古典力学 Complete Note

Purpose

This is a test note written as a normal LaTeX file inside the project. It should appear in the Complete notes section without manually moving it into the Astro content folder.

Action Principle

The starting point is the action

$$S[q] = \int_{t_1}^{t_2} L(q,\dot q,t)\,dt.$$

The physical path is selected by the stationary condition

$$\delta S = 0.$$

Euler—Lagrange Equation

For generalized coordinates $q_i$, this gives

$$\frac{d}{dt}\frac{\partial L}{\partial \dot q_i} - \frac{\partial L}{\partial q_i} = 0.$$

Map Connections

This complete note should connect to the atomic notes on Lagrangian mechanics and relativity through its metadata.