From Derivatives to Action: How Physics Describes Change
Building intuition from local laws, differential equations, and variational principles
🚧 This post is under construction. First draft 🚧
TL;DR
- Action \(S = \int_{t_1}^{t_2} L(x,\dot x,t),dt\)
- Lagrangian \(L\)
Table of Contents
- Introduction
- Why derivatives and differential equations in physics?
- Why the heat equation, the wave equation and the Schrödinger equation look so similar?
- A deeper modern perspective
- Show me how, using the principle of stationary action (least action), we can recover \(F=ma\)?
- Why do we call it Action in Physics?
- Why \(L = T - V\)?
- Mise en application sur une balle qui tombe sous l’effet de la gravité
- We can explain why a ball falls vertically while accelerating?
- How Maxwell’s equations can be derived from a variational principle?
- All of this is quite reminiscent of Feynman integrals. No?
- Conclusion
- Webliography
Space: the final frontier.
These are the voyages of a curious mind, on a lifelong quest to decode the deep architecture of reality.
Its ongoing mission: to explore why the laws of physics are written in the language of change… to uncover how motion itself is governed by the beauty of variation… and to trace the hidden principle that dictates every path through the cosmos.
To look beyond equations as mere symbols, to see in them the signature of a deeper order… where, among all possible worlds, nature chooses the one that speaks of balance.
Introduction
It’s on the way, just taking a little longer than expected.
Why derivatives and differential equations in physics?
Can we take the time to explain why we so often encounter first- or second-order differential equations, as well as derivatives, in physics formulas? I understand \(F=ma\), but I don’t understand how it was derived, from where it comes from… Same thing for the wave equation or heat propagation equation, etc.
Great question Marty! The short answer is: derivatives appear in physics because nature evolves continuously in time and space, and differential equations express how a system changes locally. But it’s worth unpacking this slowly from first principles.
1. Physics tries to describe how things change
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
Mathematically, the cleanest way to describe change is with a derivative.
For a quantity \(x(t)\):
-
Velocity: \(v = \frac{dx}{dt}\)
-
Acceleration: \(a = \frac{d^2x}{dt^2}\)
So the moment we want to describe motion**, derivatives appear naturally.
2. Why equations are often first or second order
This is not arbitrary. It comes from how many levels of change influence the system.
First derivative → flow / transport
If a phenomenon depends only on the rate of change**, we get first-order equations.
Example: radioactive decay
\[\frac{dN}{dt} = -\lambda N\]Interpretation:
The speed at which nuclei disappear is proportional to how many remain.
This law is local and simple:
- more atoms → more decay events per second
This leads to exponential decay.
Second derivative → dynamics / inertia
Second derivatives appear when acceleration matters.
Example:
\[F = ma\]But acceleration is
\[a = \frac{d^2x}{dt^2}\]So Newton’s law becomes
\[F = m \frac{d^2x}{dt^2}\]Why second order?
Because inertia exists. A body remembers its velocity.
To know the future motion, we must know:
- position
- velocity
Two initial conditions → second order equation.
3. But how did Newton discover \(F = ma\)?
Important point:
Newton did not guess the equation randomly.
It emerged from experiments + symmetry principles.
Observations known before Newton
Galileo had already measured that:
- objects fall with constant acceleration**
- motion without forces → constant velocity
This implies:
\[a = \text{constant when force is constant}\]Then experiments showed:
- double the force → double the acceleration
- double the mass → half the acceleration
So empirically:
\[a \propto \frac{F}{m}\]Which gives
\[F = ma\]So it is really an empirical law extracted from experiments.
4. Why physics prefers local laws**
Most physical laws describe local behavior:
Instead of saying
“The temperature everywhere depends on the whole system”
Nature usually says something simpler:
“What happens here depends on what is happening just next to it.”
This locality leads directly to derivatives.
Example:
Temperature \(T(x,t)\)
The heat flowing through a point depends on the temperature gradient:
\[\text{Heat flow} \propto - \nabla T\]That gradient is a derivative.
5. Example: heat equation
The heat equation is
\[\frac{\partial T}{\partial t} = \kappa \nabla^2 T\]This looks mysterious but comes from two simple physical statements.
1 Heat flows from hot to cold
(Fourier law)
\[q = -k \nabla T\]Heat flux depends on temperature gradient.
2 Energy is conserved
Change of temperature = heat entering − heat leaving.
Mathematically this gives
\[\frac{\partial T}{\partial t} = - \nabla \cdot q\]Substitute \(q\):
\[\frac{\partial T}{\partial t} = \kappa \nabla^2 T\]So the second derivative appears because:
- heat flow depends on gradient**
- conservation introduces divergence**
Gradient of gradient → Laplacian.
6. Why differential equations appear everywhere?
Almost every physical theory has three ingredients:
1. Locality
What happens here depends on nearby values.
→ derivatives in space
2. Continuous time evolution
Future depends on current rate of change.
→ derivatives in time
3. Conservation laws
Energy, momentum, charge, mass…
These are expressed mathematically using divergences and derivatives.
