Variational principles

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The calculus of variations

Pierre-Louis Moreau de Maupertuis
Pierre-Louis Moreau de Maupertuis , 1698-1759, French physicist and theologian.

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
Galileo Galilei , 1564-1642
Fermat, Pierre de
Fermat, Pierre de [ferma], 1661-1665
Riemann, Georg Friedrich Bernhard
Bernoulli, Jacob [bernuli], 1654-1705, Bernoulli, Johann [bernuli], 1667-1748
Variational principles

Problem of brachistochrone was solved (y.1696) by Johann Bernoulli. He is also the author of a general formulation of the principle of virtual displacements. He - together with his brother Jacob - studied Leibniz calculus and applied it to a number of physical problems.
The further development have contributed: Leonard Euler (1707-1783), D'Alembert, Jean-Baptiste Le Rond (1717-1783), Karl Friedrich Gauss (1777-1855), Pierre Simon Laplace (1749-1827), Ampere (1775-1836), Jacobi (1804-1851), J.A.Serret (1819-1885), M.V.Ostrogradský (1801-1861), Weierstrass (1815-1897), Ritz (1878-1909), ...

Lagrange, Josepf Louis
Lagrange, Josepf Louis, Comte [lagránž], 1736-1813,

Variational principles are divided into differential and integral.

Hamilton W.Rowan
Hamilton W.Rowan , 1805-1865

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.

Poincaré, Jules Henri
Poincaré, Jules Henri , 1854-1912

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,...).

Michael William Ovenden
Michael William Ovenden , 1926-1987
The mechanism of interaction

It seems to be evident that the bodies are actually tend to avoid each other. Also the mechanism which operates this tendency is quite clear: At the moment of conjunction are bodies - due to conservation of angular momentum - depending on the circumstances - accelerated or braked, so then the next conjunction moves to another location of the path.

In contrast, we do not know to what configurations (in case of more bodies ..) the development leads, in what configurations it comes to an end. Are there always final configurations (?) etc.

We do not know exactly how we understand the interaction of bodies: While gravitational force always acts on the connecting lines of participating bodies, repulsion act in times of conjunctions and rather in the direction of tangents to the respective orbits. Such repulsive forces can quite certainly affect placement (relative phase shift) of bodies on orbits with fixed characteristics.

But the question is whether the 'repulsive forces' can make changes of particular orbital characteristics (distance from center, eccentricity, inclination, orbital period). This question has crucial importance. The possibility of influencing of orbital characteristics would imply e.g. that the so-called Titus-Bode series which indicated approximately exponential distribution of the distances of the planets in Solar systems, may not be coincidental.

If we have admitted that repulsive forces are real and acting on the connecting lines of objects, we would have to count them into equations of the motion.

E.g. Let us have two bodies moving around the center. Repulsive forces dependent on time t continuously act among the bodies: I01(t), I02(t) a I12(t). For simplicity we neglect the variability of these forces and replace them with their average values: I01, I02 and I12. For each satellite we will write balance condition of centripetal (Fd) and centrifugal force (Fo):

· Fd1-Fo1+I12-I01 = 0

· Fd2-Fo2+I12-I02 = 0

Orbital speed of the body (1) is then coming out slightly higher, and of the body (2) slightly smaller than in the case without consideration of repulsive forces.

Action functions

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


Imagine that we are in his apartment, and we need to get to the station at the other end of town. We have a choice of four options:

     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

What path we choose depends on the viewpoint, i.e. on our "action function".

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 rijk (k=2,3,1/2,...) or logaritmic functions of masses ln(mi∙mj) etc. Action functions (i.e. generally functions f(mi,mj,rij))) 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 φ1212, e.g. f() = mi∙mj/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



‹f›=(1/2π) ∫ f Note



‹r12›=2r1/ln((r1+r2)/(r2-r1), for r1<r2




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/|r1k-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 rij. 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.

Largest area

First let us solve a simpler problem that has character similar to searching of positions (phase shifts) of bodies on the track (orbit).
We will search for the phase shift φ of curves sin(x) and sin(x-φ) that makes largest area (in interval 0-2π). Vertical distance of curves is determined by action function f(x,φ)=|sin(x)-sin(x-φ)|. Integral of this function is area and we're looking for what value φ is value of integral extreme .

Solution: First, we find the coordinates of points P1 and P2 (are always no more than two), where given curvers intersects, i.e. where sin(x)=sin(x-φ). Solution is: x=(φ+;π+2kπ)/2. Total content is the sum of the three surfaces, i.e. three integrals of action functions in intervals (0,(φ+π)/2),((φ+π)/2,(φ+3π)/2) a (φ+3π)/2,2π).
After integration and adjustment we get: F() = S1+S2+S3 = 4∙[cos((φ-π)/2)-cos((φ+π)/2)]
This function becomes extreme for: dF()/dφ = 0, i.e. for sin((φ-π)/2)-sin((φ+π)/2) = 0.
The curves define the extreme area in case φ=k∙π.


In the previous example, both functions have the same period. Now consider functions sin(n1.x) and sin(n2.x-φ) (i.e. functions with periods 2π/n1 resp. 2π/n2). So we get (n1+n2) intersections that divide the whole area into (n1+n2+1) parts. Coordinates of intersections: x=(φ+;π+2kπ)/(n1+n2).
E.g. planets Jupiter and Saturn, n1:n2≈5:2, n1+n2 = 7. The entire (synodic) cycle (2π) takes on average (J,S)=19.859 years. In the case φ=0 we get the following 7 points:
π/7( 1.418 years), 3π/7( 4.255 years), 5π/7( 7.092 years), 7π/7( 9.930 years), 9π/7(12.766 years), 11π/7(15.603 years),13π/7(18.440 years)

(π/7 = 1.418 years = 517.93 days makes approximately 2 Mayan cycles tzolkin, i.e. 2∙260 days).

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) {               
            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/r12k

Let F() = 1/(r2k-r1k) +1/r1k + 1/r2 k, r1!= From equation dF()/dr1 = k.r1k-1/(r2k -r1 k)²- k/r1k+1 =0, we get r1=r2/21/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/r12k

Let F() = ∑∑ 1/(rj k -ri k) + ∑ 1/rik, 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() = ∑∑ mimj/(rjk -ri k) + ∑ m0mi/rik. 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.

Depiction of equations
Mentioned equation resemble in principle Kirchoff's laws known from electricity. Also, there is a simple graphical representation for them. f(r1,r2)+f(r1,r3)-f(r1) = 0
          0           1               2            3
          *- f(r1)---→*←-- f(r1,r2) --* 
                      *←----------- f(r1,r3) ------+
          0            1              2             3
          *            *---f(r1,r2)--→*←--f(r2,r3)--* 
          *------------- f(r2)-------→*
Geometric average of periods

Consider function F() = ∑(mi∙(x+ri)/(x-ri), i=1,2 and look for those x=r3, aby platilo dF()/dx= 0. We get equation m1.r1/(x²-r1²) + m2.r2/(x²-r2²) = 0.

Its solution is:

x=r3 = √(r1r2 (r1m2+r2m1)/(r1m1+r2m2)), specially for m1=m2=1: x=r3 = √(r1r2).
In the Solar System e.g these analogous relationships can be found:

· T(Europa)=3.5518 days ≈ √(Tio∙Tganymedes) = 3.5577 days

· 4∙S = 117.83 years ≈ √(U∙N) = 117.67 years

· 4∙Mr = 234.6 days ≈ √(V∙Vr) = 233.7 days (≈ 2∙(V,E)/5 = 233.6 days)