Foundations — Generalized momenta and generalized forces

This page assumes nothing. If the parent note wrote a symbol, we build it here from a picture first. Read it once and you will never be stuck on notation again.

0. The three characters of the story

Before any symbol, meet the three quantities the whole subject juggles:

The rest of this page defines the right column, one symbol at a time, always tied to a picture.

1. The vector ${\vec{r}}_i$ — "where is particle number $i$?"

The tiny arrow on top, $\stackrel{⃗}{}$, is a promise: "this thing has both a length and a direction." A plain number (like temperature) has no direction; a position does — so it earns the arrow.

Why the topic needs it. All of physics starts with "where are things." Newton wrote everything in terms of these arrows. The whole game of the parent note is to stop writing arrows and write single letters $q_j$ instead — but we must know what we are replacing first.

2. Generalized coordinate $q_j$ — "one honest number per freedom"

Look at a pendulum. The bob lives at some $(x,y)$, but $x$ and $y$ are not free: the rod forces $x^2+y^2={\ell }^2$. So there is really only one independent number — the swing angle.

Degrees of freedom $n$ = the number of dials. See Generalized coordinates and constraints for the full story of how constraints reduce $n$.

Because each particle's Cartesian arrow is fixed once the dials are set, we can write ${\vec{r}}_i={\vec{r}}_i(q_1,\dots ,q_n,t).$ Read this out loud: "the position of particle $i$ is a recipe that takes the dial settings (and maybe the clock time $t$) and returns an arrow."

3. The dot: ${\dot{q}}_j$ and $\dot{x}$ — "rate of change"

Why a derivative and not just "difference"? Because motion is smooth and continuous; we want the instantaneous rate at one moment, not the average over a chunk of time. The derivative is precisely the tool that answers "how fast, right now?" — see Euler-Lagrange Equation for where derivatives dominate.

Why the topic needs it. Energy of motion depends on speed, and speed of a dial is ${\dot{q}}_j$. Every momentum formula will contain these dots.

4. The partial derivative $\frac{\partial }{\partial q_j}$ — "wiggle just one dial"

Here is the single most-feared symbol on the parent page. It is gentle once you see it.

Why the topic needs it — everywhere.

• $\frac{\partial {\vec{r}}_i}{\partial q_j}$ = "which way, and how far, does particle $i$ slide when I turn dial $j$ a hair." This is the bridge arrow between the two coordinate worlds.
• $p_j=\frac{\partial L}{\partial {\dot{q}}_j}$ and $Q_j=-\frac{\partial V}{\partial q_j}$ both live on this symbol.

5. The dot product $\vec{A}\cdot \vec{B}$ — "how much do two arrows agree?"

Why this operation and not ordinary multiplication? Because we often want the part of one arrow that lies along another — for example, only the part of a force that acts along the direction of motion does work. The $\cos \varphi$ factor extracts exactly that overlap:

• arrows aligned ($\varphi =0$): $\cos =1$, full agreement, biggest result;
• perpendicular ($\varphi =90^{\circ }$): $\cos =0$, zero — a sideways force does no work;
• opposed ($\varphi =180^{\circ }$): $\cos =-1$, negative — the force fights the motion.

Why the topic needs it. Work is $\delta W={\sum }_i{\vec{F}}_i\cdot \delta {\vec{r}}_i$ and generalized force is $Q_j={\sum }_i{\vec{F}}_i\cdot \partial {\vec{r}}_i/\partial q_j$. Both are dot products: "how much does the push agree with the allowed motion?"

6. The summation sign $\sum$ — "add up over all the things"

Why the topic needs it. A system has many particles (${\sum }_i$) and many dials (${\sum }_j$). $\delta W={\sum }_j{Q}_j\delta q_j$ says "total work = add up (each generalized force) × (its dial's wiggle)."

7. The virtual displacement $\delta {\vec{r}}_i$ — "an imaginary, time-frozen nudge"

Why the topic needs it. This is how D'Alembert's Principle and the definition of $Q_j$ get built. Forgetting that $\delta$ freezes time (keeping a stray $\partial /\partial t$) is a classic error flagged on the parent page.

8. Energies $T$, $V$ and the Lagrangian $L$

9. Putting the four new symbols together

Now the parent page's headline boxes read as plain sentences:

The link to real momentum and torque is spelled out in Angular momentum; the deep reason a cyclic coordinate conserves its momentum is Noether's Theorem; and where all this heads next is Hamiltonian Mechanics.

Prerequisite map

Equipment checklist

Test yourself — cover the right side and answer out loud.