Higher order covariant Lagrangian

Question

I'm in search of examples of Lagrangian, which are at least second order in the derivatives and are covariant, preferable for field theories. Up to now I could only find first-order (such at Klein-Gordon-Lagrangian) or non-covariant (e.g. KdV) ones. Also some pointers to the literature about general properties of such systems are welcome. Thanks

Answer 1

I) As user Vibert mentions in a comment, the Euler-Lagrange equations are not modified$^1$ by adding total divergence terms to the Lagrangian density

$$ \tag{1} {\cal L} ~ \longrightarrow ~{\cal L} +d_{\mu}F^{\mu}. $$

Adding total divergence terms leads to an inexhaustible source of higher-order Lagrangians.

II) Generically, without some cancellation mechanism in place [such as, that part of the Lagrangian density is (secretly) a total divergence] an $n$-order action would lead to $2n$-order Euler-Lagrange equations.

III) Example. The Einstein-Hilbert (EH) Lagrangian density

$$\tag{2} {\cal L}_{EH}~\sim~\sqrt{-g} \left\{g^{\mu\nu} R_{\mu\nu}(\Gamma_{LC},\partial\Gamma_{LC})-2\Lambda\right\} $$

depends on both second-order temporal and spatial derivatives of the metric $g_{\mu\nu}$. This is of course an important example. Here $\Gamma_{LC}$ refer to the Levi-Civita (LC) Christoffel symbols, which in turn are first order derivatives of the metric $g_{\mu\nu}$. However, it is possible to add a total divergence term to render the Lagrangian density first order, as user drake mentions in a comment. Thus the Euler-Lagrange equations for the Einstein-Hilbert action $S_{EH}[g_{\mu\nu}]$, i.e. the Einstein field equations (EFE), are not of fourth order, as one may naively have expected, but still of second order.

IV) Higher-order Lagrangians are also discussed in many Phys.SE posts, see e.g. here and here.

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$^1$ Note that adding total divergence terms (1) may affect consistent choices of boundary conditions for the theory.

Answer 2

You can look at the Lagrangian for the galileon particles for instance in this paper. It has the property that the equations of motion remains 2nd order in the derivatives and covariant.