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Dmytro Taranovsky

February
10, 2001; Modified January 7, 2002

In this paper, *a≡b*
means *a* is defined to be equal to *b*.

The world is modeled by a finite set of
point objects. *Location*, **r**, is an ordered triple, (*x,
y, z*), of real numbers. *Time*, *t*, is a real number.
Each object is characterized by its *trajectory* **r**(*t*)
and its time-independent properties that include *mass*, a
positive real number *m* that can be interpreted as resistance
to acceleration. The physical meaning of trajectory is location of
the object in terms of time. No two objects can have the same
location at the same time. For an object of trajectory **r**(t)
and mass *m*: *velocity*, **v**≡**r'**(t); *speed*≡
||**v**||; *momentum*, **p** ≡ * m***v**;
*acceleration*, **a**** **≡** r"**(t); *net force*
on the object, **F**≡d**p**/dt =m**a**; *angular
momentum* (a measure of rotation), **L**≡cross(**r**,
**p**). Net force on the given object is equal to the sum of the
forces for each of the other objects on the given object for each
interaction. Let __i__ and __j__ denote any two different
objects, and *A* denote any interaction. Let the trajectory of
object *i* be recorded as **r _{i}**(t), and mass as

For __i__, __j__ and *A*, let
**F _{ijA}** be the force of

where F

Conservation and Relativity Theorem:

F

_{d}is the same for F_{ijA}and F_{jiA}; |F_{d}|=||F_{ijA}||.F

_{ijA}(t) =-F_{jiA}(t).F

_{ijA}is parallel to**r**._{j}-r_{i}Total momentum and total angular momentum are conserved.

Physical laws are invariant under

*translations*in space and time (that is ((*r*_{i}*t-t*)_{0}*+c*)-->(*r*_{i}*t*)), rotations in space, reflections in space (such as*-r*(_{i}*t*)-->(*r*_{i}*t*)), and transformation**r**(t)-v*t-->_{i}**r**(t) for all objects (v, c, and t_{i}_{0}must be invariant of object, time, and location).

Notes: *Total momentum* is the sum
of momentum for all objects. *Total angular momentum* is the sum
of angular momentum for all objects. 'Invariant' means 'not depending
on'. The equation for F_{ijA} is used to prove the theorem.

*Gravitational force* between __i__
and __j__ has F_{d}=-G*m _{i}m_{j}/d_{ij}^{2}*
where G is approximately 6.67*i0

G is called universal gravitational constant.

Each object __i__ is characterized by
its electric charge *q _{i}*, a real number.

*Ideal spring* between __i__ and
__j__ of length *x* (*x*>0) and spring constant k (k
is not Coulomb's constant) causes force between __i__ and __j__
of F_{d} = k*(*x*-d_{ij}). Usually, k>0.

*Kinetic energy*, E_{K} ≡
m_{i}||**v _{i}**||

'd*e' can be abbreviated as 'de';
'square root' is abbreviated 'sqrt'.

Let **r**=(x, y, z),**
r _{1}**=(x

c

Dot product of

Cross product of

Norm (also called magnitude) of

df/dt ≡ f'(t) ≡ limit of (

Integral of