Therefore, by Proposition 2, the second particle is also acted upon by a central force F2(r).
If confirmed by peer review, this will mark the second particle discovered at the Large Hadron Collider in the last few months.
If the path of the first particle is described in the form r = g θ1), the path of the second particle is given by the function r = g θ2/k), since.
For example, let the path of the first particle be an ellipse : \frac{1}{r} = A + B \cos \theta_1 where A and B are constants; then, the path of the second particle is given by : \frac{1}{r} = A + B \cos \left( \frac{\theta_2}{k} \right).
Given any increment of time, the distances traveled by the first particle in successive increments form a geometrically decreasing sequence.
Newton showed that the motion of the second particle can be produced by adding an inverse-cube central force to whatever force F1(r) acts on the first particle F_2(r) - F_1(r) = \frac{L_1^2}{mr^3} \left( 1 - k^2 \right) where L1 is the magnitude of the first particle's angular momentum, which is a constant of motion (conserved) for central forces.
The areal velocity of the second particle equals that of the first particle multiplied by the same factor k : h_2 = 2 \frac{dA_2}{dt} = r^2 \frac{d\theta_2}{dt} = k r^2 \frac{d\theta_1}{dt} = 2 k \frac{dA_1}{dt} = k h_1 Since k is a constant, the second particle also sweeps out equal areas in equal times.
If the orbit rotates at an angular speed Ω, the angular speed of the second particle is faster or slower than that of the first particle by Ω; in other words, the angular speeds would satisfy the equation ω2 = ω1 + Ω.
Thus Calosi (forthcoming) shows that one can treat e.g., a fully entangled three particle system by positinga binary entanglement relation between the first particle and the composite system made up of the second and third particles.
Though the $600 million collider opened about a year ago at the lab in Upton, N.Y., the achievement of the first particle collisions was delayed by technical problems.
