We revisit the notion of parametrization invariance while introducing certain weakened notions of invariance in the calculus of variations. In this work, we employ a straightforward approach in the classical setting and mostly restrict attention to functionals on one-dimensional domains. We establish a connection between parametrization invariant functionals and functionals embodying a weaker notion of invariance of their Lagrangian; we term this notion as T -Lagrangian analogous to the wellknown idea of null Lagrangian. However, the Euler-Lagrange operator of a T -Lagrangian vanishes only along the tangential direction in the configuration space. On one-dimensional domain and for first- and second-order theories, we show that functional described by such a Lagrangian is necessarily a parametrization invariant functional modulo null Lagrangian. Keeping the motivation for partial differential equations, we also introduce and explore the notion of N -Lagrangian, with an invariance complementary to the case of T -Lagrangian, whose Euler-Lagrange operator vanishes along normal directions. We find that in a one-dimensional setting, every N -Lagrangian is simply a null Lagrangian
Mathematics and Mechanics of Solids
Dharmavaram, Sanjay and Lal Sharma, Basant. "Generalizing Parametrization Invariance in the Calculus of Variations." (2023) .