7. Why order 2 is extremely common
Most fundamental laws in physics are second order because:
- systems have inertia**
- energy depends on velocity squared**
- conservation of momentum naturally leads to acceleration
Examples:
- Newtonian mechanics
- wave equation
- Schrödinger equation (first order in time but second in space)
- Maxwell equations (effectively second order)
8. Summary
We see derivatives everywhere because physics tries to answer:
How does the state of the system change locally**?
Derivatives are simply the mathematical language of change.
And differential equations express:
The rate of change of something depends on the current state of the system.
A good mental model is:
Physics laws are like rules of evolution.
They don’t tell us the whole trajectory.
They only say:
If the system is like this right now**, then it will start changing like that.
That is exactly what a differential equation describes.
Why the heat equation, the wave equation and the Schrödinger equation look so similar?
Once we see the common structure, a lot of physics suddenly becomes less mysterious.
We’ll go step by step.
1. The common mathematical structure
Many physical phenomena involve a quantity that depends on space and time:
\[u(x,t)\]Examples:
- temperature \(T(x,t)\)
- displacement of a string \(y(x,t)\)
- quantum wavefunction \(\psi(x,t)\)
In all three cases, the evolution of the system takes the form: v \(\text{time change} = \text{spatial variation}\)
The spatial variation is usually measured by the Laplacian:
\[\nabla^2\]which is basically the second derivative in space.
Why the second derivative? Because it measures curvature.
2. The physical meaning of the second spatial derivative
Imagine a function (u(x)).
The second derivative tells us whether the value at a point is:
- higher than its neighbors
- lower than its neighbors
Graphically:
- (u’’ > 0) → valley
- (u’’ < 0) → hill
So the second derivative tells us how different a point is from its surroundings.
And many physical processes try to reduce those differences or react to them.
That’s why the Laplacian appears everywhere.
3. Example 1: heat diffusion
The heat equation is
\[\frac{\partial T}{\partial t} = \kappa \nabla^2 T\]Interpretation:
- if a point is hotter than its neighbors**, heat flows away
- if a point is colder than neighbors**, heat flows in
So temperature differences smooth out over time.
The Laplacian measures exactly that difference.
Result: the system diffuses.
4. Example 2: waves
The wave equation is
\[\frac{\partial^2 u}{\partial t^2} = c^2 \nabla^2 u\]Example: vibrating string.
Here the interpretation is different.
If a point on the string is curved**, tension pulls it back toward equilibrium.
More curvature → stronger restoring force.
That force produces acceleration**, which is why the time derivative is second order.
So:
- curvature → force
- force → acceleration
Thus:
\[\text{acceleration} \propto \text{curvature}\]This produces waves instead of diffusion.
5. Example 3: quantum mechanics
The Schrödinger equation is
\[i\hbar \frac{\partial \psi}{\partial t} = -\frac{\hbar^2}{2m} \nabla^2 \psi + V\psi\]This also contains the Laplacian.
Why?
Because the Laplacian corresponds to kinetic energy.
In fact, the kinetic energy operator is:
\[-\frac{\hbar^2}{2m}\nabla^2\]So again:
- spatial curvature determines how the wavefunction evolves.
The mathematics ends up looking surprisingly similar to diffusion or waves.
6. The deep unifying idea
In many physical systems:
\[\text{time evolution} = \text{spatial curvature}\]The curvature represents how different a point is from nearby points.
Depending on the physics:
| System | Equation | Behavior |
|---|---|---|
| Heat | \(\partial_t T = \kappa \nabla^2 T\) | Diffusion |
| Waves | \(\partial_t^2 u = c^2 \nabla^2 u\) | Oscillations |
| Quantum | \(i\partial_t \psi = -\nabla^2 \psi + V\psi\) | Probability waves |
Same mathematical ingredient, different physics.
7. Why nature loves local curvature laws**
There are several deep reasons.
1. Local interactions
Atoms only interact with nearby atoms.
So the state at a point depends on neighbors**, not the whole universe.
The Laplacian is precisely the operator that compares a point to its neighbors.
2️. Symmetry
Physical laws must be invariant under:
- translations
- rotations
The simplest operator with those symmetries is the Laplacian.
3️. Energy minimization
Many systems evolve to minimize energy.
Energy often contains terms like
\[(\nabla u)^2\]When we minimize such an energy functional, the resulting equation contains
\[\nabla^2 u\]So the Laplacian appears naturally from variational principles.
8. A powerful intuition
We can think of the Laplacian as:
difference between a point and the average of its neighbors**
In fact, on a discrete grid:
\[\nabla^2 u(x) \approx u_{\text{neighbors}} - u(x)\]So the equations basically say:
- heat: move toward neighbor average
- waves: accelerate toward neighbor average
- quantum: evolve based on neighbor difference
9. Why physics equations look simple
Another surprising fact:
The real world equations are often the simplest possible ones.
