Generalised Coordinates and the Lagrangian
Generalised coordinates that respect the constraints let ideal constraint forces drop out of the equations, because their virtual work at fixed time is zero even though the forces are still present. The Lagrangian is kinetic minus potential energy written in those coordinates. A pendulum needs one angle rather than two positions plus a tension.
What a learner can do afterwards
- Picks a set of generalised coordinates for a constrained system and counts the degrees of freedom
- Writes kinetic and potential energy in those coordinates and forms the Lagrangian
- Explains that ideal constraint forces drop out of the equations because their virtual work at fixed time is zero, while the physical forces remain present
1 · Read
Rolling needed three equations and gyroscopes needed torque bookkeeping. The Lagrangian method sidesteps that grind: it rewrites mechanics around energy instead of forces. You pick generalised coordinates that respect the constraints, write kinetic minus potential energy in them, and fixed recipes produce the motion. In the basic mechanical case, L equals T minus V. Adding a constant to V changes nothing, since the equations of motion depend only on derivatives of L.
Counting degrees of freedom first tells you how many coordinates you need: a simple pendulum swings one way, so one angle suffices instead of two positions plus a string rule. Take a planar simple pendulum with a fixed pivot, a point bob of mass m, a massless string of fixed length l, and theta the angle from the downward vertical. The bob speed is l times the angular rate, so T is one half m l squared times rate squared. Potential V is the other half: gravity qualifies, so V is minus m g l cos theta up to a constant.
For the pendulum you form script L equals T minus V with the single angle theta. Gravity qualifies for V since its work is path independent, so V is minus m g l cos theta. The explicit Lagrangian is script L equals half m l squared theta dot squared plus m g l cos theta. Ideal springs also admit a potential. Friction does not, so if air drag acts on the bob it enters as a generalised force in the Euler-Lagrange equation rather than inside V.
Constraint forces like string tension drop out once the coordinates respect the constraint, but they never disappear from the physics. The mechanism is virtual work: a hypothetical nudge at fixed time. For the planar pendulum the bob must stay at distance l from the pivot, so every allowed virtual displacement is tangent to the circle, while tension points radially inward. A radial force dotted with a tangential displacement gives zero, so tension does zero virtual work. The tension is still present in a force diagram. Actual work can differ: a support that moves in time may do real work while the constraint force still does zero virtual work.
The formalism reaches further than the textbook cases. A bead on a hoop needs one angle along the hoop, not two positions plus a radial rule. Nonconservative applied forces like friction enter as generalised forces. An ideal moving support does not by itself break the recipe: a coordinate map r of q and t may give an explicitly time-dependent kinetic energy and Lagrangian.
For a conservative system, pick coordinates the constraints allow, write script L equals T minus V, and ideal constraint forces drop out through zero virtual work while remaining present in the physics.
Three readings, six flashcards and no extra questions were written for this Lagrangian topic. The linked video is an MIT OpenCourseWare lecture on YouTube. No video transcript is copied.
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