Classical mechanics

Level UniversityDifficulty ★★★★★Application⌖ Open in the map

What is it?

Newton's second law $F = m\ddot x$ turns forces into a second-order ODE for the motion. Gravity, springs, friction and collisions are all modelled this way; energy and momentum are the conserved quantities that check a simulation.

Formulas

$m\,\ddot x = F(x, \dot x, t), \qquad F_{\text{grav}} = -\frac{G\,m_1m_2}{r^2}\,\hat r$; $E = \tfrac12 m\norm{\dot x}^2 + U(x), \qquad \frac{\dd E}{\dd t} = 0$
energy is conserved when the force is conservative, $F = -\nabla U$

The mathematics behind it

Ordinary differential equations★★★★★fundamental
Newton's second law is a second-order ODE.
Second-order linear equations (oscillations)★★★★★fundamental
The harmonic oscillator is the model of every small vibration around an equilibrium.
Gradient★★★★★frequent
Conservative forces are minus the gradient of a potential: $F = -\nabla U$ .
Line integrals★★★★★frequent
Work and potential energy; energy conservation for conservative forces.

Where is it used?

Computing topics reachable from here, through the chain of ideas that leads to them:

⚛ Physics and simulation

Physics engines★★★★★
Physics engines→N-body gravitational simulation★★★★★
Physics engines→Fluid dynamics and CFD★★★★★
Physics engines→Fluid dynamics and CFD→Weather and climate modelling★★★★★

What depends on it

Physics engines

Exercises

1Computation

A ball is thrown up at 20 m/s ( $g = 9.8$ m/s²). Using calculus, find the maximum height and the time to reach it.

Solution

$y(t) = 20t - 4.9t^2$ ; $y'(t) = 20 - 9.8t = 0 \Rightarrow t \approx 2.04$ s, $y \approx 20.4$ m.

This page has the essentials. A fuller treatment (intuition, formal definition, worked example) is on the way.