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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 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
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 ...
The two dots on top of the x position vectors denote their second derivative with respect to time, or their acceleration vectors. Adding and subtracting these two equations decouples them into two one-body problems, which can be solved independently. Adding equations (1) and results in an equation describing the center of mass motion.
The formula for the acceleration A P can now be obtained as: = ˙ + + (), or = / + / +, where α is the angular acceleration vector obtained from the derivative of the angular velocity vector; / =, is the relative position vector (the position of P relative to the origin O of the moving frame M); and = ¨ is the acceleration of the origin of ...
The Euler three-body problem is known by a variety of names, such as the problem of two fixed centers, the Euler–Jacobi problem, and the two-center Kepler problem. The exact solution, in the full three dimensional case, can be expressed in terms of Weierstrass's elliptic functions [ 2 ] For convenience, the problem may also be solved by ...
The problem is also important because some more complicated problems in classical physics (such as the two-body problem with forces along the line connecting the two bodies) can be reduced to a central-force problem. Finally, the solution to the central-force problem often makes a good initial approximation of the true motion, as in calculating ...
The solution of these equations of motion provides a description of the position, the motion and the acceleration of the individual components of the system, and overall the system itself, as a function of time. The formulation and solution of rigid body dynamics is an important tool in the computer simulation of mechanical systems.