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Point dynamics in a fixed coordinate system

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Physics formulas related Point-dynamics in fixed coordinate systems
  • Force, (angular) momentum and energy
  • Conservative force fields
  • Gravitation
  • Orbital equations
  • Kepler's orbital equations

Force, (angular)momentum and energy

Newton's 2nd law connects the force on an object and the resulting acceleration of the object where the momentum is given by p = mv:

p = mv:

Newton's 3rd law is given by: F action = F reaction

For the power P holds: P = W = F v. For the total energy W, the kinetic energy T and the potential energy U holds: W = T + U; T = U with T = (1/2)mv2.

The kick S is given by:

The work A, delivered by a force, is

The torque torque is related to the angular momentum angular momentum and
angular momentum. The following equation is valid:



Hence, the conditions for a mechanical equilibrium are: .
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.

Conservative force fields

A conservative force can be written as the gradient of a potential: . From this follows that . For such a force field also holds:


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):



The gravitational potential is then given by V = -Gm/r. From Gauss law it then follows: .

Orbital equations

If V = V (r) one can derive from the equations of Lagrange for the conservation of angular momentum:



For the radial position as a function of time can be found that:



The angular equation is then:



If F = F(r): L =constant, if F is conservative: W =constant, if F perpendicular to V then and .

Kepler's orbital equations

In a force field F = kr-2, the orbits are conic sections with the origin of the force in one of the foci (Kepler's 1st law). The equation of the orbit is:


with

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 is the excentricity of the orbit. Orbits with an equal are of equal shape. Now, 5 types of orbits are possible:
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 .
If the surface between the orbit covered between t1 and t2 and the focus C around which the planet moves is A(t1; t2), Kepler's 2nd law is


Kepler's 3rd law is, with T the period and Mtot the total mass of the system:


The virial theorem

The virial theorem for one particle is:


The virial theorem for a collection of particles is:


These propositions can also be written as: 2E(kin) + E(pot) = 0.
 
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