Early undergraduateApplied MathematicsLesson available
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
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 with non-zero angular momentum under an inverse-square attractive force are ellipses (Kepler's first law); with zero angular momentum the motion is along a line.
Common misconceptions
- Thinking a force is needed to keep an object moving at constant velocity.
- Confusing mass with weight.
Leads to
Optional discoveries
- Smooth manifolds Configuration spaces are manifolds, and Hamiltonian mechanics lives on cotangent bundles with their symplectic structure.
- ← Conic sections
Learning pathways
Further reading
- V. I. Arnold, Mathematical Methods of Classical Mechanics
- John R. Taylor, Classical Mechanics (2005)