Okay so a couple of questions. Firstly I realise that in order to study the dynamics of one particle (classically), we define the Lagrangian and Hamiltonian to be the maps from the tangent and cotangent bundle of its configuration space to some subset of the real numbers, correct?

$$ \mathscr{L} : TM \longrightarrow \mathbb{R} \ , \ H: T^{*}M \longrightarrow \mathbb{R} $$

So how is this generalised to relativistic mechanics. Is the configuration space of one particle now just spacetime? Or is it something more complicated.

Also I realise that we require the Lagrangian and the Hamiltonian to be Lorentz invariant. Well that should be easy enough for the Lagrangian because the Minkowski metric is a metric on the tangent bundle so I suppose it's easy to check that a function on the tangent bundle is invariant but how exactly is this done for the Hamiltonian directly? Does the cotangent bundle inherit a metric as well?

  • $\begingroup$ Actually the Hamiltonian can't be Lorentz-invariant because it is related to the energy, which is a frame-dependent concept. If you try to define a "relativistic Hamiltonian", you will find that it vanishes identically. See, e.g., Goldstein, Poole, Safko. $\endgroup$ – Ben Niehoff Mar 11 '17 at 16:59
  • $\begingroup$ Ah okay so basically we only deal with Lagrangians in relativistic physics? $\endgroup$ – Nameless Paladin Mar 11 '17 at 17:46
  • $\begingroup$ Mostly, yes. You can choose a frame and go to the Hamiltonian, but you'll have to break manifest Lorentz symmetry to do so. Note however, Hamiltonians are necessary for quantization, so this results in a lot of interesting consequences for quantizing relativistic systems. $\endgroup$ – Ben Niehoff Mar 11 '17 at 18:27

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