Now this is a bit of a mix of a math and a physics question, but I think it is best asked here.(adsbygoogle = window.adsbygoogle || []).push({});

Assume we are given a Lorentzian manifold ##(Q, g)## together with a metric connection ##\nabla##. Naturally we define geodesics ##\gamma## via

$$\nabla_{\dot \gamma} \dot \gamma = 0 \quad ,$$

leading locally to

$$0=\ddot \gamma^k + \Gamma^k{}_{(ij)} \, \dot \gamma^i \dot \gamma^j \quad ,$$

where ##\dot \gamma = \frac{d \gamma}{d \tau}##.

Alternatively, we could have defined the Lagrangian ##L \in C^\infty(TQ,\mathbb{R})##, i.e. a function on the tangent bundle, by

$$L(q,\dot q) = \frac{1}{2} g_{q}(\dot q, \dot q)$$

for ##(q,\dot q) \in TQ## and then varied the action

$$S(\gamma) = \int_{\gamma} L (\gamma, \dot \gamma) \, d t$$

leading to the same local geodesic equation.

Now, in physics, one makes the step to "field theory" by considering a function ##\phi

\in C^\infty(Q, \mathbb R)## and then taking a new functional

$$\tilde S (\phi) = \int_Q \mathcal L (\phi, d \phi) \, \mu$$

where ##\mu \in \Omega^n(Q)## is the canonical volume form. ##\mathcal L## is a function from some space of smooth functions cartesian product with their exterior derivative to ##\mathbb R## and is called the Lagrange density.

My question is:

How do the two formalisms connect? What justifies us in calling ##\tilde S## the action as well? Is there a mathematical discipline dealing with such systems ##(Q,g, \mathcal L)##?

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# Lagrangian density

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