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The covariant phase space idea, in one sentence, is that there is a natural symplectic structure on the space of the classical trajectories of a system and that the usual $(q,p)$ coordinates just happen to be a convenient parametrization of this space. The advantage is that you don’t have to use this parametrization: there’s a construction of the phase space without any unnatural choices for field theory or for theories with higher derivatives.

I understand the pitch above. I’ve also seen several review articles 50+ pages long which describe the relevant (and unsurprisingly heavy) mathematics. However, I find myself lost reading them because I don’t know what to expect on the journey. Why should there be such an object? What are the crucial points leading to it? (And what’s the technical stuff I can skip on first reading—i. e. all the rest?)

Can anyone give a overview of this story that’s short enough that one can see all the logic at once?

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  • $\begingroup$ See also this Phys.SE answer. $\endgroup$ – Qmechanic Feb 26 '16 at 20:28
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Given a dynamical system $\dot x=F(x)$ with continuously differentiable $F$, there is a canonical bijection between initial conditions at a fixed time and solution trajectories.

If the dynamical system is Hamiltonian, the phase space is the space of initial conditions, hence is canonically isomorphic to the space of solution trajectories. Considering the latter is an advantage if one does not want to single out a particular time.

This is the case in relativistic field theory, where the choice of a time coordinate breaks manifest Poincare symmetry. The space of solution trajectories is manifestly covariant, while standard phase space is not. For those who prefer manifest covariance this is a decisive advantage.

See also this thread: Symplectic form on covariant phase space

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