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In classical physics, we have the Newton equation for a particle F = dp/dt,

where F is force and p the linear momentum (p = m dr/dt). Assuming that mass m is constant, we can rewrite the N. eq. as F = m d2r/dt2.

Knowing F on the particle, the study of motions consists in the integration of the Newton's second order differential equation, obtaining

r = r(t), called motion equation, that describes the position of the particle in function of the time: to study the motion of the particles means to find this r(t). (Note that, in the integration of N. eq., two constants appear: you can calculate them by imposing the boundary conditions, e.g. r(0) and dr(0)/dt). For example, you may study the motion of a particle with F = gmx, being g a constant and x a versor. It's g = d2x/dt2 → ∫g dt = ∫d2x/dt2 dt → gt - gt0 = dx(t)/dt - dx(0)/dt; if t0 = 0 and dx(0)/dt = v0, it's dx(t)/dt = v0 + gt → ∫dx/dt dt = ∫v0 + gt → x(t) - x(0) = v0t + gt2/2;

if x(0) = x0, we have the motion equation of the particle: x(t) = x0 + v0t + gt2/2.

If there are N particles, you will have to solve a N. eq. for each one: a system of N equations! (See statistical mechanics).

Otherwise, you can use Euler-Lagrange equation instead of Newton equation, that doesn't need F, but kinetic and potential energy, T and V (using scalars, and not vectors). (See analytical mechanics).

In quantum mechanics the situation gets more complex, involving the Schroedinger equation.

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17y ago

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