Variational principles

The calculus of variations

Configurations of bodies in Solar system (at least those staying here for a sufficiently long time) are not and can not be accidental. This suggests that all movement happens so that some (summary) characteristics is extreme.

According to de Maupertuis nature realizes a purpose and behaves so that certain quantities are minimized (i.e. saved up). So there is something in addition to the usual causal understanding of things. He proposed a universal law of nature according to which the bodies are moved that the total consumption of certain quantities (so called "action") is as small as possible.

This idea is not new - it is the foundation of variational calculus, math disciplines, which is used to formulate a general physical problems since the 17th century.
At the birth of the calculus of variations were: Italian Galileo Galilei (1564-1642) and Frenchman Pierre de Fermat (1601-1665).

• Galileo Galilei formulated (y.1638) issue of the curve of the fastest fall (i.e. brachistochrone). The two points in a gravitational field have to be joined by such curve, that fixed point following it 'slides' down it the shortest possible time.
• According to Pierre de Fermat beam of light spreads between two points such a way that light travels it in the shortest time. This principle explains the law of reflection and also refraction law.

Applications in astronomy

H.Poincaré has found that disturbing function in a resonant gravitational system (averaged with respect to critical or resonance parameter) has a local minimum.

The possibility of the gradual emergence of resonant configurations was shown by means of numeric integration of the motion of bodies (Hills, 1970). The synchronization must according to some analyzes occur even under very weak forces between bodies (A.M.Molčanov).

M.Ovenden, (?inspired by ideas R.Basse from y.1958?), suggested (before y.1972) function, whose extreme should be searched: mean potential energy. However, this so called Ovenden's principle of minimal interaction, could be applied only to certain groups of bodies (for moons in Laplace's resonance,...).

• Astronomer A.E.Roy dedicated his book (Orbital motion, y.1973) just to M.Ovenden and ranked in it the brief outline of the principle of minimal interaction.
• Caxton C.Foster (see) tried to build a computer program, verifying and demonstrating Ovenden's principle. In the system of three bodies he gradually shifted orbits, in compliance with the law of conservation of energy and momentum. Doing so he sought such configurations, whose mean potential energy is minimal.

Action functions

We will call action function the function whose (mean) value has become extreme.

     Journey                        Viewpoint
     -------------------------------------------
     1 / Walk through the park      the cheapest way
     2 / By bus                     a cheap fast path
     3 / Taxi                       the fastest route 
     4 / Helicopter                 the shortest path

Ovenden's principle is based on the assumption that the action function is the potential energy. Let's try to validate other possibilities, e.g. functions, v nichž figurují mocniny distance r_ij^k (k=2,3,1/2,...) or logaritmic functions of masses ln(m_i∙m_j) etc. Action functions (i.e. generally functions f(m_i,m_j,r_ij))) are denoted by lowercase letter f().

We distinguish three variants of function according to treatment with distances:

Functions dependent on the distance (linear, quadratic, general power, ..)
If r1 and r₂ are distances from the centre S of satellites P₁ and P₂ and φ₁₂ their angle distance at a given moment, is actual distance:
r₁₂ = r(1,2) = sqrt(r₁²+r₂²-2r₁r₂ cos φ₁₂) (i.e. cosine theorem in triangle SP₁P₂)

Functions independent of the distance ,
i.e. functions, whose value varies only with the angle φ₁₂=φ₁-φ₂, e.g. f() = m_i∙m_j/g(φ₁₂). In this case, function g() must remove the singularity at conjunction of bodies (i.e. for φ₁₂=0); e.g. g()=sin(φ₁₂/2),g()=1+cos(φ₁₂), etc.)

Functions deforming distances (logarithmically,...) If we notice stratification of matter in space between the Sun and Jupiter, we get the impression that if the Sun and Jupiter had influence inversely proportional to distance, there is no way to justify why are Venus and Earth so close to the sun. It seems that the action functions for the Sun somehow differs from the action functions for Jupiter. Or we have to use different assumptions to measure distances in the Solar System: For example, if we replace all distances from the center by their logarithms, we get Venus and the Earth - in numerical world of logarithms - closer to Jupiter.

Dependence on the weight - linear, logarithmic ...

Central body

Although it may seem that the interaction is only a matter of satellites, it may be necessary (at least in some cases) to count action functions also for relations between the central body and orbiters. The opposite case may lead to the result, where the optimal configuration of n bodies will have n-1 objects in close proximity to the central body. (In the case that we omit the central body from calculations, a 'force' is missing which would push the bodies further from the center ...)

Integrals of action functions

We are interested in the sum of all values action function during a specific interval. Integrals of action function (i.e. funkcionals) will be denoted by capital letter F(), mean values of action functions by (). Mean value ‹f›() of action function f() will be gained by division of integral F() by width of integration interval w (usually w=2π): ‹f›()= ∫ f()/w = F()/w

Compensation for integrals

We can generally get very complicated functions by integrating of action function. We will therefore for further calculations use also some simpler functions F() although they were not formed by integration, but that are in some sense analogous to the more complex functions. These replacement functions have to be (as well as integrals) symmetrical with respect to a corresponding variables of individual satellites.
E.g. functions F=1/|r₁^k-r₂ ^k | is symmetric with regard to r₁ and r₂, kεR.

Optimal configurations

Extremes of action integrals

Suppose system of n bodies with centre 0 and (n-1) satellites 1,2,..,n-1. We are looking for such configuration of bodies in which the integral of action function becomes extreme. According to the searched parameters determining the configuration we distinguish the following cases:

· Position on the track
Track of all bodies are given, but the position on the track is given for only one or for several selected objects. We are looking for a suitable position (phase shifts) of other bodies.

