Back to Dmytro Taranovsky's home page.

Dmytro Taranovsky
February 10, 2001; Modified January 7, 2002
In this paper, a≡b means a is defined to be equal to b.

Classical Mechanics

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, vr'(t); speed≡ ||v||; momentum, p mv; acceleration, a r"(t); net force on the object, F≡dp/dt =ma; 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 ri(t), and mass as mi. Distance between i and j, dij ≡ ||rj-ri||. (Distance between i and i is 0.)

Important Properties of All Interactions

For i, j and A, let FijA be the force of i on j through A. The force obeys:
FijA=FD(dij, ||vj-vi||, dot(rj-ri, vj-vi), {i, j}, A)*(rj-ri)/dij
where FD is a continuous function (unique for the universe) that returns a real number Fd. ({} indicate that the order of the enclosed arguments is irrelevant.) FijA is called repulsive if its Fd>0 and attractive if its Fd<0. Interactions are specified by stating their FD.
Theorem: If at time t0, for each object its r(t0) and v(t0) are specified and if FD is given, then the development of the system in time is uniquely determined for all times (both future (t>t0) and past (t<t0)). The system ends existence when and only when two objects approach arbitrarily close to each other.

Conservation and Relativity Theorem:

  1. Fd is the same for FijA and FjiA; |Fd|=||FijA||.

  2. FijA(t) =-FjiA(t).

  3. FijA is parallel to rj-ri.

  4. Total momentum and total angular momentum are conserved.

  5. Physical laws are invariant under translations in space and time (that is (ri(t-t0)+c)-->ri(t)), rotations in space, reflections in space (such as -ri(t)-->ri(t)), and transformation ri(t)-v*t-->ri(t) for all objects (v, c, and t0 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 FijA is used to prove the theorem.

Important Interactions (Gravity, Electricity, and Springs)

Gravitational force between i and j has Fd=-Gmimj/dij2 where G is approximately 6.67*i0-iimeters3/(kg*sj ).
G is called universal gravitational constant.

Each object i is characterized by its electric charge qi, a real number. Electric force between i and j has Fd=kqiqj/dij2 where k>0. 'k' is called Coulomb's constant and is the same for all cases.

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 Fd = k*(x-dij). Usually, k>0.


Kinetic energy, EK ≡ mi||vi||2/j for i. Total EK is the sum of EK for all objects. An interaction is called conservative if its Fd for i and j is dependent only on dij. For example, gravity and electricity are conservative interactions. For a conservative A, potential energy between i and j, U{i, j}A ≡ integral(FD(x,...,{i,j},A)dx, dij, infinity) where x is the first argument of FD and other arguments do not change. (If the integral diverges, make the upper bound equal one instead of infinity.) Total potential energy is the sum of all potential energies for all unordered pairs of objects (that is if {i, j} is counted, then {j, i} is excluded; {i, i} is not counted) for all conservative interactions. Total mechanical energy is the sum of total kinetic and total potential energy. Power of FijA, PijA≡dot(FijA, vj). Total power on i, Pi, is the sum of powers for all forces on i.
Theorem: When all interactions are conservative, total mechanical energy is conserved (that is invariant of time).
Theorem: When all interactions are conservative, physical laws are invariant under reversal of time, t-->-t.
Theorem: Pi=dEK/dt, where EK is the kinetic energy of i.

Appendix: Mathematical Background

'd*e' can be abbreviated as 'de'; 'square root' is abbreviated 'sqrt'.
Let r=(x, y, z), r1=(x1, y1, z1), and r2=(x2, y2, z2), c be any real number.
r1+r2 ≡ (x1+x2, y1+y2, z1+z2); 0≡(0, 0, 0)
cr ≡ (cx, cy, cz); r*c ≡ cr; r/c ≡ r*(1/c); -r ≡ -1*r; r1-r2r1+-r2.
Dot product of r1 and r2, dot(r1, r2) ≡ x1x2+y1y2+z1z2
Cross product of r1 and r2, cross(r1, r2) ≡ (y1z2-y2z1, z1x2-z2x1, x1y2-x2y1)
Norm (also called magnitude) of r, ||r|| ≡ sqrt(x2+y2+z2); |x| ≡ sqrt(x2).
r1 is parallel to r2 when r1=cr2 or r2=0; if r is not 0, then direction of r is r/||r||.
df/dt ≡ f'(t) ≡ limit of (f(t+a)-f(t))/a as a approaches 0 for any f(t) for which the limit exists. r'(t) ≡ (x'(t), y'(t), z'(t)); d2f/dt2 ≡ f"(t) ≡ (f'(t))'(t).
Integral of f(x) with respect to x of lower bound x1 and upper bound x2, integral(f(x)dx, x1, x2) = F(x2)-F(x1) where F'(x) = f(x); F(infinity) ≡ limit of F(x) as x approaches infinity.