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The problem of two fixed centers conserves energy; in other words, the total energy is a constant of motion.The potential energy is given by =where represents the particle's position, and and are the distances between the particle and the centers of force; and are constants that measure the strength of the first and second forces, respectively.
The three-body problem is a special case of the n-body problem, which describes how n objects move under one of the physical forces, such as gravity. These problems have a global analytical solution in the form of a convergent power series, as was proven by Karl F. Sundman for n = 3 and by Qiudong Wang for n > 3 (see n -body problem for details).
Consequently, the acceleration is the second derivative of position, [7] often written . Position, when thought of as a displacement from an origin point, is a vector: a quantity with both magnitude and direction. [9]: 1 Velocity and acceleration are vector quantities as well. The mathematical tools of vector algebra provide the means to ...
There are two main descriptions of motion: dynamics and kinematics.Dynamics is general, since the momenta, forces and energy of the particles are taken into account. In this instance, sometimes the term dynamics refers to the differential equations that the system satisfies (e.g., Newton's second law or Euler–Lagrange equations), and sometimes to the solutions to those equations.
The n-body problem considers n point masses m i, i = 1, 2, …, n in an inertial reference frame in three dimensional space ℝ 3 moving under the influence of mutual gravitational attraction. Each mass m i has a position vector q i. Newton's second law says that mass times acceleration m i d 2 q i / dt 2 is equal to the sum of the ...
The acceleration of a falling body in the absence of resistances to motion is dependent only on the gravitational field strength g (also called acceleration due to gravity). By Newton's Second Law the force F g {\displaystyle \mathbf {F_{g}} } acting on a body is given by: F g = m g . {\displaystyle \mathbf {F_{g}} =m\mathbf {g} .}
The n-body problem is an ancient, classical problem [19] of predicting the individual motions of a group of celestial objects interacting with each other gravitationally. Solving this problem – from the time of the Greeks and on – has been motivated by the desire to understand the motions of the Sun, planets and the visible stars.
Classical mechanics is the branch of physics used to describe the motion of macroscopic objects. [1] It is the most familiar of the theories of physics. The concepts it covers, such as mass, acceleration, and force, are commonly used and known. [2] The subject is based upon a three-dimensional Euclidean space with fixed axes, called a frame of ...