Physicists usually assume:
- locality
- symmetry
- smoothness
- conservation laws
When we write the simplest equation compatible with these principles**, we often end up with:
- first or second derivatives
- Laplacians
- linear terms
Which explains why the same equations appear everywhere.
10. Summary
Derivatives appear in physics because:
- physics describes change**
- laws are local**
- systems react to differences with neighbors**
- curvature (second derivative) measures those differences
That’s why the same structures appear in:
- heat
- waves
- quantum mechanics
- electromagnetism
- fluid dynamics
A deeper modern perspective
It is important to understand that:
The principle of least (or stationary) action is one specific instance of a much broader idea: variational principles.
Let’s unpack that carefully.
1. What is a variational principle (in general)?
A variational principle is any statement of the form:
“The physical solution is the one that makes a certain quantity stationary (usually an extremum this means a minimum or a maximum).”
Mathematically:
\[\delta \mathcal{F} = 0\]where \(\mathcal{F}\) is some functional (a function of functions).
So the structure is:
- We define a quantity depending on a function (trajectory, field, shape…)
- We require that small variations do not change it at first order
This idea exists far beyond physics.
2. The principle of least action = a specific variational principle
The principle of least action is just the case where:
\[\mathcal{F} = S = \int L\,dt\]So:
\[\delta S = 0\]This gives:
- Newton’s laws
- Maxwell’s equations
- Schrödinger equation
- General relativity
It’s extremely powerful but conceptually, it’s just one member of a larger family.
Historically, it was developed by people like Pierre-Louis Maupertuis and later formalized by William Rowan Hamilton.
3. Other variational principles in physics
There are many important examples that are not phrased as “action minimization”**, even though some can be reformulated that way.
1. Fermat’s principle (optics)
Associated with Pierre de Fermat
\[\delta \int n(s)\,ds = 0\]Interpretation:
Light follows the path that extremizes travel time.
This explains:
- refraction
- reflection
This is actually an action principle for light**, but historically it came first and looks different.
2. Principle of minimum potential energy
In statics:
\[\delta U = 0\]A system at equilibrium minimizes its potential energy.
Examples:
- a hanging chain
- elastic structures
- equilibrium configurations
No time involved — this is not an “action over time”, just a spatial variational principle.
3. Principle of virtual work
Used in mechanics and engineering:
\[\sum F_i \cdot \delta x_i = 0\]Interpretation:
For equilibrium, virtual displacements produce no net work.
This is another variational formulation of mechanics.
4. Least dissipation / entropy principles
In thermodynamics and statistical physics:
- minimum entropy production (near equilibrium)
- Onsager’s principle
These are variational principles involving irreversible processes**, not classical action.
5. Rayleigh–Ritz method
Used in quantum mechanics and engineering:
\[E[\psi] = \frac{\langle \psi | H | \psi \rangle}{\langle \psi|\psi\rangle}\]Minimizing this gives approximations of energy levels.
This is a variational method**, not a fundamental law, but it uses the same idea.
4. So what makes the action special?
This is a good question because, in effect, not all variational principles are equal. The action principle is special because:
1. It applies to dynamics (time evolution)
Many variational principles describe equilibrium.
The action principle describes how systems evolve in time.
2️. It is extremely general
It works for:
- particles
- fields
- relativity
- quantum theory
3. It encodes symmetries
Through Emmy Noether’s theorem:
Every symmetry of the action corresponds to a conservation law.
Examples:
- time invariance → energy conservation
- space invariance → momentum conservation
- rotation invariance → angular momentum
This is a deep structural reason physicists love the action.
5. Modern viewpoint
In modern physics, the hierarchy is roughly:
-
Variational principles (very general idea)
-
Action principle (central, universal in fundamental physics)
- specific Lagrangians → specific theories
-
So yes:
The principle of least action is a particular realization of a much broader conceptual framework.
6. A deeper intuition
Why do variational principles appear at all?
One way to think about it:
Instead of describing physics as:
“local cause → local effect”
We describe it as:
“global constraint on all possible histories”
Then the real trajectory is the one that satisfies that constraint.
It’s a very different viewpoint:
- Newton: local differential equation
- Lagrange/Hamilton: global optimization over paths
Yet they give the same results.
7. Subtle but important point
“Least action” is slightly misleading.
In reality:
\[\delta S = 0\]means:
- minimum
- maximum
- or saddle point
So the correct name is:
principle of stationary action**
8. Big picture
We can think of it like this:
- Differential equations → local description**
- Variational principles → global description**
They are two equivalent ways of encoding the same physics.
Final takeaway
- Variational principles are a general mathematical framework**
- The principle of least action is one specific (and extremely powerful) example**
- Many other principles (optics, statics, thermodynamics) fit into the same pattern
- Modern physics is largely built on the action because of its universality and symmetry properties**
Show me how, using the principle of stationary action (least action), we can recover \(F=ma\)?
This is one of the most beautiful things in theoretical physics: Newton’s law can be derived from a variational principle. I’ll go carefully and start from the basics so every step makes sense.
