Generalised Coordinates and the Lagrangian
Choosing coordinates that already respect the constraints removes the constraint forces from the problem, and the Lagrangian is kinetic minus potential energy written in those coordinates. A pendulum needs one angle rather than two coordinates and 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 why constraint forces do not appear once the coordinates respect the constraint
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 already respect the constraints, write kinetic minus potential energy in them, and fixed recipes produce the motion. Counting degrees of freedom first tells you exactly how many coordinates you need: a simple pendulum swings one way, so one angle suffices instead of two positions plus a string rule.
Kinetic energy T is half the job: turn half m v squared into your coordinates and rates. For a pendulum length L, 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 L equals T minus V with the single angle theta. That one line carries the whole system: swing rate inside T and height inside V. Friction has no potential and needs separate treatment outside the basic recipe, which is why textbook systems stick to gravity and ideal springs.
Constraint forces like string tension never appear once the coordinates respect the constraint. The angle cannot move against the string, so tension does no work in this description. That disappearance is the payoff for choosing good coordinates.
Pick coordinates the constraints allow, write T minus V, and let tension vanish.
2 · Watch
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Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.