Foundations — Derivation of Euler-Lagrange equations from D'Alembert's principle

Before you can read a single line of the parent derivation, you must be able to read its alphabet. This page builds every symbol from zero, in an order where each one only uses symbols already defined. Nothing here is assumed.

0. The stage: particles and positions

The picture. Plant an origin. Every particle is a dot; the arrow from the origin to that dot is ${\vec{r}}_k$. As the particle moves, the tip of the arrow traces its path.

Why the topic needs it. Newton's law talks about where things are and how that changes. ${\vec{r}}_k$ is the raw "where." Everything else is built by watching how ${\vec{r}}_k$ changes.

The picture. If ${\vec{r}}_k$ is the position arrow, then ${\dot{\vec{r}}}_k$ is a small arrow tangent to the path showing the direction of travel; two dots ($\ddot{}$) is the arrow showing how that travel-arrow itself is turning or stretching.

1. The reading-frame for changes: $\partial$, $\frac{d}{dt}$

Why the topic needs it. The Euler–Lagrange equation literally is $\frac{d}{dt}\frac{\partial L}{\partial \dot{q}}-\frac{\partial L}{\partial q}=0$: one $\frac{d}{dt}$ wrapped around one $\partial /\partial \dot{q}$, minus another $\partial /\partial q$. If you cannot feel the difference between these two operators, the equation is unreadable.

2. Combining vectors: the dot product $\cdot$

The picture. Shine a light straight down onto $\vec{B}$; the shadow $\vec{A}$ casts along $\vec{B}$, times the length of $\vec{B}$, is the dot product. Key fact we will lean on constantly:

• If the two vectors are perpendicular ($\varphi =90^{\circ }$), then $\cos 90^{\circ }=0$, so $\vec{A}\cdot \vec{B}=0$.

Why the topic needs it. The whole magic of D'Alembert's principle is that constraint forces point perpendicular to the allowed motion. Perpendicular means the dot product is zero, so those forces vanish from the equation. The dot product is the eraser.

3. Force, momentum, and Newton

Why the topic needs it. The entire derivation is a machine for making ${\vec{F}}_k^{(c)}$ disappear so only ${\vec{F}}_k^{(a)}$ (and inertia) remain. See Generalized forces and potentials.

4. Constraints and generalized coordinates

The picture. A bead on a bent wire lives in 3D, but you only need one number — how far along the wire it has slid — to pin it down. That number is $q$. The wire is the constraint; $q$ is the freedom that survives.

5. Virtual displacement $\delta \vec{r}$ — the star of the show

The picture. Pause the movie. Ask: "In this frozen frame, which tiny nudges could I give the particle without breaking any constraint?" Those permitted nudges are the virtual displacements. For a bead on a wire, the only allowed $\delta \vec{r}$ points along the wire — never off it.

Why the topic needs it. Combine sections 2 and 5: constraint force $\perp$ allowed motion $\Rightarrow {\vec{F}}_k^{(c)}\cdot \delta {\vec{r}}_k=0$. Sum over particles and the constraint forces are gone. That single fact is D'Alembert's principle.

6. Generalized force $Q_i$ and the potential $V$

7. Putting the alphabet together

Here is how every symbol feeds the final Euler–Lagrange equation. Read top to bottom.

Every arrow is a dependency: you cannot understand a node until you understand its parents. This is exactly the order the parent derivation walks. Onward to Hamilton's principle (least action) and Noether's theorem once this is solid.

Equipment checklist

Cover the right side and see if you can answer each before revealing.