The key idea comes from the Pierre-Louis Maupertuis principle**, later generalized by William Rowan Hamilton and Joseph-Louis Lagrange.
The principle says:
The trajectory of a physical system is the one that makes the action stationary.
1. Define the action
We introduce a quantity called the action:
\[S = \int_{t_1}^{t_2} L(x,\dot x,t),dt\]where:
- \(x(t)\) = position
- \(\dot x = dx/dt\) = velocity
- \(L\) = Lagrangian**
For a particle in a potential \(V(x)\):
\[L = T - V\]where:
- \(T = \frac{1}{2} m \dot{x}^2\) (kinetic energy)
- \(V(x)\) (potential energy)
So:
\[L(x,\dot x) = \frac12 m\dot x^2 - V(x)\]2. The principle of stationary action
Nature selects the path (x(t)) such that the action is stationary:
\[\delta S = 0\]Meaning: small variations of the trajectory do not change the action at first order.
We therefore consider a slightly modified path:
\[x(t) + \epsilon \eta(t)\]where:
- \(\epsilon\) is small
- \(\eta(t)\) is an arbitrary function
- \(\eta(t_1)=\eta(t_2)=0\) (endpoints fixed)
3. Compute how the action changes
The action becomes
\[S(\epsilon) = \int_{t_1}^{t_2}L(x+\epsilon\eta,\dot x+\epsilon\dot\eta,t) \, dt\]Now differentiate with respect to (\epsilon):
\[\frac{dS}{d\epsilon} = \int\left(\frac{\partial L}{\partial x}\eta+\frac{\partial L}{\partial \dot x}\dot\eta\right) \, dt\]Then we impose:
\[\frac{dS}{d\epsilon}=0\]4. Remove the derivative on \(\dot{\eta}\)
We integrate by parts:
\[\int \frac{\partial L}{\partial \dot x}\dot\eta dt = \left[\frac{\partial L}{\partial \dot x}\eta \right]_{t_1}^{t_2} - \int\frac{d}{dt}\left(\frac{\partial L}{\partial \dot x} \right)\eta dt\]The boundary term vanishes because:
\[\eta(t_1)=\eta(t_2)=0\]So the variation becomes
\[\delta S = \int \left(\frac{\partial L}{\partial x} - \frac{d}{dt} \frac{\partial L} {\partial \dot x} \right) \eta(t) dt\]5. Euler–Lagrange equation
Because (\eta(t)) is arbitrary, the only way the integral can always vanish is if:
\[\frac{d}{dt} \frac{\partial L}{\partial \dot x} - \frac{\partial L}{\partial x} = 0\]This is the Euler–Lagrange equation.
This equation is the core of Lagrangian mechanics.
6. Insert the Lagrangian
Recall:
\[L = \frac12 m\dot x^2 - V(x)\]First term
\[\frac{\partial L}{\partial \dot x} = m\dot x\]So
\[\frac{d}{dt}(m\dot x) = m\ddot x\]Second term
\[\frac{\partial L}{\partial x} = -\frac{dV}{dx}\]7. The resulting equation
Insert into Euler–Lagrange:
\[m\ddot x - (-\frac{dV}{dx}) = 0\]which gives
\[m\ddot x = -\frac{dV}{dx}\]But the force in a potential is
\[F = -\frac{dV}{dx}\]So we obtain:
\[F = m\ddot x\]Which is exactly Newton’s second law.
8. Why this is profound
This derivation shows something remarkable:
Newton’s laws are not fundamental, they are consequences of a deeper principle.
The deeper principle is:
Physical systems extremize the action.
This framework generalizes to almost all physics:
- classical mechanics
- optics
- quantum mechanics
- quantum field theory
- general relativity
9. Intuition for why the action works
The action accumulates the difference:
\[T - V\]over time.
The system chooses the trajectory that balances:
- kinetic energy
- potential energy
so that small deviations do not reduce the action.
In some sense, the system finds the most economical path in spacetime.
10. Summary
- Define the action
- Use the Lagrangian
- Require stationary action
- Obtain Euler–Lagrange equation
- Substitute \(L\)
→ gives
\[F = m\ddot x\]Why do we call it Action in Physics?
This is not a weird question at all and this actually goes straight into the history and philosophy of physics.
1. Where the word “action” comes from
The term “action” was introduced in the 18th century, mainly by Pierre-Louis Moreau de Maupertuis (around 1744).
He proposed an early version of a least-action principle and used the word action to describe a quantity that measures the “effort” or “activity” of nature along a path.
At the time, physics was still heavily influenced by philosophical ideas like:
- Nature acts in the most efficient way
- God or nature minimizes effort or “expenditure”
So “action” was meant to sound like:
how much “doing” or “effort” happens during a motion
2. What Maupertuis meant by “action”
His original definition wasn’t exactly \(\int (T - V) \, dt\). He defined action roughly as:
\[\text{action} \sim \int p , dq\]- (p) = momentum
- (q) = position
So already, action was something that:
- depends on the whole path**
- accumulates along motion
- measures something like “quantity of motion × distance”
3. Who formalized it
The modern definition came later with:
- Joseph-Louis Lagrange**
- William Rowan Hamilton**
Hamilton (1830s) is the one who really established:
\[S = \int (T - V) \, dt\]and turned “action” into a central, precise mathematical object.
