Foundations — Lagrangian L = T − V

Before you can read the parent note Lagrangian $L = T-V$, you must own every symbol it throws at you. We build them here from nothing — plain words, then a picture, then why the topic can't live without it. Each block leans on the one before.

1. Position, and a "configuration"

Picture it. A single bead on a wire: one number (how far along the wire) fixes it completely. A pendulum bob: one angle fixes it. A free particle in a plane: two numbers $(x,y)$.

Why the topic needs it. The whole Lagrangian machine asks "which sequence of configurations does nature trace out in time?" So we first need a way to name a single configuration.

2. Generalized coordinate $q$

Picture it. Look at the pendulum in the figure above. You could track the bob with $(x,y)$, but those two numbers are not independent — the string forces $x^2+y^2={\ell }^2$. Instead use one honest free number: the angle $\theta$. Here $q=\theta$.

Why "generalized"? Because the Euler–Lagrange machinery works for whatever number you pick — Cartesian, polar, angle, arc-length. You get to choose the coordinate that makes the problem simplest. That freedom is the whole payoff, and it is why the parent note never draws force arrows.

• When there are several coordinates we write $q_1,q_2,\dots$ or just $q_i$ (the little $i$ is a counter: "the $i$-th coordinate").

See Generalized Coordinates and Constraints for the full story of why one angle can legally replace two coordinates.

3. The dot: velocity $\dot{q}$

Picture it. If $q$ is your position along a road and time ticks forward, $\dot{q}$ is the speedometer reading: big when you cover distance fast, zero when parked, negative when reversing.

Why the topic needs it. Moving-energy (kinetic energy) depends on how fast things move, not just where they are. So $L$ must be fed both $q$ and $\dot{q}$.

4. The partial derivative $\frac{\partial }{\partial q}$

Picture it. Stand on a hillside whose height is $f(q,\dot{q})$. Walking only in the $q$-direction (a fixed compass bearing) and measuring your slope gives $\partial f/\partial q$. Walking only in the $\dot{q}$-direction gives $\partial f/\partial \dot{q}$. Two different slopes at the same spot.

Why this tool and not an ordinary derivative $\frac{d}{dt}$? The Lagrangian depends on several things at once ($q$, $\dot{q}$, maybe $t$). An ordinary derivative would change everything together. We specifically need "wiggle one knob, freeze the rest" — that is exactly what the partial does, and it is why the Euler–Lagrange equation has $\partial L/\partial q$ and $\partial L/\partial \dot{q}$ as two separate pieces.

5. Kinetic energy $T$

Picture it. $v^2$ means the energy shoots up faster than the speed: double the speed, four times the $T$. In a coordinate $q$, the speed is often $\dot{q}$ (or $\ell \dot{\theta }$ for a swinging bob), so $T$ is built from $\dot{q}$.

Why the topic needs it. $T$ is the first half of the Lagrangian. It carries all the velocity information.

6. Potential energy $V$

Picture it. A ball high on a shelf has large gravitational $V=mgy$ (high $y$); on the floor, small $V$. A stretched spring has $V=\frac{1}{2}k{x}^2$ — the more you pull, the more it stores. $V$ depends on $q$, not on $\dot{q}$.

The crucial link. Force is the downhill slope of the potential: $F=-\frac{dV}{dq}$. The minus sign says "things roll toward lower $V$." This is the fact the parent note uses in Step 6 to recover Newton.

7. Putting the halves together: $L=T-V$ and total energy $E=T+V$

Why the minus in $L$? The parent note proves it: only $T-V$ makes the machinery spit out $F=m\ddot{x}$ with the correct sign. If you used $T+V$, the force would come out backwards (anti-physics). Keep the two straight — same ingredients, opposite sign, completely different jobs. Total energy $E=T+V$ reappears later as the Hamiltonian.

8. The integral ${\int }_{t_1}^{t_2}dt$ and the action $S$

Picture it. Chop the time axis into thin slivers of width $dt$. On each sliver the quantity is roughly constant; multiply value × width to get a thin strip's area, then stack all strips. The total shaded area is the integral.

Why the topic needs it. To ask "which path is best?" we must first score each path with a single number. Integrating the scalar $L$ over time does exactly that — see Principle of Least Action.

9. The variation $\delta$ and "stationary"

Picture it. Hold a piece of string between two nails. The true path is the resting string; a variation $\delta q$ is you plucking it slightly in the middle. "Fixed endpoints" = the nails don't move.

Why "stationary" and not "minimum"? At the very bottom of a bowl the slope is flat; that flatness — not "lowest" — is what the maths demands. Nature picks the path where $S$ can't be improved by wiggling. This single condition, $\delta S=0$, generates the Euler–Lagrange Equation.

10. How it all feeds the topic

Read it top-down: positions give you a coordinate $q$; its time-rate gives $\dot{q}$; those build $T$ and $V$; their difference is $L$; summing $L$ over time gives the action $S$; demanding $S$ be stationary (with the partial-derivative tool) yields the Euler–Lagrange equation and the motion.

Equipment checklist

Cover the right side and test yourself.

Connections

• Parent: Lagrangian $L=T-V$ — where every symbol here gets used.
• Generalized Coordinates and Constraints — the deep story of choosing $q$.
• Principle of Least Action — the meaning of stationary $S$.
• Euler–Lagrange Equation — the equation these foundations produce.
• Generalized Momentum and Conservation — $p=\partial L/\partial \dot{q}$ uses the partial derivative built here.
• Hamiltonian H = T + V — where $E=T+V$ takes centre stage.
• Newton's Second Law — the $F=m\ddot{x}$ we ultimately recover.