Questions 1 to 8 carry one mark each, questions 9to18carry two marks each, questions 19to27carry three marks each and questions 28to30 carry five marks each. However, an internal choice has been provided in one question of two marks, in one question of three marks and all three questions of five marks each. You have to attempt only one of the given choice in such questions 1. Name the fundamental force which is responsible for the stability of the nucleus.
In general, no closed-form solution for such a problem exists, and the time evolution of the system is believed to be chaotic.
The use of computershowever, makes solutions of arbitrarily high accuracy over a finite Paper solved span possible using numerical methods for integration of the trajectories.
Restricted three-body problem[ edit ] Circular[ edit ] In the circular restricted three-body problem, two massive bodies move in circular orbits around their common center of massand the third mass is negligible with respect to the other two.
It can be useful to consider the effective potential. Periodic solutions[ edit ] InLeonhard Euler found three families of periodic solutions in which the three masses are Paper solved at each instant. InLagrange found a family of solutions in which the three masses form an equilateral triangle at each instant.
Together, these solutions form the central configurations for the three-body problem.
These solutions are valid for any mass ratios, and the masses move on Keplerian ellipses. These four families are the only known solutions for which there are explicit analytic formulae.
In the special case of the circular restricted three-body problem, these solutions, viewed in a frame rotating with the primaries, become points which are referred to as L1, L2, L3, L4, and L5, and called Lagrangian pointswith L4 and L5 being symmetric instances of Lagrange's solution.
Special-case solutions[ edit ] Configuration of the Sitnikov problem InMeissel stated what is now called the Pythagorean three-body problem: Burrau  further investigated this problem in In Victor Szebehely and coworkers established eventual escape for this problem using numerical integration, while at the same time finding a nearby periodic solution.
InWilliam Duncan MacMillan found one special solution. InSitnikov improved this solution. Broucke each found a set of solutions that form part of the same family of solutions: In this family the three objects all have the same mass and can exhibit both retrograde and direct forms.
In some of Broucke's solutions two of the bodies follow the same path. This solution has zero total angular momentum. The quantum three-body problem is studied in university courses of quantum mechanics.
Some mathematical research is still dedicated either to finding a good numerical solution  or finding ways to reduce the problem to a more simple system that can be solved analytically, such as the Hartree—Fock method and the Franck—Condon principle. For a special case of the quantum three-body problem known as the hydrogen molecular ionthe system energies are solvable analytically see discussion in quantum mechanical version of Euler's three-body problem in terms of a generalization of the Lambert W function.
However, the solution is possible only by making the assumption that the two "nuclei" are fixed relative to each other, which essentially reduces the problem to a single-body problem within an energy potential.
However, these problematic initial conditions are not generic, since they have Lebesgue measure zero. An important issue in proving this result is the fact that the radius of convergence for this series is determined by the distance to the nearest singularity. Therefore, it is necessary to study the possible singularities of the 3-body problems.
As it will be briefly discussed below, the only singularities in the 3-body problem are binary collisions collisions between two particles at an instant and triple collisions collisions between three particles at an instant. Collisions, whether binary or triple in fact, any numberare somewhat improbable, since it has been shown that they correspond to a set of initial conditions of measure zero.
However, there is no criterion known to be put on the initial state in order to avoid collisions for the corresponding solution. So Sundman's strategy consisted of the following steps: Using an appropriate change of variables to continue analyzing the solution beyond the binary collision, in a process known as regularization.
Proving that triple collisions only occur when the angular momentum L vanishes.Paper Mario, known as Mario Story (マリオストーリー Mario Sutōrī) in Japan, is a role-playing game developed by Intelligent Systems for the Nintendo 64, and was later re-released for the iQue Player and Virtual Console..
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