4. Why the name stuck
The name “action” survived even though the meaning became more abstract, for a few reasons:
- It still represents something accumulated over motion**
- It connects to energy and dynamics**
- It has the flavor of a global measure of what happens along a trajectory
Even today, physicists sometimes loosely interpret action as:
- “how much dynamical activity happens along a path”
- or “the cost of a trajectory”
5. A deeper modern perspective
In modern physics, “action” is less about “effort” and more about:
- a functional that encodes the entire dynamics of a system
- the central object from which all equations of motion can be derived
And in quantum mechanics (as we noticed with Feynman), it becomes even more fundamental:
- every path is weighted by \(e^{iS/\hbar}\)
So “action” is really:
the quantity that controls how nature evolves, both classically and quantum mechanically
6. Summary
- The word “action” was introduced by Maupertuis in the 18th century
- It originally meant something like “effort” or “amount of motion”**
- It was later formalized by Lagrange and Hamilton into \(S = \int (T - V) \, dt\)
- The name stuck even as the concept became more abstract and central
Why \(L = T - V\)?
It has been stated that the Lagrangian \(L = T - V\), where \(T\) is kinetic energy and \(V\) is potential energy. Why? Why do we minimize the difference rather than the sum? Why do we minimize the difference between the values rather than the difference between their squares?
Can we revisit the origins of the definition of what is known as the action in physics?
1. What the action is
En mécanique classique, on veut comprendre comment un objet bouge d’un point A à un point B. Au lieu de regarder seulement sa position à un instant donné, on peut regarder tout le chemin qu’il prend.
C’est là qu’intervient l’action, notée \(S\). C’est une sorte de “score” qu’on calcule pour chaque chemin possible. Plus concrètement, pour chaque trajectoire que l’objet pourrait suivre entre \(t_1\) et \(t_2\), on associe un nombre \(S\) qui résume “combien d’énergie il dépense pour se déplacer”.
Pour calculer ce score, on utilise le Lagrangien**, noté \(L\) :
\[L = T - V\]- \(T\) est l’énergie cinétique : l’énergie due au mouvement, qui dépend de la vitesse.
- \(V\) est l’énergie potentielle : l’énergie due aux forces qui “poussent ou tirent” l’objet (comme la gravité ou un ressort).
On fait ensuite l’intégrale de \(L\) sur le temps entre \(t_1\) et \(t_2\) :
\[S = \int_{t_1}^{t_2} L \, dt = \int_{t_1}^{t_2} (T - V) \,dt\]Cette intégrale donne un seul nombre pour chaque chemin (encore une fois, pense à un score, à une note globale, une note “intégrale”). Le principe de moindre action dit que le chemin que l’objet choisit réellement rend ce nombre “stationnaire” (souvent minimum) par rapport à tous les autres chemins possibles.
Donc, plutôt que de suivre directement les forces à chaque instant comme avec \(F = ma\), on peut penser en termes de chemin global et l’objet “choisit” la trajectoire qui rend l’action spéciale.
More precisely, in classical mechanics, the action \(S\) is a functional, a quantity that depends on the whole path a system takes between two times:
\[S[q(t)] = \int_{t_1}^{t_2} L(q, \dot{q}, t) dt\]where \(L = T - V\) is the Lagrangian**, \(T\) is kinetic energy, \(V\) is potential energy, and \(q(t)\) describes the configuration of the system over time.
The principle of least action (more accurately, principle of stationary action) states that the physical path a system follows makes \(S\) stationary (usually a minimum) compared to nearby paths:
\[\delta S = 0\]This yields Euler–Lagrange equations**, which are exactly Newton’s laws in disguise.
2. Why \(L = T - V\) and not \(T + V\)
Historically, this comes from Hamilton’s reformulation of mechanics:
- Kinetic energy \(T\) represents motion.
- Potential energy \(V\) represents “stored” energy due to forces.
- The Lagrangian \(L = T - V\) turns out to encode the dynamics correctly: when we apply \(\delta S = 0\), the resulting equations of motion reproduce Newton’s second law \(F = ma\).
If we tried \(T + V\), the Euler–Lagrange equations would lead to completely wrong dynamics. Mathematically, \(T - V\) naturally appears when we try to generalize d’Alembert’s principle**, which itself expresses that the difference between applied forces and inertial forces “vanishes” along the physical path.
So it’s not arbitrary; it’s derived to make Newton’s laws emerge from a variational principle.
3. Why minimize (or make stationary) instead of maximize?
- In many simple cases, the action is actually minimized**, like in free particle motion or harmonic oscillators.
- More generally, the principle is “stationary action,” meaning \(\delta S = 0\). This includes minima, maxima, and saddle points. The key is that the first variation vanishes**, we don’t always strictly minimize in a global sense.
