- #1
Geometry_dude
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Now this is a bit of a mix of a math and a physics question, but I think it is best asked here.
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)##?
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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