← All parts of this equation

Equation 4 · Part 5 · Physics Optimizes, Evolution Satisfices: Two Kinds of Perfection

=

x¨μ+Γαβμx˙αx˙β=0\ddot{x}^\mu + \Gamma^\mu_{\alpha\beta}\dot{x}^\alpha\dot{x}^\beta = 0
=

What this part means

The expressions on both sides represent the same quantity under the stated assumptions.

Its job in the formula

The equals sign connects the complete expression on the left with the complete expression on the right. Both sides must have compatible units.

The passage around this formula

Snell’s law, in other words, is not an independent fact about glass. It is what falls out of a single global stipulation — least time — applied to one path among infinitely many. Pierre-Louis Maupertuis generalized the same move to all of mechanics in the 1740s, proposing that nature always acts by the smallest possible “action,” and Euler, Lagrange, and Hamilton spent the following century turning that guess into the working machinery of classical physics: a moving body’s entire trajectory falls out of minimizing one integrated quantity over the whole path, not from adding up forces moment to moment. General relativity is this idea pushed as far as it goes. Einstein’s field equations recast…

Read this part in the article →

Learn the underlying idea

An equals sign says that the expression on its left and the expression on its right have the same value under the stated definitions and assumptions.

Open the illustrated equality: what the equals sign claims guide →

Sources cited in the surrounding passage

These citations provide research context; check each source for the exact claim it supports.