Numerical integration in the context of Numerical analysis

Molecular dynamics (MD) is a computer simulation method for analyzing the physical movements of atoms and molecules. The atoms and molecules are allowed to interact for a fixed period of time, giving a view of the dynamic "evolution" of the system. In the most common version, the trajectories of atoms and molecules are determined by numerically solving Newton's equations of motion for a system of interacting particles, where forces between the particles and their potential energies are often calculated using interatomic potentials or molecular mechanical force fields. MD simulations are widely applied in chemical physics, materials science, and biophysics.

Because molecular systems typically consist of a vast number of particles, it is impossible to determine the properties of such complex systems analytically; MD simulation circumvents this problem by using numerical methods. However, long MD simulations are mathematically ill-conditioned, generating cumulative errors in numerical integration that can be minimized with proper selection of algorithms and parameters, but not eliminated.

In the branch of mathematics known as real analysis, the Riemann integral, created by Bernhard Riemann, was the first rigorous definition of the integral of a function on an interval. It was presented to the faculty at the University of Göttingen in 1854, but not published in a journal until 1868. For many functions and practical applications, the Riemann integral can be evaluated by the fundamental theorem of calculus or approximated by numerical integration, or simulated using Monte Carlo integration.

In geophysics and physical geodesy, a geopotential model is the theoretical analysis of measuring and calculating the effects of Earth's gravitational field (the geopotential).The Earth is not exactly spherical, mainly because of its rotation around the polar axis that makes its shape slightly oblate. However, a spherical harmonics series expansion captures the actual field with increasing fidelity.

If Earth's shape were perfectly known together with the exact mass density ρ = ρ(x, y, z), it could be integrated numerically (when combined with a reciprocal distance kernel) to find an accurate model for Earth's gravitational field. However, the situation is in fact the opposite: by observing the orbits of spacecraft and the Moon, Earth's gravitational field can be determined quite accurately. The best estimate of Earth's mass is obtained by dividing the product GM as determined from the analysis of spacecraft orbit with a value for the gravitational constant G, determined to a lower relative accuracy using other physical methods.

5261 Eureka is the first Mars trojan discovered. It was discovered by David H. Levy and Henry Holt at Palomar Observatory on 20 June 1990. It trails Mars (at the L₅ point) at a distance varying by only 0.3 AU during each revolution (with a secular trend superimposed, changing the distance from 1.5–1.8 AU around 1850 to 1.3–1.6 AU around 2400). Minimum distances from Earth, Venus, and Jupiter, are 0.5, 0.8, and 3.5 AU, respectively.

Long-term numerical integration shows that the orbit is stable. Kimmo A. Innanen and Seppo Mikkola note that "contrary to intuition, there is clear empirical evidence for the stability of motion around the L₄ and L₅ points of all the terrestrial planets over a timeframe of several million years".