Favorite Physics Equations

Here are some of my favorite equations (no particular order).

1) The Drake Equation$$ N = R_* · f_p · n_e · f_1 · f_i · f_c · L$$

What it is/why I love it: Theoretically computes the number of civilizations (with which communication might be possible) in our galaxy. Doesn't have much practical use but always gets me wondering.

Variables:

• $N$ : Number of civilizations in our galaxy where communication might be possible
• $R_*$: Rate of star formation in our galaxy
• $f_p$ : Fraction of stars in our galaxy with planets in their system
• $n_e$ : Fraction of planets (per each star with associated planets) that could potentially support life
• $f_1$ : Fraction of planets that could support life which actually develop life
• $f_i$ : Fraction of planets with intelligent life to planets with any life
• $f_c$ : Fraction of civilizations that broadcast detectable signals into space to all civilizations
• $L$ : Length of time such f_c civilizations broadcast

2) Newton's Second Law$$F = m · a$$

What it is/why I love it: Calculates net force on an object. When I studied physics, my professors would relate everything back to this equation. No matter how complicated, it seemed like every equation could be related in some way to this simple equation. Also, if you know the mass and the acceleration vector of something, in a way, you literally know everything about it.

Variables:

• $F$: Net force
• $m$ : Mass of the object
• $a$ : Acceleration of the object

3) Heisenberg Uncertainty Principle$$\Delta x · \Delta p \ge \frac{h}{4 · \pi}$$

What it is/why I love it: This equation literally has people questioning life itself. Some people refuse to believe in this because it means that there is always an inherent uncertainty surrounding specific particles. Their position and momentum can not both be known exactly. Ever.

Variables:

• $\Delta x$ : Uncertainty in position
• $\Delta p$ : Uncertainty in momentum
• $h$ : Planck constant

4) Second Law of Thermodynamics$$\Delta S \ge 0 $$

What it is/why I love it: This equation means (net) entropy, "chaos" if you will, always increases in the universe. Awesome...

Variables:

• $\Delta S$ : Change in entropy

5) Definition of complex numbers$$ i^2 = -1 $$

What it is/why I love it: Opens up an entire new area of mathematics necessary in fields like quantum mechanics.

Variables:

• $i$ : Imaginary number defined as the square root of -1

6) Shapley Value$$\phi _i (v) = \sum_{S \subseteq (N \setminus {i})} \frac{\lvert S \rvert ! · (n - \lvert S \rvert - 1)!}{n!} · (v(S \cup \{i\}) - v(S)))$$

What it is/why I love it: Game theory, in a sense the study of decision making, has two sub-fields: cooperative (participants are working towards a common goal) and non-cooperative (participants are competing for something). The Shapley value, calculated in cooperative decision making, produces the theoretically ideal value for the payoff for each player in the coalition of participants. It produces the fairest distributions of amassed assets AKA the best communal outcome. I am very interested in the science of decision making (see this article of mine), so I love the fact that there is a beautiful, thorough equation that can be used to make a theoretically-irrefutable best decision.

Variables:

• $\phi _i$ : The value player $i$ gets using the "fairest" distribution of total assets gathered.
• $v(S)$ : A function that takes a subset of the overall set of players $N$ and produces a real number. Function $v(S)$ produces a real number, called the "worth" of subset $S$, representing the total expected payoff if all members in subset $S$ work together. Note $v$ must satisfy $v(\emptyset ) = 0$.
• $N$ : A set of $n$ players.
• $n$ : The total number of players.

Honorable mentions

• Taylor's theorem
• Schrödinger equation
• Pythagorean theorem
• Maxwell's equations
• Fundamental theorem of calculus