Why is that in the Lagrangian formulation the Taylor's serie is limited to the first order? – physics.stackexchange.com

For the Hamilton's principle: $$\delta s =\int_{t_1}^{t_2}L(\mathbf {q+\delta q},\mathbf {\dot q+\delta \dot q},t) dt-\int_{t_1}^{t_2}L(\mathbf {q},\mathbf {\dot q},t) dt=0.\\$$ In the textbooks, ...

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