This is a special case of this question of mine, which, I think, might have drawn little attention because it was too general. In this question, I would like to consider a specific case.

Take a classical system given by the action $$ S[x] = \int dt \left[ \frac{\alpha}{2} \ddot{x}^2 + \frac{\beta}{2} \dot{x}^2 + \frac{\gamma}{2} x^2 \right]. $$

This action is Lagrangian, but it is not what we are usually dealing with in physics, because the Lagrangian contains second time derivatives.

The equations of motion are:

$$ \alpha \ddddot{x} - \beta \ddot{x} + \gamma x = 0. $$

This can be solved as usual by employing Fourier transform:

$$ x(t) = a_1 e^{i \omega_1 t} + a_1^{*} e^{-i \omega_1 t} + a_2 e^{i \omega_2 t} + a_2^{*} e^{-i \omega_2 t}, $$

where $\omega_1^2$ and $\omega_2^2$ are the two roots of $\alpha \omega^4 + \beta \omega^2 + \gamma$, given by

$$ \omega_{1,2}^2 = \frac{- \beta \pm \sqrt{\beta^2 - 4 \alpha \gamma}}{2 \alpha} .$$

Now by definition the phase space is the space of solutions of equations of motion. In this case it is parametrized by two complex co-ordinates $a_{1,2}$, meaning that it is 4-dimensional.

I would like to know if there's a natural way to associate the symplectic structure (Poisson bracket) on this phase space. I.e.,

$$ \{a_1, a_1^{*}\} = ? $$ $$ \{a_1, a_2^{*}\} = ? $$ $$ \{a_2, a_1^{*}\} = ? $$ $$ \{a_2, a_2^{*}\} = ? $$ $$ \{a_1, a_2\} = ? $$ $$ \{a_1^{*}, a_2^{*}\} = ? $$


FWIW, the Ostrogradsky formalism provides a systematic recipe for obtaining a Hamiltonian phase space formulation of an arbitrary Lagrangian theory with higher-time derivatives. However, it famously leads to quantum mechanical instabilities, cf. e.g. this Phys.SE answer by user genneth.

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