CRITICERS REVIEWERS please be patient. We still have a lot of medical problems. I have to spend a lot of time on my wife's medical poroblems which put me way behind on my academic teaching work. Teaching and students have the highest priorety before any publiction or research. I need stil to contact the software peole here at Syracuse University to help me with the enciclopedic article below. I finally submitted a paper entitled Depth Perseption with one Eye or Camera to OPTICAL ENGINEERING - an SPIE publication. I am a member of SPIE. A paper on a similar subject submitted to PHOTONICS NORTH 2012 SPIE Symposium was accepted. Such a camera wou;d eliminate the need for the LIDAR on the roof of Autonomous vehicles like the Google blind man's car. I have to find out how to write some math expressions appearing in the text below. In the mean time please READ in the September 2011 issue of "ATOMOBILE magazine the article entitled "Delta Wing" discussing conservative incremetal change approach vs big step change. Today is Friday 30 March 2012.
Philipp Kornreich
REACTION MECHANICS is a mathematical model of nature that describes the motion of discrete point objects coupled by non instantaneous interactions. Most often the delay is caused by the interaction between objects propagating with a finite velocity. In many cases this is the velocity of light. An object coupled by interactions propagating with a finite velocity experiences an interaction radiated by other objects in the past. The object also experiences a recoil reaction because it, too, radiates an interaction, hence the name Reaction Mechanics. This model is valid for systems with masses small enough where the curvature of space caused by large masses or energy concentrations can be neglected and the system is large enough so that quantum mechanical effects are not important. Reaction Mechanics is consistent with special relativity.
In the limit of no delay Reaction Mechanics reverts to Newtonian Classical Mechanics [1]. In the limit of infinitesimal delay Reaction Mechanics reverts to the General Theory of Relativity [2],
Neither Newtonian Classical Mechanics nor the Theory of General Relativity consider interactions over more than an infinitesimal distance that propagate with a finite velocity.
At first glance it appears that Reaction Mechanics involves non causal terms, apparent terms involving things happening in the future and detected at present. This is only a peculiarity of the mathematics. The physics of Reaction Mechanics is completely causal. All physical events either happened in the past or are occurring at the present. For example, one object senses an interaction at the present that was radiated by another object in the past, a so called retarded interaction [3][4][5]. The first object also experiences a recoil interaction at the present because it radiates an interaction at the present that might be sensed by some other object in the future, a so called advanced interaction [6] Causality requires that the first object experience the recoil interaction at the present if the radiated interaction ever reaches its destination or not. This is different than the interpretation given by Wheeler and Feynman[7] in their Absorber Theory. The third of Newton’s laws of motion actio et reactio requires that the recoil interaction be equal and opposed to the interaction the second object will experience in the future. Unfortunately, in the mathematical formulation the future interaction appears.
To describe the motion of objects couplet by a delayed interaction one proceeds as follows: First a causal Lagrangian File:Lagrangian formula.png has to be formed[8]. The Lagrangian at time step k is a function of the present coordinate vector with components xµk, the present velocity vector with components xµk, a past coordinate vector with components xµk, and a past velocity vector with components xµk. The µ’s velocity vector component at time step k, xµk, is the derivative of the µ’s component, xµk, of the position vector with respect to a time like parameter τ that all the parameters in the system are functions of.
File:1-1a formula.png (1.1)
τ is not necessarily the time measured by clocks. It is a parameter that all the parameters in the system, including the time measured by clocks, is a function of. The time step k is best defined by time differences File:Tk-tk.png that represent finite time delays. For systems with no delay, the Lagrangian is equal to the difference of the kinetic energy and potential energy. Next, a sum S of Lagrangians is formed.
File:1-2 formula.png (1.2)
In order to calculate the motion of the objects the sum of Lagrangians is inserted into the equation of motion of Reaction Mechanics:
File:1-3a formula.png (1.3)
Here Z is the sum of constraints
a)File:1-5 formula.png where b)File:1-4 formula.png (1.4)
This requires that each constraint File:1-3 formula 2.png be equal to its own constant Wk. Since equation 1.4a is equal to zero it can be multiplied by a constant, α a so called Lagrange multiplier, and added to the homogeneous equation of motion without changing it. It is, of course possible to have more than one constraint. Only the Lagrangians File:Lagrangian formula.png and File:Lagrangian 2.png contain present position vector and present velocity vector components. Since all derivatives in the equation of motion are with respect to present position vector and present velocity vector components only these Lagrangians in the sum need be considered in the equation of motion.
