From Derivatives to Action: How Physics Describes Change

/* Images */
  .reveal section img {
    max-width: 85%;
    max-height: 45vh;
    object-fit: contain;
    border-radius: 8px;
  }
  .reveal section img.img-full {
    max-height: 65vh;
    display: block;
    margin: 0 auto;
  }

/* Bumper slides */
  .reveal section.bumper {
    display: flex !important;
    flex-direction: column;
    justify-content: center;
    align-items: center;
    height: 100%;
    text-align: center;
  }
  .reveal section.bumper h1,
  .reveal section.bumper h2 {
    text-align: center;
    width: 100%;
  }
</style>

# From Derivatives to Action

## How Physics Describes Changes

---

# Why Derivatives and Differential Equations?

---
# Most physical phenomena involve change

* Position changes with time
* Temperature changes in space
* Velocity changes due to forces
* Electric fields vary in space and time

If $x(t)$ expresses where we are in function of time

* Velocity: $v = \frac{dx}{dt}$
* Acceleration: $a = \frac{d^2x}{dt^2}$

When describing changes (in space in time...) **derivatives** appear naturally.

---
# Why equations are first or second order?

Because these equations describe
- how a system's state evolves
- based on its current configuration

---
# Why first-order equations?

* Where the rate of change depends on the current value.
    * Radioactive decay
    * Heat transfer (Newton’s Law of Cooling)
    * Percentages ($\frac{dS}{dt} = r \cdot S$)
* The universe is interested in how the "now" dictates the "next moment".

---
# Example: radioactive decay

* We **measure** that the speed at which nuclei disappear is proportional to how many remain.
* Translation: $\frac{dN}{dt} = -\lambda N$

* Solution (exponential decay): $N(t) = N_0 e^{-\lambda t}$

---
# Why second-order equations?

* Because of Newton’s Second Law: $F = ma$
* Which explains that the Force changes the rate of change of the position
* Any law involving force (from gravity to electromagnetism) is naturally second-order
* In wave equations, the second derivative describes how a disturbance "snaps back" toward equilibrium, allowing energy to propagate through space.

---
# The Physics of "Memory"

* A force doesn't change a position directly, it changes the *rate of change* of the position
* $F = m \frac{dv}{dt} = m \frac{d^2x}{dt^2}$
* To predict the future state at $t + \Delta t$, we need to know
    1.  Where the object is ($x$)
    2.  How it got there (Velocity: $\dot{x}$). The "memory" of its previous motion, which dictates its momentum.
* If the universe were governed by first-order equations ($F = mv$), inertia wouldn't exist

---

# Why the same equations everywhere?

---
# Three phenomena, one structure

Different fields, same pattern.

* Temperature: $T(x,t)$
* String displacement: $y(x,t)$
* Quantum wavefunction: $\psi(x,t)$

| System | Equation | Behavior |
| ------- | -------------------------------------------- | ------------ |
| Heat | $\partial_t T = \kappa \nabla^2 T$ | Diffusion |
| Waves | $\partial_t^2 u = c^2 \nabla^2 u$ | Oscillations |
| Quantum | $i\partial_t \psi = -\nabla^2 \psi + V\psi$ | Prob. waves |

Same math ingredient, different physics.

---
# The Laplacian $\nabla^2$
## Measuring local difference

The second spatial derivative measures **curvature**: how different a point is from its neighbors.

* $\nabla^2 u > 0$ : point is colder/lower than neighbors: heat/force flows in
* $\nabla^2 u < 0$ : point is hotter/higher than neighbors: heat/force flows out

| 1D        | 3D |
| :-------: | :--------------------------------------------: |
| $\nabla^2 u = \frac{d^2u}{dx^2}$ | $\nabla^2 u = \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2}$ |

---
# Why?
## Locality + Symmetry + Conservation

Physics equations are the simplest ones compatible with:

1. **Locality**: what happens here depends on neighbors, not the whole universe
2. **Symmetry**: laws must be invariant under translations and rotations
3. **Conservation laws**: energy, momentum, charge

The simplest differential operator satisfying all three constraints is the **Laplacian** $\nabla^2$.

Which explains why the same equations appear in heat, waves, quantum mechanics, and electromagnetism.

