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What Is The Three Body Problem And Why Is It Hard To Solve?

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The branch of science that describes the natural world around us is Physics. To be as close as possible to the objective reality (i.e. to only rely on what can be described with the greatest accuracy without any personal bias of the scientist), physics relies on mathematics as its core working instrument. Thus, advances in mathematics often help advance theoretical physics, which then is verified by practical implementation. But if the mathematics involved becomes too complex to solve, the advancement physics is also affected.

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Gravitational mechanics (motion of objects due to gravity) is one of the oldest branches of physics which relies heavily on mathematics. One such problem in gravitation is the Three Body Problem.

The Problem Statement

Newton’s theory of gravity explains with reasonably high accuracy the interaction between two gravitational sources. But since the real world consists of systems with more than two bodies (for example, the solar system), an equation for more than two bodies is needed.

The Three Body Problem in physics is concerned with the evolution (change with time) of a closed system (no external forces present) with three gravitational sources (three planets, three stars, or a combination thereof). The aim is to arrive at a solution which would give the value of position and momentum of the three bodies at any instant of time.

Orbits. Sun earth, moon orbits diagram. Orbit movements with directions and angles. Elliptical arrows showing trajectory directions. Physics, astronomy illustration. White background. Vector graphic

The interplay between the earth, the sun and the moon is a 3-body problem which must be solved accurately for sending satellites to the moon (Photo Credit : grayjay/Shutterstock)

Type of Solution Needed

The aim is to arrive at an equation (or a set of equations) which would specify the position and momentum of each body in the system at any time, using Newtonian theory of gravitation.

These are the requirements for the solution to be feasible:

  • The solution has to be general, i.e. work for all possible starting configurations of the three bodies. More specifically, a solution in the form an equation (or a set of equations) is required which would give the position and momentum of all the three bodies at any later time by just plugging in the value of time.
  • For example, the equation y(t) = u t + 0.5 a t2 is a general solution for where the answer can be calculated by just plugging in the value of t. The equation will be valid for any value of u and a (initial conditions).
  • The solution has to be closed form. A closed form solution is an expression where the solution could be obtained using a finite number of mathematical operations (+, -, , , ,  etc.) between the variables.
  • For example, the equation y = mx + c is a closed form solution because a single multiplication and a single addition are used to express y. Similarly, the equation, y = e+ is a closed form expression because there is one exponentiation, one addition and one square rooting. But the expression, y = 1 + x + x2 + x3 + …… is not a closed form solution because there are infinitely many additions required to express y.
equations of linear motion with constant acceleration

Kinematics equations are examples of closed-form expressions. For any value of time, the position can be predicted with absolute accuracy. (Photo Credit : zizou7/Shutterstock)

If a solution does not possess the mentioned attributes, that solution is discarded because:

  • Lack of a general solution means that for each and every unique starting configuration, a new equation must be derived. Since there could be infinitely many starting positions for three bodies, an infinitely many number of solutions have to be derived.
  • Lack of a closed form solution means that infinitely many terms have to computed, without any indication of when to stop. Thus, the accuracy cannot be determined (since there’s no end-point) and computing power used up is enormous.

Definition,Of,The,Euler's,Constant,In,Two,Different,Ways

Equations like these are generally not preferred because they converge quite slowly, which is undesirable since a lot of computational power and time is used up. (Photo Credit : benjaminec/Shutterstock)

Mathematical Dead-End and Physical Interpretation

According to Newton, the force of attraction between two bodies, separated by distance r, due to gravitation is given by:

    where,

ma, mb = masses of bodies and respectively,

= unit vector (vector of length 1) along the line joining the centres of the two bodies.

Newton's,Law,Of,Universal,Gravitation

Newton’s law of gravitation (Photo Credit : Nasky/Shutterstock)

Let three masses, ma, mb and mc be placed in space in any arbitrary (random) configuration. Let the distances between the each of three bodies be rab, rac and rbc respectively. We wish to find the positions ra, rb and rc and momenta pa, pb and pc respectively for the three masses at any instant of time t.

Each mass is affected by the gravitational attraction due to the other two. This can be expressed mathematically as:

  • Force experienced by ma due to mb and mc:     
  • Force experienced by mb due to ma and mc:     
  • Force experienced by mc due to ma and ma:     

where , and are the accelerations of masses a, b and c respectively.

The above equations are a system of differential equations which is actually Newton’s IInd law applied to gravitation (gravitational force is equal to rate of change of momentum).

Since the three masses affect each other, the above is a coupled system of equations. The energy of the system is given by the following system of partial differential equations:

           

where,

H = Total Energy of the system (kinetic + potential)

pi = momentum of each mass

ri = position of each mass

Equation (i) conveys that the change of position with time (i.e. velocity) is equal to ratio of change of energy with change in momentum (velocity = change in total energy  change in momentum)

Equation (ii) conveys that the change of momentum with time (i.e. force) is equal to ratio of change of energy with change in position (force = change in total energy  displacement of mass)

This is where we run into a problem.

The above system of equations is non-integrable and hence it is impossible to find a general closed form solution which would predict the position and momentum at any instant of time indefinitely into the future (or the past).

Physically, this is because the motion of each body depends on the motion of the other two, and the centre of mass of the system constantly shifts position. Since it is impossible to measure with complete accuracy the initial position and momentum of the bodies (accuracy maybe 99.99%, but not 100.00%), there always exists a minute uncertainty (0.01% or more or less) in the measurement of initial conditions. Since the final state depends on the initial conditions, any uncertainty in final state gets multiplied in this system as it evolves from the initial to the final state. Longer the duration, greater the uncertainty in the final state. If sufficient time has passed, the actual final state maybe totally different from the theoretically computed state.

(Check this out if you want a rigorous mathematical study of non-integrability)

Implications of Non-Integrability and Alternate Solutions

This critical dependence of the final state on the initial conditions makes the system chaotic. For example, even a 1mm error in measurement of initial condition leads to a huge increase in uncertainty of the final state after millions of years. Thus, the objective of finding an expression which would give the position and momentum indefinitely into the future fails.

Double pendulum with labels

The Double Pendulum is an example of a chaotic system. Small change in initial conditions translates to an exponential change in the final condition. (Photo Credit : Der Messer/Wikimedia commons)

But wait, you might wonder how NASA makes predictions about comets passing by Earth hundreds of years into the future, or the future state of solar system thousands of years into the future?

The answer is Numerical Analysis. Approximate solutions are computed (a numerical value is assigned) at each instant of time. For example, if the state of the system is required to be computed at t=50 seconds, then successive solutions are computed from t=0, 1, 2, 3, …… 50. For each instant of time, there is a numerical value associated with the position and momentum, and that value is used to calculate the solution at the next time step.

Numerical integration illustration

Approximate solutions are obtained using numerical algorithms. The accuracy of the solutions is determined by the frequency of calculation of results (Photo Credit : Krishnavedala/Wikimedia commons

The post What Is The Three Body Problem And Why Is It Hard To Solve? appeared first on Science ABC.

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