File:1-11 formula.png (1.5)
This is the equation of motion of Reaction Mechanics. The derivation of this equation will be discussed in section iV. The derivation uses a variational method[9] similar to the one used for the derivation of the Euler Lagrange equation[10] of motion of classical mechanics and the equation of motion of the General Theory of Relativity. The least action principle in combination with the discrete Nagumo equation[11] is used to derive this mathematical model. In the limit of no delay the equation of motion derived here reverts to the Newtonian equation of motion. It is interesting to note that before this, for incremental delays this equation of motion reverts to the equation of motion of the Theory of General Relativity. In this mathematical model it is necessary to know some features of the point objects of the system over a span of time so that initial causal Lagrangians can be formulated. In order to obtain a solution of the equation of motion, equation 1.5, initial conditions will have to be specified. In this case it is required that one not just specify the values of the coordinates and their derivatives at an instant of time but specify the coordinates and their derivatives over a continuous range of values before and after the starting point.
Next, the connection between Reaction Mechanics and the General Theory of Relativity is investigated. The equation of motion of the General Theory of Relativity[12] has the following form:
File:2-1 formula.png (2.1)
where gμν is the geodesic tensor. Here summation over repeated Greek indices is used. The geodesic tensor is used to specify the length of a distance element dρ in the curved space used in this model. The geodesic tensor describing the curved space is mass and energy dependent.
File:2-2a formula.png (2.2)
By rewriting the second term of equation 2.1 and multiplying by the inverse geodetic tensor gμν one obtains the equation of motion of the General Theory of Relativity:
File:2-2 formula.png (2.3)
where the Christoffel symbol of the second kind is:
File:2-3 formula.png (2.4)
The General Theory of Relativity also reverts to Newtonian mechanics in the limit of the absence of gravitational and other potentials. The absence of gravitational and other potentials results in a flat space. In this case the geodetic tensor is diagonal and constant. This leaves only the acceleration equal to zero in equation 2.3 in agreement with Newtonian mechanics. Both Newtonian mechanics and the General Theory of Relativity describe the motion of classical systems under different assumption. Therefore, Reaction Mechanics in some limit must give the same description as the General Theory of Relativity. This is, of course, true in the limits of the absence of all potentials delayed or otherwise. However, in the limit of very small delay times θ the equation of motion of Reaction Mechanics assumes a form related to the equation of motion of the General Theory of Relativity, equation 2.3. This can be demonstrated as follows: The homogenious equation of motion of Reaction Mechanics, equation 1.5 without constrains and where it is assumed that the coordinates have four components as conventionally used in the relativistic case, with x4 = jct is:
File:2-4 formula.png (2.5)
Consider the special case where the Lagrangians Lk is only a function of the current velocity components and the current and past coordinate components. Than the Lagrangian Lk+1 is a function of the future velocity components and the current and future coordinate components. For small delays θ, one can expand the sum of Lagrangians to lowest order:
File:2-5 formula.png (2.6)
The first order terms cancel. Here the zero order Lagrangian is:
File:2-6 formula.png (2.7)
One can define a second rank tensor:
File:2-7 formula.png (2.8)
The tensor hμν = hνµ is symmetric, The components of the hμν tensor have dimensions of mass. By substituting equation 2.8 into equation 2.6 and the resulting expression into the equation of motion, equation 2.5 one obtains:
File:2-8 formula.png (2.9)
The first two terms are the Euler Lagrange equation of classical mechanics operating on the zero order Lagrangian and must thus be equal to zero. The sixth, eighth, ninth and tenth terms are higher order terms and can be neglected in this approximation. This leaves:
File:2-9 formula.png (2.10)
where the Christoffel symbol is:
File:2-11a formula.png (2.11)
By multiplying equation 2.10 by the inverse tensor hμν as was done in the derivation of equation 2.3 one obtains:
File:2-12a formula.png (2.12)
where the Christoffel symbol of the second kind in this case is:
File:2-13a formula.png (2.13)
Equation 2.12 has the same form as the equation of motion of the Theory of General Relativity, equation 2.3. The components of the hμν tensor are proportional to the elements gμν of the geodetic tensor.
