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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.
That is, there is no way to start from the differential equations implied by Newton's laws and, after a finite sequence of standard mathematical operations, obtain equations that express the three bodies' motions over time. [53] [54] Numerical methods can be applied to obtain useful, albeit approximate, results for the three-body problem. [55]
One may instead change to a coordinate frame fixed in the rotating body, in which the moment of inertia tensor is constant. Using a reference frame such as that at the center of mass, the frame's position drops out of the equations. In any rotating reference frame, the time derivative must be replaced so that the equation becomes
2 Explanation and derivation. 3 See also. 4 References. ... Euler's laws of motion are equations of motion which extend Newton's laws of motion for point particle to ...
In three spatial dimensions, this is a system of three coupled second-order ordinary differential equations to solve, since there are three components in this vector equation. The solution is the position vector r of the particle at time t , subject to the initial conditions of r and v when t = 0.
Traditionally the Newton–Euler equations is the grouping together of Euler's two laws of motion for a rigid body into a single equation with 6 components, using column vectors and matrices. These laws relate the motion of the center of gravity of a rigid body with the sum of forces and torques (or synonymously moments) acting on the rigid body.
In fact, Appell's equation leads directly to Lagrange's equations of motion. [3] Moreover, it can be used to derive Kane's equations, which are particularly suited for describing the motion of complex spacecraft. [4] Appell's formulation is an application of Gauss' principle of least constraint. [5]
In this context Euler equations are usually called Lagrange equations. In classical mechanics, [2] it is equivalent to Newton's laws of motion; indeed, the Euler-Lagrange equations will produce the same equations as Newton's Laws. This is particularly useful when analyzing systems whose force vectors are particularly complicated.