So the focus is on “stationary” not strictly “minimum”.
4. Why not the difference of squares or other functions?
- The Lagrangian is chosen to be linear in energies \(T - V\) because it matches Newtonian dynamics.
- Using \(T + V\) or \((T - V)^2\) gives different Euler–Lagrange equations that do not reproduce physical trajectories.
- It turns out that (T - V) is the simplest combination leading to second-order differential equations in time, which is what we observe for real motion.
5. Historical genesis
- Maupertuis (1744) first proposed a kind of “least action” principle: the integral of momentum along the path is minimized.
- Euler and Lagrange formalized it, showing \(S = \int L \, dt\) gives Newton’s equations.
- Hamilton gave the modern formulation with \(L = T - V\) in 1834.
- So the action \(S\) is not just “something we made up”; it arises naturally from trying to express mechanics in terms of variational principles**, which is elegant and generalizable (to fields, quantum mechanics, relativity…).
6. Summary
- Action \(S = \int (T - V) \, dt\) because this combination reproduces Newtonian dynamics via a variational principle.
- We look for stationary points of \(S\), not strictly minima, because the actual physical path makes \(\delta S = 0\).
- Other choices like \(T + V\) or \((T - V)^2\) fail to give the correct equations of motion.
- Historically, it comes from the effort to express mechanics as a global extremum principle**, starting with Maupertuis and culminating in Hamilton’s formulation.
Mise en application sur une balle qui tombe sous l’effet de la gravité
1. Définir les énergies
- Énergie cinétique \(T\) : dépend de la vitesse de la balle. Si la balle tombe verticalement, \(T = \frac{1}{2} m v^2\).
- Énergie potentielle \(V\) : due à la gravité, \(V = m g h\), où \(h\) est la hauteur au-dessus du sol.
Le Lagrangien est donc :
\[L = T - V = \frac{1}{2} m v^2 - m g h\]2. Action pour un chemin
Imaginons que la balle parte du point \(h_1\) à \(t_1\) et arrive à \(h_2\) à \(t_2\). On peut imaginer plusieurs chemins possibles**, par exemple :
- Une chute droite et régulière (vitesse constante de plus en plus grande).
- Un chemin “zigzag” (imaginons que la balle saute en l’air avant de tomber).
- Un chemin très bizarre qui monte d’abord puis descend.
Pour chaque chemin**, on calcule l’action :
\[S = \int_{t_1}^{t_2} \left( \frac{1}{2} m v(t)^2 - m g h(t) \right) dt\]Chaque chemin donne un nombre différent.
3. La balle “choisit” le bon chemin
Le principe de l’action nous dit que le chemin réel suivi par la balle est celui qui rend \(S\) stationnaire (souvent minimum).
- Si tu imagines toutes les trajectoires possibles, la vraie trajectoire correspond à celle qui équilibre le plus harmonieusement énergie cinétique et énergie potentielle sur le temps.
- Dans le cas de la chute libre, ça correspond simplement à une chute rectiligne uniformément accélérée**, exactement ce que Newton nous dit avec (F = ma).
4. Intuition visuelle
- Le “score” \(S\) est comme une mesure de “coût” du chemin.
- Le chemin réel minimise ce coût global, plutôt que de décider instant par instant quelle force appliquer.
- C’est comme si la balle avait “calculé à l’avance” le meilleur trajet pour que \(T\) et \(V\) s’équilibrent sur tout le chemin.
We can explain why a ball falls vertically while accelerating?
Parfait, on va faire le raisonnement pas à pas pour une balle qui tombe verticalement sous la gravité, en utilisant le principe de l’action. On va rester simple et clair.
1. Définir les variables
On note :
- \(h(t)\) = hauteur de la balle à l’instant (t)
- \(v(t) = \dot{h}(t) = \frac{dh}{dt}\) = vitesse
- Masse de la balle = \(m\) (ne change rien au résultat)
- Accélération due à la gravité = \(g\)
L’énergie cinétique :
\[T = \frac{1}{2} m \dot{h}^2\]L’énergie potentielle : \(V = m g h\)
Donc le Lagrangien : \(L = T - V = \frac{1}{2} m \dot{h}^2 - m g h\)
2. Retrouver l’équation d’Euler-Lagrange
Ici je prends le temps d’expliquer d’où vient l’équation d’Euler-Lagrange.
1. Idée de départ
On a le principe de l’action :
\[S[h(t)] = \int_{t_1}^{t_2} L(h, \dot{h}) , dt\]La balle “choisit” le chemin (h(t)) qui rend \(S\) stationnaire par rapport à de petites variations du chemin.
On imagine donc un chemin légèrement modifié :
\[h(t) \to h(t) + \epsilon \eta(t)\]- \(\eta(t)\) est une petite fonction arbitraire qui s’annule aux extrémités \(t_1\) et \(t_2\) (on ne change pas les points de départ et d’arrivée).
- \(\epsilon\) est un petit nombre pour mesurer la variation.