File:2-14a formula.png (2.14)
where the proportionality constantcan be thought of as a test mass. Here an incremental distance is:
File:2-15a formula.png (2.15)
The equation of motion, equation 2.12, can be extended in the conventional way to represent the effect of gravity on the form of the coordinate system etc. Returning to equation 2.9. It was here assumed that the first two terms of equation 2.9, the Newtonian equation of motion, are equal to zero. This is consistent if one considers Newtonian mechanics, the Theory of General Relativity and Reaction Mechanics are approximate mathematical models that, to good approximations, describe how nature functions in certain regions. Thus all these mathematical models can coexist. The motions described by each theory give results that approximate nature to a certain accuracy.
In order to derive the Hamiltonian formalism for Reaction Mechanics a reaction mechanical momentum with components pµk at the present time, at time step k, is defined as follows:
File:3-1 formula.png (3.1)
By substituting equation 1.2 into equation 3.1 one obtains for the momentum components:
File:3-2 formula.png (3.2)
Observe that the momentum is a function of the retarded, present and advanced coordinate and velocity components. Thus the momentum pµk of the non local interactions includes both the momentum of the point objects as well as the momentum of an interaction field. The Hamiltonian sum H is the Legender transform[13] of the sum S of Lagrangians with respect to all the velocity vector components.
File:3-3 formula.png (3.3)
Since the sum of Lagrangians S is a function of all the xµk and xµk the change dS in the sum of Lagrangians can be written as:
File:3-4 formula.png (3.4)
From equation 3.3 the change in the Hamiltonian sum H is:
File:3-5 formula.png (3.5)
By substituting equation 3.4 into equation 3.5 and making use of equation 1.5 with the constraints set equal to zero one obtains:
File:3-6 formula.png (3.6)
By collecting terms one obtains:
File:3-7 formula.png (3.7)
This shows that the Hamiltonian sum is a function of the pµk's and xµk's. Thus the change of the Hamiltonian sum can be written as:
File:3-8 formula.png (3.8)
By comparing equations 3.7 and 3.8 one obtains the Hamiltonian equations of motion of Reaction Mechanics[13]:
File:3-9 formula.png (3.9)
Next, consider the time derivative of some quantity Ak(xµk,pµk)
File:3-10 formula.png (3.10)
By substituting equations 3.9 into equation 3.10 one obtains Poisson's bracket of Reaction Mechanics[14].
File:3-11 formula.png (3.11)
A variation method[9] similar to the one used for the derivation of the Euler Lagrange Equation of motion of classical mechanics is used for the derivation of the equation of motion of Reaction Mechanics. As discussed in section I, In order to derive the mathematical model of Reaction Mechanics for systems of discrete point objects a causal Lagrangian File:Lagrangian formula.png is postulated. As described in section I it is a function of the present coordinate vector with components xµk, the present velocity vector with components xµk, a past coordinate vector with components xµk Ţ , and a past velocity vector with components xµk Ţ. It is assumed that the time differences, as for example between tk and tk-1, differ by more than infinitesimal quantities. The vectors can have any number of dimensions. The coordinate component x4 = jct as used in special relativity; where File:Velocity of light.png, c is the velocity of light in free space and t is the time can also be a vector component. The components of the generalized velocity vectors are:
[file:1-1a formula.png]] (4.1)
where τ is not necessarily the time measured by clocks. It can be a parameter that all the parameters in the system, including the time measured by clocks, are a function of. Except for accommodating the non instantaneous interactions a conventional method similar to the Hamiltonian method is followed for deriving the equation of motion of Reaction Mechanics. One can define a sum S of such Lagrangians.
File:1-2 formula.png (4.2)
In order to derive an equation of motion an action integral I is formulated.
