Illustrated Physics
physics, drawn
Concept

Lagrange multiplier — where it appears

A number introduced to find the largest or smallest value of a quantity under a constraint, by adding the constraint to the quantity with an unknown coefficient. At the optimum the coefficient measures how much the best value would change if the constraint were relaxed.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

The reaction that reaches zero, and where the bead lets go. The force the sphere pushes back with, in units of the bead's weight, against the angle from the top. It starts at 1.000 and falls, because the speed the bead has gained needs more centripetal force than gravity's component along the radius can supply. At 48.19° it reaches zero, and past that the surface would have to pull inward to keep the bead on it — which a surface cannot do. So the bead leaves there, and the departure angle is a statement about the sign of a constraint force rather than about a speed or a height. The multiplier is what carries that sign: solve the motion in the angle alone and the reaction is absent from every equation, so nothing in the solution knows that the constraint has stopped holding, and the bead is drawn happily circling a sphere it has already left.

The force a coordinate cannot see

Writing a pendulum in terms of its angle is the first good move anybody learns, and it deletes the tension from every equation that follows. The string still breaks. Recovering the force that the clever choice of coordinate threw away turns out to be a computation with a sign in it, and the sign is where the bead leaves the sphere.

mechanics · Least action Range as a map of launch velocities. The plane of launch velocities — horizontal component across, vertical up — with the curves of equal range drawn on it. Launched and landing at one height, the range is 2vₓvᵧ/g, so every curve of equal range is a hyperbola vₓvᵧ = constant, drawn here at ranges of 0.25, 0.50, 0.75, 1.00 v²/g. A thrower who can produce one speed in any direction can reach any point on the half-circle of radius v, and the best throw is where that circle touches the highest hyperbola it meets — at 45°, where the hyperbola vₓvᵧ = ½ is tangent to it, because a circle centred on the origin is symmetric about the diagonal and so is the hyperbola. The famous angle is a property of the shape of the set of throws.

The best throw is a tangency

Shot putters release at about 37°, long jumpers take off at about 20°, a ball thrown forward from a moving truck should be aimed steeply and flies flat, and a golf ball's drag alone moves its best angle to 38°. Each is usually explained as an exception to 45°. None of them is. Drawn as a map over launch velocities, range has curves of equal value, a thrower has a set of throws they can make, and the best throw is always where the set first touches a curve.

mechanics · Projectile

Named alongside it

The objects these essays reach for when they reach for this one.

BifurcationConstraintConstraint forceDegrees of freedomDragEffective potentialGeneralised coordinatesHolonomic constraintLagrangianNormal forceOptimisationProjectile

All concepts