
Geometric Derivation of EulerLagrange EquationIntuitive geometric derivation.
StatementGiven a functional : EulerLagrange says that the function at a stationary point of the functional obeys: Where . This result is often proven using integration by parts – but the equation expresses a local condition, and should be derivable using local reasoning. We will explore an alternate derivation below. Motivating ExamplePhysical systems in stable equilibrium will move to a configuration that locally minimizes their potential energy. For example, consider a chain draped over two pulleys (at height , separated by distance ), with excess chain resting on the ground. The chain will take a shape between the two pulleys that minimizes its gravitational potential energy. The space is interesting: If the chain is taut, then it will be high above the ground, and have high energy. If the chain is very saggy, it will pull up lots of chain from the ground, and have high energy. In between is the optimal shape. The potential energy of a given shape of the chain between the pulleys is: We have integrated only along the section between the pulleys, because we can define at ground level, and ignore constants (the energy of the chain dangling outside the pulleys) and constant factors (). Now we want to find the function that minimizes (subject to the boundary conditions ). Geometric DerivationFor to be a stationary point of , small perturbations to the function must not alter the value of (to first order). So consider an infinitesimal segment of (grey), discretized around . We will consider perturbing at by an amount (changing from the blue –> red), and examine its effect on . Recall: Perturbing this point affects in two ways:
The first contribution is simply: The second contribution is in two parts: increasing the derivative on the left, and decreasing the derivative on the right. Based on the figure, we have (before perturbation): And after perturbation: So the change in derivatives is: This affects by an amount: Since , we can write: So finally, the effect of changing the derivatives is: And the net effect of perturbing the point is: So clearly, requiring for small perturbations anywhere along the function implies the EulerLagrange equation: And now both terms have meaning:
At a stationary point, these effects must exactly cancel. Final CommentsFor completeness, we derive the solution to the example, and extend it to the case of a fixedlength chain. SolutionIn this case: Applying EulerLagrange directly, we find: We could solve this, but the simpler approach is to use a theorem: Thm: If , then is a constant. Proof: Where the last step is applying the original form of the EulerLagrange equation. Since our particular is independent of , we can apply this theorem: For constants chosen to satisfy boundary conditions. This gives us the familiar catenary curve. ExtensionWe know the catenary is also the shape formed if we hang a fixedlength chain between two fixed endpoints. This is no coincidence. The fixedlength chain problem is a constrained minimization problem, with the same potential energy functional , but the additional constraint that the arclength of is some specified constant. Specify constrained problems by – separation distance and chain arclength. And unconstrained problems by – separation distance and height. We will show that the solution to this problem is also a catenary, by showing that: Theorem: For a given constrainedproblem , there is a corresponding unconstrainedproblem that contains the constrainedproblem as a subproblem. Therefore the constrained problem must also be a catenary. Lemma: For any , we can always find some unconstrained configuration such that the length of chain between and is exactly . Proof: Refer to the family of unconstrained solutions we found. With centered choice of coordinates, , and the family takes the form: The boundary condition determines the parameter . Now, clearly the arclength between and is a monotone decreasing function of , with extremes at and . (eg, for , the arclength turns out to be ) So any value of in this range is achievable for some – and further, there is some that gives rise to this (in particular, works). Result: Now, given a constrainedproblem , construct an unconstrainedproblem such that the length of chain between and is exactly . (which Lemma guarentees we are able to do). Over this span, the unconstrained problem is identical to the constrained problem: arclength , and endpoints separated by . If one situation's optimalsolution (say A) had lower potential energy (over the span) than the other (B), then B could assume the form of A over the span, without violating any constraints but resulting in lower energy. Contradiction of optimality, so both problems must have identical solutions over the span – and the constrained optimal solution is also a catenary. So we have also derived the solution to the constrained problem. (no Lagrange multipliers necessary!) 