Le principe de l’action dit :
\[\frac{d}{d\epsilon} S[h(t) + \epsilon \eta(t)] \Big|_{\epsilon=0} = 0\]C’est-à-dire que la dérivée de l’action par rapport à cette petite variation est nulle.
2. Développement
On remplace \(L\) :
\[S[h+\epsilon \eta] = \int_{t_1}^{t_2} L(h + \epsilon \eta, \dot{h} + \epsilon \dot{\eta}) dt\]On dérive par rapport à (\epsilon) et on évalue en (\epsilon = 0) :
\[\delta S = \int_{t_1}^{t_2} \left( \frac{\partial L}{\partial h} \eta + \frac{\partial L}{\partial \dot{h}} \dot{\eta} \right) dt = 0\]3. Intégration par parties
On veut éliminer \(\dot{\eta}\) pour ne garder que \(\eta\). On fait une intégration par parties sur le deuxième terme :
\[\int_{t_1}^{t_2} \frac{\partial L}{\partial \dot{h}} \dot{\eta} , dt = \left[ \frac{\partial L}{\partial \dot{h}} \eta \right]_{t_1}^{t_2} - \int_{t_1}^{t_2} \frac{d}{dt} \left( \frac{\partial L}{\partial \dot{h}} \right) \eta dt\]Mais \(\eta(t_1) = \eta(t_2) = 0\), donc le terme aux bornes disparaît. Il reste :
\[\delta S = \int_{t_1}^{t_2} \left( \frac{\partial L}{\partial h} - \frac{d}{dt} \frac{\partial L}{\partial \dot{h}} \right) \eta dt = 0\]4. Principe du coefficient devant \(\eta\)
Cette intégrale doit être nulle pour toute fonction arbitraire (\eta(t)). La seule façon que ce soit toujours vrai est que le coefficient de (\eta) soit nul :
\[\frac{d}{dt} \frac{\partial L}{\partial \dot{h}} - \frac{\partial L}{\partial h} = 0\]Et voilà, c’est l’équation d’Euler-Lagrange.
En résumé :
- On perturbe le chemin avec une petite variation.
- On exige que la variation de l’action soit nulle.
- Avec une intégration par parties, on obtient une condition sur (h(t)) : c’est l’Euler-Lagrange.
3. Ecrire l’équation d’Euler-Lagrange
Le principe de l’action dit : \(\delta S = \delta \int_{t_1}^{t_2} L(h, \dot{h}) dt = 0\)
L’équation d’Euler-Lagrange qu’on vient de retrouve dit que : \(\frac{d}{dt} \frac{\partial L}{\partial \dot{h}} - \frac{\partial L}{\partial h} = 0\)
Calculons les dérivées :
- \[\frac{\partial L}{\partial \dot{h}} = m \dot{h}\]
- \[\frac{d}{dt} \frac{\partial L}{\partial \dot{h}} = m \ddot{h}\]
- \[\frac{\partial L}{\partial h} = - m g\]
Donc l’équation devient : \(m \ddot{h} - (- m g) = 0 \quad \Rightarrow \quad m \ddot{h} + m g = 0 \quad \Rightarrow \quad \ddot{h} = - g\)
Voilà, exactement l’accélération verticale sous gravité.
4. Intégration pour la trajectoire
On intègre deux fois pour obtenir la hauteur :
\[\dot{h}(t) = \dot{h}_0 - g t\] \[h(t) = h_0 + \dot{h}_0 t - \frac{1}{2} g t^2\]- \(h_0\) = hauteur initiale
- \(\dot{h}_0\) = vitesse initiale (souvent 0 si on lâche la balle)
Donc pour une balle lâchée : \(\dot{h}_0 = 0 \quad \Rightarrow \quad h(t) = h_0 - \frac{1}{2} g t^2\)
C’est exactement la chute rectiligne uniformément accélérée que vous connaissez.
How Maxwell’s equations can be derived from a variational principle?
The amazing thing is that the four Maxwell equations can be obtained from one single variational principle**, exactly like \(F=ma\). The difference is that instead of a particle trajectory \(x(t)\), we now vary fields in spacetime.
The equations describe the electromagnetic field discovered by James Clerk Maxwell.
1. From particles to fields
In mechanics we had:
- variable: \(x(t)\)
- action:
In electromagnetism we instead describe fields:
- electric field \(\mathbf{E}(x,t)\)
- magnetic field \(\mathbf{B}(x,t)\)
But varying vector fields directly is inconvenient. Instead we introduce potentials:
- scalar potential \(\phi\)
- vector potential \(\mathbf{A}\)
The physical fields are defined by
\[\mathbf{E} = -\nabla \phi - \frac{\partial \mathbf{A}}{\partial t}\] \[\mathbf{B} = \nabla \times \mathbf{A}\]These automatically satisfy two Maxwell equations.
