Abstract
We consider the duality which is obtained by interchanging dependent and independent variables in a Lagrangian. This duality is studied here for a second-order Lagrangian, which could describe, e.g. a Euler- Bernoulli beam (more generally, a one-dimensional elastic body of higher grade); however, the essential theory follows purely formally for a general Lagrangian. Next, the corresponding Noether theory of transformation groups and invariants is developed, making use of duality as far as possible, and showing the limits of duality theory. Particular emphasis is laid on the difference between relations equally valid on any curve and those valid only on the solution curve, stressing the difference between “true” and “apparent” invariants.
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