Particle dynamics

Particle dynamics studies motion determined by forces through Newton’s laws.

For force minus the derivative of a potential, total kinetic plus potential energy is conserved.

A trajectory approaching a nondegenerate unstable maximum of its potential energy has a logarithmically divergent period.

Simple harmonic motion obeys $\ddot{x}+{\omega }^{2}x=0$ and has a period independent of its amplitude.

A quartic oscillator adds an anharmonic term to the quadratic potential, for example $H=(p^2+q^2)/2+ϵq^4$.

Harmonic trajectories are isochronous: all nonconstant oscillations with a fixed frequency have the same period, regardless of amplitude.

Collision mechanics uses impulse together with conservation of momentum and, for elastic collisions, kinetic energy.

A collision is an interaction between approaching bodies or particles that changes their motion. Momentum conservation and conservation of energy constrain the outcomes; inelastic collisions can transfer mechanical energy into internal degrees of freedom.

A collision cross-section is the effective area that converts a target number density and incident particle speed into a collision rate. It can depend on particle species, relative speed, and long-range forces, and need not equal a purely geometric collision cross-section.

The impact parameter is the perpendicular distance between an incoming particle's asymptotic straight-line trajectory and the centre of a scattering potential. In central-force scattering it determines the deflection angle whenever the scattering map is single-valued.

Two spheres of diameters $D_1,D_2$ collide geometrically when their projected centre separation is at most $(D_1+D_2)/2$, giving cross-section $\sigma =\pi (D_1+D_2{)}^2/4$.

For an equal-density sphere of diameter $D$, the ratio of geometric cross-section to mass isBreaking a fixed mass into smaller pieces therefore increases its total cross-section in inverse proportion to fragment size.

Gravitational focusing increases a physical collision cross-section because mutual gravity bends initially non-colliding trajectories toward one another.

An elastic collision conserves total momentum and total kinetic energy.

An inelastic collision conserves the total momentum of the isolated colliding bodies but converts some of their kinetic energy into internal energy. The decrease in translational kinetic energy drives dissipative spreading of a planetary ring.

In a frictionless oblique collision, impulse acts along the common normal and tangential velocity components remain unchanged.

Impulse is the time integral of force and equals the change in linear momentum.

Particles initially on a circle of radius $R$ with common tangential speed $v$ move freely on tangent lines. They remain on a circle of radiusrotated through $\theta (t)=\arctan (vt/R)$, while total angular momentum remains $mRv$.

The relative velocity of body 1 with respect to body 2 is ${\mathbf{v}}_{12}={\mathbf{v}}_1-{\mathbf{v}}_2$.