Distances of bodies
Distance (from the center) is given for one or several bodies. We are looking for the appropriate path of other bodies.

Action functions vary depending on the current values of r_ij. during motion along elliptical orbits. Instead of searching for an extreme of mean action functions it could make sense e.g look under what circumstances mean action functions varies least.

Numeric integration

Some action functions are difficult to integrate, or even integral is so complicated that it is difficult to find an extreme. In this case, we can estimate the numerical solution (by computer program).

Let us consider e.g. 2 satellites moving with orbital periods T₁, T₂ on excentric circles with the distance of the centers Dx₁₂ (Dy₁₂=0). And let orbit radii in relation to the orbital periods respect Kepler's third law, i.e.: r³/T² = const.

For the initial phase shift Fi₁₂ total value (integral) "sumF" of action function f()=1/r₁₂ is counted during the time interval Time.

        #define cDIVISION 100
        #define c23 0.66666666667
        float numIntg(float aTime, float aT1, float aT2, float aFi12, float aDx12) {
           int i;
           float sumF=0;
           float r1 = pow(aT1,c23);   /* Kepler */
           float r2 = pow(aT2,c23);   /* Kepler */
           float limit = aTime*cDIVISION;
           for (i=0;i<limit;i++) {
                float t = (float)i/cDIVISION;
                float angle1 = 2*pi*(t/aT1);
                float angle2 = 2*pi*(t/aT2+aFi12);
                float f1x=r1*cos(angle1);  float f1y=r1*sin(angle1);
                float f2x=r2*cos(angle2)+aDx12;  float f2y=r2*sin(angle2);
                float dx12 =  fabs(f2x-f1x);  float dy12 = fabs(f2y-f1y);
                float r12 = sqrt(dx12*dx12+dy12*dy12);
                if (r12!=0) {               
                        sumF=sumF+(1.0/r12);
                }
            }
            
            return F;
        }

We repeatedly perform numerical integration for various values of the initial phase shift φ₁₂ and we find for which φ₁₂ an extreme occurs.

Two bodies around the center

Functions of type 1/r₁₂^k

Let F() = 1/(r₂^k-r₁^k) +1/r₁^k + 1/r₂ ^k, r₁!= From equation dF()/dr₁ = k.r₁^k-1/(r₂^k -r₁ ^k)²- k/r₁^k+1 =0, we get r1=r₂/2^1/k.
E.g. for r₂=100:

Functions of type 1/r₁₂

Let F()=ln((r₁+r₂)/(r₁-r₂))/2/r₁ +1/r₁+ 1/r₂, r₁ Let us denote q=r₁/r₂. Then (r₁+r₂)/(r₂-r₁)=(1+q)/(1-q) and ln((1+q)/(1-q))= 2 argtgh q.

We rewrite the function to F()=(argtgh(r₁/r₂)+1)/r₁ + 1/r₂.

From equation dF()/dr₁=0 it follows q/(1-q²)-argtgh(q)-1 = 0 (*).

Function argtgh(q)=q+q³/3+q⁵/5+...; in the first approximation argtgh(q)=q and equation (*) goes into shape q³+q²-1 = 0.

We find numerically that the function F() takes a minimum for q≈0.7865; i.e. for r1=100 is r₂≈78.7.

Three bodies around center

Functions of type 1/r₁₂^k

Let F() = ∑∑ 1/(r_j ^k -r_i ^k) + ∑ 1/r_i^k, r₁<R₂. Numeric solution, e.g. for r₃=100:

These values disregard the weight of solids, i.e. it is assumed, the mass of the center is the same as the mass of planet.

Now, let us have: F() = ∑∑ m_im_j/(r_j^k -r_i ^k) + ∑ m₀m_i/r_i^k. In the case m₀ = 10, m₁=m₂=m₃=1 we get:

The bigger mass of center is, the further from the centre planets are pushed, i.e. in other words, the lighter planet are (relative to the center), the more they coalesce closer together.

Case k=1

Let F() = 1/(r₂-r₁)+1/(r₃-r₂)+1/(r₃-r1)+1/r1+1/r₂+1/r₃, r1<r₂
From equation
dF()/dr1 = 1/(r₂-r₁)²+1/(r₃-r₁)²-1/r₁² = 0, dF()/dr2 = -1/(r₂-r₁)²+1/(r₃-r₂) ²-1/r₂² = 0
it follows:
1/(r₃-r₁) ² + 1/(r₃-r₂)² = 1/r₁² + 1/r₂². The equation satisfies r₁+r₂ = r₃. Denote q=r₁/r₂. Hence must be true: (1-q²)(1-q) ² = q², i.e. q⁴-2q³+q²+2q-1=0.

Numerically we get solution q=0.47 (i.e.32/68, see the previous table).

Now we simplify the equation.
Suppose, that it has to be true: [dF()/dr₁ =] 1/(r₂-r₁)+1/(r₃-r₁)-1/r₁ = 0, [dF()/dr₂ =] -1/(r₂-r1)+1/(r₃-r₂)-1/r₂ = 0.
Hence 1/(r₃-r₁) + 1/(r₃-r₂) = 1/r₁ + 1/r₂. Similarly also here - equation is satisfied by: r₁+r₂ = r3. From the first equation is 1/(r₂-r₁) = 1/r₁-1/r2, i.e. r₁²-3r₁r₂+r₂². In the designation q=r₁/r₂: q²-3q+1=0. Hence q=0.382 (=((√5-1)/2) ².

          0           1               2            3
          *- f(r1)---→*←-- f(r1,r2) --* 
                      *←----------- f(r1,r3) ------+

          0            1              2             3
          *            *---f(r1,r2)--→*←--f(r2,r3)--* 
          *------------- f(r2)-------→*