¶Differential Calculus and Rates of Change

Calculus, independently developed by Isaac Newton and Gottfried Wilhelm Leibniz in the late 17th century, is the mathematical study of continuous change. Differential calculus focuses on the concept of the derivative, which measures the instantaneous rate of change of a quantity with respect to another.

¶The Derivative

If a function $s(t)$ represents the position of an object moving along a line at time $t$, the average velocity over a time interval $\Delta t$ is $\frac{\Delta s}{\Delta t}$. The instantaneous velocity $v(t)$ is defined as the limit of this average velocity as the time interval approaches zero: $v(t) = \lim_{\Delta t \to 0} \frac{s(t + \Delta t) - s(t)}{\Delta t} = \frac{ds}{dt}$

¶Higher-Order Derivatives

Derivatives can themselves be differentiated. Acceleration $a(t)$ is the rate of change of velocity, and therefore the second derivative of position: $a(t) = \frac{dv}{dt} = \frac{d^2s}{dt^2}$

¶Numerical Integration (Euler's Method)

In computational physics, analytical solutions to complex differential equations (like chaotic multi-body gravitational systems) are often impossible. Instead, algorithms perform numerical integration to approximate the solution over small discrete time steps ($\Delta t$).

Euler's method is the simplest first-order numerical procedure: $s(t_{n+1}) \approx s(t_n) + v(t_n) \Delta t$ $v(t_{n+1}) \approx v(t_n) + a(t_n) \Delta t$

While Euler's method provides the foundation for understanding computational integration, modern spacecraft simulators utilise more stable and accurate higher-order algorithms like Runge-Kutta (RK4) to minimize accumulated truncation errors over time.

Reference: Stewart, J. (2015). Calculus: Early Transcendentals (8th ed.). Cengage Learning.

¶References

• Calculus (4th Edition) (Michael Spivak)
• Calculus: Early Transcendentals (James Stewart)
• MIT OCW: 18.01 Single Variable Calculus