### Tag: Galilean transformation

How Hamiltonians vary under coordinate changes

## How Hamiltonians vary under coordinate changes

TL;DR – Hamiltonians are not invariant: they change as the time component of a covector (i.e. covariant component) in phase space. In the previous post we saw how momentum varies as covariant components and that keeps the Hamiltonian equations unchanged under coordinate transformations. We have also seen, though, that under coordinate transformations that mix time …

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Galilean transformations are not compatible with Hamiltonian mechanics

## Galilean transformations are not compatible with Hamiltonian mechanics

TL;DR Galilean boosts do not leave Hamilton’s equations and phase space volumes unchanged. They give good approximations only for small changes in velocity. When studying physics, you learn that relativistic mechanics is more “correct” and non-relativistic mechanics is just an approximation. But you may also get a sense that non-relativistic mechanics is self-consistent: it just …

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