In physics, one only sees Lagrangian that are scalar-valued:
$$ L:t,q,\dot{q}\to \mathbb{R} $$
whose integral over time is the action
$$ S=\int_0^tL(t,q,\dot{q})dt $$
In my own amusement research I have ended up with a Lagrangian that is vector valued:
$$ L:t,q,\dot{q}\to \mathbb{R}^n $$
For instance
$$ \mathbf{L}=e_0L_t(t,q,\dot{q})+ e_1L_x(t,q,\dot{q})+e_2L_y(t,q,\dot{q})+e_3L_z(t,q,\dot{q}) $$
A-priori, there doesn't seem to be any reason why this shouldn't be workable in some sense. But, what would be the interpretation of such a thing: each orthogonal axis are 'independent' sub-systems and the master system cannot 'rotate' between them?
Is there any literature which investigate such Lagrangian with perhaps physical applications?
Does the Euler-Lagrangian equations apply to it - can I get $n$ independent equations of motions via:
$$ e_1\frac{\partial L_1}{\partial q}(t,q,\dot{q})-e_1\frac{d}{dt}\frac{\partial L_2}{\partial \dot{q}}(t,q,\dot{q}) =0\\ \vdots\\ e_2\frac{\partial L_2}{\partial q}(t,q,\dot{q})-e_2\frac{d}{dt}\frac{\partial L_2}{\partial \dot{q}}(t,q,\dot{q})=0 $$
An example of a Lagrangian which is not a scalar but a 4-vector can be found here A. Sudbery, A vector Lagrangian for the electromagnetic field, J. Phys. A: Math. Gen. 19 L33 (1986). So the idea is both mathematically and physically correct, although seems unexplored. Actually, I have arrived at the same question by considering two independent 1-dimensional Lagrangian systems (eg. two independent free particles or harmonic oscillators). Both a sum and a difference (actually, any linear combination) of the 1D Lagrangians plays a role of a Lagrangian for the 2D system. However, the resulting 2D Lagrangians have different symmetries (O(2) for sums, and O(1,1) for differences). Yet, Noether theorem implies the same conserved quantity (angular momentum) which nevertheless leads to different Lie algebras (o(2) or o(1,1) respectively) because the link between velocity and canonical momentum depends on the Lagrangian. Hence the question: How to combine two different Lagrangians into a single Lagrangian if the standard procedure "add them" is non-unique? The answer that comes immediately is: do not form linear combinations but pairs, i.e. replace L=L1+L2 by L=(L1,L2)... So I started to google for a "vector Lagrangian" and have found this very interesting paper from 1986. The problem seems open, but you ask a good question.