本科低年级Applied Mathematics计划中

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

显示到达此主题所需的全部内容

本主题的课程尚未编写。下面的提纲是课程地图的一部分，尚未经过审校。

前置知识

Ordinary differential equations Multivariable and vector calculus

38 项必需的前置知识（一直追溯到底）

你将学会

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.

定义

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.

定理

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).

常见误解

Thinking a force is needed to keep an object moving at constant velocity.
Confusing mass with weight.

可选发现

Smooth manifolds Configuration spaces are manifolds, and Hamiltonian mechanics lives on cotangent bundles with their symplectic structure.

学习路径

Applied mathematics