I = File:1-3 formula.png (4.3)
It is assumed that the Lagrangians, their first and second derivatives with respect to the coordinate and velocity components are continuous in the interval τ1 to τ2. The Calculus of Variation[9] will be used for determining a relation among the coordinate components for which the action integral I has an extremum value. In this process one can also include constraints File:1-3 formula 2.png. The constrains are limited to once for which a sum Z can be formed
File:1-4 formula.png (4.4)
and where the sum Z of constraints is equal to a sum of constants Wk.
File:1-5 formula.png (4.5)
This requires that each constraint File:1-3 formula 2.png be equal to each own constant Wk. Since equation 4.5 is equal to zero it can be multiplied by a constant, α a so called Lagrange multiplier, and added to equation 4.3 without changing the action integral.
I = File:1-5 formula 3.png (4.6)
It is, of course possible to have more than one constraint. Here the subscript j labels the particular constraints. By substituting equation 4.2 for the sums of Lagrangians and the constraints of equation 4.4 into the action integral of equation 4.6 one obtains:
File:1-7 formula.png (4.7)
where dummy variables ηk are introduced so that one can vary the coordinate components to obtain an optimum path. The path through the multi-dimensional space defined by these coordinates will be optimum if all the
File:1-8 formula.png (4.8)
By substituting equation 4.7 into equation 4.8 one obtains:
File:1-9 formula.png (4.9)
where summation over repeated Greek indices is used. Note that only the Lagrangian Lk and Lk+1 and constraints Cjk and Cjk+1 contain the current coordinate vector components xµk and the current velocity vector components ҳμḳ. Equation 4.9 can be integrated by parts:
File:1-10 formula.png (4.10)
It is assumed that the variation of the coordinate components xµk with the dummy variables ηk, File:Coordinate variation.png goes to zero at the limits τ1 and τ2.
File:4-11a formula.png (4.11)
Also, the fundamental lemma of the Calculus of Variation states that since the variation File:Coordinate variation.png of the coordinates xµk with the dummy variables ηk, are arbitrary the integral will go to zero if the curly bracket is equal to zero. This yields the following equation of motion:
File:1-11 formula.png (4.12)
Equation 4.12 is a combination of the conventional Euler Lagrange equation {} and the discrete Nagumo equation. The Lagrange multipliers αj have to be evaluated by substituting the coordinate components calculated from the equation of motion, equation 4.12 into the constraint equations, equation 4.5. Here Lk is a function of the current and the past coordinates and velocities while Lk+1 is a function of the current and future coordinates and velocities .
File:1-12 formula.png (4.12)
Equation 4.12 is the equation of motion of Reaction Mechanics equation 1.5 discussed in section I. It contains retarded, current and advanced terms.
Reaction Mechanics has been successfully used to calculate the contribution to the small yearly increase in the major axis of the Lunar and binary star 596B orbits due to the delay in the gravitational interaction. The loss in orbital energy due to the gravitational interaction propagating with the speed of light between objects is transferred to the center of mass motion. It results in an acceleration of the center of mass. The calculated results agree with the value of 38 mm measured by NASA[15] of the annual increase of the Lunar orbit. The calculation also give values of 1.077 pico meter per second squared for the acceleration of the center of mass of the binary star system 596B[16] which is in agreement with values of indirectly inferred acceleration of stellar objects.
The following definitions are useful in the understanding of Reaction Mechanics:
Retarded Interaction
A RETARDED INTERACTION is an interaction that occurred in the past and its consequences are sensed at present. An example is a fortification wall absorbing the energy and momentum of a canon ball impact at the present doe to the fact that the canon ball was fired by a gun in the past.
Advanced Interaction
An ADVANCED INTERACTION is an interaction that will occur in the future but its consequences are sensed at present. Usually, the mechanism sensing the consequences at the present of the future interaction senses a recoil reaction at the present that is equal and opposed to the future interaction. An example of this is a canon that fires a canon ball at the present that will hit a fortification wall in the future. The canon ball will transfer the momentum and energy to the wall in the future. The recoil reaction the gun experiences at the present is equal to the reaction the wall will experience in the future. In our real world the same recoil reaction will occur if the ball actually hits the wall or not.