Early undergraduateApplied MathematicsPlanned
Classical mechanics
Newton's laws as differential equations, conservation of energy and momentum, oscillators, central forces and Kepler's laws, and an introduction to Lagrangian mechanics.
math.applied.classical-mechanics
The lesson for this topic has not been written yet. The outline below is part of the curriculum map and has not been reviewed.
Builds on
Required prerequisites, all the way down: 38
You will learn to
- Solve the equations of motion of projectiles and of harmonic oscillators, with and without damping.
- Use conservation of energy and angular momentum to analyse motion under a central force and derive Kepler's laws.
- Derive equations of motion from a Lagrangian using the Euler–Lagrange equations.
Definition
- Newton's second law: F = ma, where the acceleration a is the second derivative of position.
- A force field F is conservative if F = −∇V for some potential V.
- The Lagrangian of a system is L = T − V, kinetic energy minus potential energy.
Theorem
- For conservative forces the total energy T + V is conserved.
- Under a central force, angular momentum is conserved and the motion stays in a plane.
- Bound orbits under an inverse-square attractive force are ellipses (Kepler's first law).
Common misconceptions
- Thinking a force is needed to keep an object moving at constant velocity.
- Confusing mass with weight.
Optional discoveries
- Smooth manifolds Configuration spaces are manifolds, and Hamiltonian mechanics lives on cotangent bundles with their symplectic structure.
Learning pathways
Further reading
- V. I. Arnold, Mathematical Methods of Classical Mechanics
- John R. Taylor, Classical Mechanics (2005)