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In classical mechanics, the central-force problem is to determine the motion of a particle in a single central potential field.A central force is a force (possibly negative) that points from the particle directly towards a fixed point in space, the center, and whose magnitude only depends on the distance of the object to the center.
A two-point-particle model of such a system nearly always describes its behavior well enough to provide useful insights and predictions. A simpler "one body" model, the "central-force problem", treats one object as the immobile source of a force acting on the other. One then seeks to predict the motion of the single remaining mobile object.
The Hamilton–Jacobi equation is a formulation of mechanics in which the motion of a particle can be represented as a wave. In this sense, it fulfilled a long-held goal of theoretical physics (dating at least to Johann Bernoulli in the eighteenth century) of finding an analogy between the propagation of light and the motion of a particle.
In Cartesian coordinates the Lagrangian of a non-relativistic classical particle in an electromagnetic field is (in SI Units): = ˙ + ˙, where q is the electric charge of the particle, φ is the electric scalar potential, and the A i are the components of the magnetic vector potential that may all explicitly depend on and .
The original Langevin equation [1] [2] describes Brownian motion, the apparently random movement of a particle in a fluid due to collisions with the molecules of the fluid, = + (). Here, v {\displaystyle \mathbf {v} } is the velocity of the particle, λ {\displaystyle \lambda } is its damping coefficient, and m {\displaystyle m} is its mass.
The unit vector u ρ travels with the particle and always points in the same direction as r(t). Unit vector u θ also travels with the particle and stays orthogonal to u ρ. Thus, u ρ and u θ form a local Cartesian coordinate system attached to the particle, and tied to the path travelled by the particle. [20]
Calculating the Minkowski norm squared of the four-momentum gives a Lorentz invariant quantity equal (up to factors of the speed of light c) to the square of the particle's proper mass: = = = + | | = where = is the metric tensor of special relativity with metric signature for definiteness chosen to be (–1, 1, 1, 1).
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.
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