## #004: See how much of relativity is already there in the particle equation

*Category: Classical mechanics - Tags: Differential geometry, Riemannian geometry, Symplectic geometry, General relativity*

Following the conventions set up here, apply the Hamiltonian constraint on the symplectic form to see whether there is a dependence between the EM field and the metric tensor.

*Mathematical problem.* In the linked note, we find the expression of the
symplectic form in terms of position and velocity. The expression lives on the
eight dimensional phase space (i.e. \(\{t, q^i, E, p_i\}\)), which is redundant
given the Hamiltonian constraint \(\mathcal{H} = \frac{1}{2} m u^\alpha g_{\alpha\beta} u^\beta\).
The idea is to use the constraint to lower the dimensionality of the form to a seven
dimensional non-symplectic form, and to understand whether, on that submanifold,
there is a relationship between the \(A_\alpha\) and \(g_{\alpha\beta}\) fields.

*Physical significance.* The Hamiltonian constraint plays a double role: it is the
generator of the affine parameter (i.e. “time” evolution) and is a conserved
quantity during evolution (specifically, the rest energy of the particle). What
we want to understand is whether this double role puts a constraint on how the
\(A_\alpha\) and \(g_{\alpha\beta}\) fields can vary across phase-space.

If there is indeed a connection, it would mean that there are some hints of general relativity in plain particle dynamics. If there isn’t, then the link between curvature and energy is a separate assumption from those required to derive particle dynamics under scalar and potential forces.