2. The electromagnetic action
For fields, the action becomes an integral over spacetime:
\[S = \int L \, d^3x \,dt\]The Lagrangian density for electromagnetism is
\[L = \frac{\epsilon_0}{2}(E^2 - c^2 B^2) * \rho \phi - \mathbf{J}\cdot \mathbf{A}\]where
- \(\rho\) = charge density
- \(\mathbf{J}\) = current density
Interpretation:
| Term | Meaning |
|---|---|
| \(E^2\) | electric field energy |
| \(B^2\) | magnetic field energy |
| \(\rho \phi\) | coupling to charge |
| \(J\cdot A\) | coupling to current |
So the action contains field energy + interaction with charges.
3. Apply the variational principle
The rule is exactly the same as before:
\[\delta S = 0\]But now we vary the fields:
\[\phi \rightarrow \phi + \delta\phi\] \[\mathbf{A} \rightarrow \mathbf{A} + \delta\mathbf{A}\]From this we obtain Euler–Lagrange equations for fields.
For a field \(q\):
\[\partial_\mu \left( \frac{\partial L}{\partial (\partial_\mu q)} \right) - \frac{\partial L}{\partial q} =0\]This is the field version of the Euler–Lagrange equation.
4. Vary the scalar potential
When we vary \(\phi\), the calculation gives:
\[\nabla \cdot \mathbf{E} = \frac{\rho}{\epsilon_0}\]This is Gauss’s law.
5. Vary the vector potential
Varying \(\mathbf{A}\) gives:
\[\nabla \times \mathbf{B} - \frac{1}{c^2} \frac{\partial \mathbf{E}}{\partial t} = \mu_0 \mathbf{J}\]This is Ampère–Maxwell law.
6. The other two equations appear automatically
Because of how (E) and (B) were defined from the potentials, two equations are automatically satisfied:
\[\nabla \cdot B = 0\](no magnetic monopoles)
and
\[\nabla \times E = -\frac{\partial B}{\partial t}\](Faraday’s law).
So all four Maxwell equations emerge.
7. The four Maxwell equations
The full set is:
Gauss law
\[\nabla \cdot E = \frac{\rho}{\epsilon_0}\]Gauss law for magnetism
\[\nabla \cdot B = 0\]Faraday law
\[\nabla \times E = -\frac{\partial B}{\partial t}\]Ampère–Maxwell law
\[\nabla \times B = \mu_0 J + \frac{1}{c^2}\frac{\partial E}{\partial t}\]All from one action.
8. Why this is conceptually powerful
The variational approach reveals deep structure.
1️. Symmetry
The action is invariant under gauge transformations:
\[A \rightarrow A + \nabla \chi\] \[\phi \rightarrow \phi - \partial_t \chi\]This symmetry explains charge conservation.
2️. Relativity
The electromagnetic action is naturally written in relativistic form.
That is why electromagnetism fits perfectly with special relativity.
3️. Field theory structure
The same framework works for:
- electromagnetism
- quantum fields
- the Standard Model
- gravity
So Maxwell theory is actually the first modern field theory.
9. A beautiful consequence: light
If we remove charges ((\rho=0), (J=0)) and combine Maxwell equations, we obtain
\[\nabla^2 E = \frac{1}{c^2} \frac{\partial^2 E}{\partial t^2}\]This is the wave equation.
Meaning:
electromagnetic fields propagate as waves.
The wave speed is
\[c = \frac{1}{\sqrt{\mu_0\epsilon_0}}\]which turned out to be exactly the speed of light.
This is how Maxwell predicted that light is an electromagnetic wave.
10. Summary
Just like \(F=ma\) comes from extremizing
\[S=\int (T-V),dt\]Maxwell’s equations come from extremizing the electromagnetic field action**
\[S=\int L(E,B,\rho,J),d^3x,dt\]Nature seems to follow a universal rule:
The laws of physics come from making an action stationary.
All of this is quite reminiscent of Feynman integrals. No?
What we see with the principle of action is indeed very similar to the idea behind Feynman integrals in quantum mechanics. Let’s try to establish the connection:
1. Classique vs quantique
- Classique : l’objet suit un chemin unique qui rend l’action \(S\) stationnaire.
- Quantique : une particule explore tous les chemins possibles entre deux points, pas seulement celui qui minimise l’action. Chaque chemin contribue avec un poids complexe :
- \(S\) = action pour ce chemin
- \(\hbar\) = constante de Planck réduite
Le chemin classique émerge comme celui dont les contributions des chemins voisins s’ajoutent de façon constructive**, alors que les chemins “bizarres” s’annulent par interférences.
2. Intuition
- En classique, \(\delta S = 0\) sélectionne le chemin réel.
- En quantique, tous les chemins existent, mais le chemin classique correspond au maximum d’interférences constructives, ce qui explique pourquoi les lois classiques émergent à grande échelle.
3. Donc oui
On peut voir le principe de l’action comme la version “limite classique” de l’intégrale de chemin de Feynman.
- Les deux partent de la même idée : une action associée à chaque chemin.
- La différence : le classique choisit un chemin, le quantique somme sur tous.
Conclusion
Coming soon, just a bit delayed.
Webliography
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