechanics/pointdynamics25.gif)
If
![F = F(r):](/media/stories/equations/classicalmechanics/pointdynamics26.gif)
![F perpendicular to V](/media/stories/equations/classicalmechanics/pointdynamics27.gif)
![](/media/stories/equations/classicalmechanics/pointdynamics28.gif)
![](/media/stories/equations/classicalmechanics/pointdynamics29.gif)
Kepler's orbital equations
In a force field![F = kr-2](/media/stories/equations/classicalmechanics/pointdynamics30.gif)
![](/media/stories/equations/classicalmechanics/pointdynamics31.gif)
with
![](/media/stories/equations/classicalmechanics/pointdynamics32.gif)
a is half the length of the long axis of the elliptical orbit in case the orbit is closed. Half the length of the short axis is
![](/media/stories/equations/classicalmechanics/pointdynamics33.gif)
![](/media/stories/equations/classicalmechanics/pointdynamics34.gif)
1. k < 0 and= 0: a circle.
2. k < 0 and 0 << 1: an ellipse.
3. k < 0 and= 1: a parabole.
4. k < 0 and> 1: a hyperbole, curved towards the centre of force.
5. k > 0 and> 1: a hyperbole, curved away from the centre of force.
Other combinations are not possible: the total energy in a repulsive force field is always positive so
![](/media/stories/equations/classicalmechanics/pointdynamics35.gif)
If the surface between the orbit covered between
![t1](/media/stories/equations/classicalmechanics/pointdynamics36.gif)
![t2](/media/stories/equations/classicalmechanics/pointdynamics37.gif)
![A(t1; t2)](/media/stories/equations/classicalmechanics/pointdynamics38.gif)
![](/media/stories/equations/classicalmechanics/pointdynamics39.gif)
Kepler's 3rd law is, with T the period and
![Mtot](/media/stories/equations/classicalmechanics/pointdynamics40.gif)
![](/media/stories/equations/classicalmechanics/pointdynamics41.gif)
The virial theorem
The virial theorem for one particle is:![](/media/stories/equations/classicalmechanics/pointdynamics42.gif)
The virial theorem for a collection of particles is:
![](/media/stories/equations/classicalmechanics/pointdynamics43.gif)
These propositions can also be written as:
![2E(kin) + E(pot) = 0](/media/stories/equations/classicalmechanics/pointdynamics44.gif)