Power Expantion in Lagrangian Derivation

[tharchin]

In Mechanics by Landau-Lifgarbagez there is a step during the derivation of the Lagrangian where..

$$\int_{t_1}^{t_2} L(q+\delta q, \dot q + \delta \dot q, t ) \, \mathrm{d}t - \int_{t_1}^{t_2} L(q, \dot q, t ) \, dt$$

then they write "when this difference is expanded in powers of $$\delta q$$ and $$\delta \dot q$$ in the integrand, the leading terms are of first order."

The don't show this expansion and I was hoping someone could point me to a reference where they do. Thanks.

 

[dgOnPhys]

In Mechanics by Landau-Lifgarbagez there is a step during the derivation of the Lagrangian where..

$$\int_{t_1}^{t_2} L(q+\delta q, \dot q + \delta \dot q, t ) \, \mathrm{d}t - \int_{t_1}^{t_2} L(q, \dot q, t ) \, dt$$

then they write "when this difference is expanded in powers of $$\delta q$$ and $$\delta \dot q$$ in the integrand, the leading terms are of first order."

The don't show this expansion and I was hoping someone could point me to a reference where they do. Thanks.

Hi tharchin,

I don't have the book at hand but he is probably just talking of a Taylor expansion of the integrand
$$L(q+\delta q, \dot q + \delta \dot q, t ) = L(q, \dot q, t ) + \frac{\partial L}{\partial q}\delta q +\frac{\partial L}{\partial \dot q}\delta \dot q$$