---

# From local to global: variational principles

---
# Two languages for the same physics

**Local (differential equations):**
> If the system is like *this* right now, it will start changing like *that*.

Step-by-step evolution from one instant to the next.

**Global (variational principle):**
> Among all possible paths connecting A to B, which one does nature choose?

The entire trajectory is considered at once.

Both languages are **equivalent**, they encode the same physics.

---
# What is a variational principle?

Any statement of the form:

> *The physical solution is the one that makes a certain quantity stationary.*

$$\delta \mathcal{F} = 0$$

Historical first. Fermat's principle (~1660):

> *Light follows the path of stationary travel time.*

This explains both reflection and refraction from a single principle.

From optics to mechanics: Maupertuis, Euler, Lagrange, Hamilton all searched for the same idea.

---

# The principle of stationary action

---
# Action: a "score" for each trajectory

For any possible path, we compute a single number, the **action**:

$$S = \int_{t_1}^{t_2} L(x, \dot{x}, t)\, dt$$

* $L = T - V$ is the **Lagrangian** (kinetic minus potential energy)
* $T = \frac{1}{2}m\dot{x}^2$ and $V(x)$ = potential energy

The principle:

> *The real trajectory is the one that makes $S$ stationary: $\delta S = 0$*

Not arbitrary. $L = T - V$ is the **unique** combination that reproduces Newton's laws.

---
# Example: $F = m\ddot{x}$

Starting from $L = \frac{1}{2}m\dot{x}^2 - V(x)$ and requiring $\delta S = 0$:

1. Vary the path: $x(t) \to x(t) + \varepsilon\eta(t)$, with $\eta(t_1) = \eta(t_2) = 0$
2. Expand to first order, integrate by parts (boundary terms vanish)
3. Require $\delta S = 0$ for **any** variation $\eta$

This gives the **Euler-Lagrange equation**:

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

Substituting $L$ it comes $\quad  F = -\frac{dV}{dx} = m\ddot{x}$

---
# Noether's theorem (1915)

> *Every symmetry of the action gives a conservation law.*

| Symmetry | Conservation law |
| -------- | ---------------- |
| Time invariance | Energy |
| Space invariance | Momentum |
| Rotation invariance | Angular momentum |

Newton's laws do not *explain* why energy is conserved. The action framework does.

---
# One principle, all of fundamental physics

Starting from $\delta S = 0$ with the right Lagrangian:

* **Newton's laws**: classical mechanics
* **Maxwell's equations**: electromagnetism
* **Schrodinger equation**: quantum mechanics
* **Einstein's field equations**: general relativity

Light itself is a consequence: combining Maxwell's equations from the EM action gives a wave equation with speed $c = \frac{1}{\sqrt{\mu_0\varepsilon_0}}$.

---

# The quantum connection

---
# Feynman path integral
## All paths contribute

Classical: one path, the one where $\delta S = 0$.

Quantum (Feynman, ~1948): a particle "explores" **all** possible paths.

Each path contributes an amplitude: $e^{iS/\hbar}$

* $S$ = action for that path
* $\hbar$ = reduced Planck constant (very small)

The total amplitude is the **sum over all paths**.

---
# Why the classical path survives?

When $\hbar \to 0$ (macroscopic scale):

* Near the stationary-action path, phases vary slowly, **constructive interference**
* Away from it, phases oscillate rapidly, **destructive interference**, cancels out

The classical trajectory ($\delta S = 0$) is the one where neighboring paths interfere constructively.

Classical mechanics **emerges** from quantum mechanics as a limit.

---

# Take away

---
# Take away

* Physics uses derivatives because nature evolves **continuously**: derivatives are the language of change
* Most laws are **2nd order** because systems have inertia. They remember their velocity
* The same equations (heat, waves, quantum) arise from **locality + symmetry + conservation**
* The Laplacian $\nabla^2$ is the universal operator: it measures how a point differs from its neighbors
* Differential equations (local) and variational principles (global) are **two equivalent languages**
* The action $S = \int (T-V)\,dt$ encodes all of classical physics in a single principle: $\delta S = 0$
* **Noether's theorem**: every symmetry of the action generates a conservation law
* Quantum mechanics sums all paths weighted by $e^{iS/\hbar}$ and the classical path emerges by constructive interference