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Geometric Derivation of Euler-Lagrange EquationIntuitive geometric derivation.
StatementGiven a functional ![]() Euler-Lagrange says that the function ![]() 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
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 ![]() We have integrated only along the section between the pulleys, because
we can define Now we want to find the function Geometric DerivationFor
Recall: Perturbing this point affects
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 ![]() Since ![]() So finally, the effect of changing the derivatives is: ![]() And the net effect of perturbing the point is: ![]() So clearly, requiring ![]() 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 fixed-length chain. SolutionIn this case: ![]() Applying Euler-Lagrange directly, we find: ![]() We could solve this, but the simpler approach is to use a theorem: Thm: If Proof: ![]() Where the last step is applying the original form of the Euler-Lagrange equation. Since our particular ![]() For constants ExtensionWe know the catenary is also the shape formed if we hang a fixed-length chain between two fixed endpoints. This is no coincidence. The fixed-length chain problem is a constrained minimization problem,
with the same potential energy functional Specify constrained problems by And unconstrained problems by We will show that the solution to this problem is also a catenary, by showing that: Theorem: For a given constrained-problem Lemma: For any Proof: Refer to the family of unconstrained solutions we found.
With centered choice of coordinates, ![]() The boundary condition So any value of Result: Now, given a constrained-problem So we have also derived the solution to the constrained problem. (no Lagrange multipliers necessary!) |