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I've heard of Lagrangian systems but what are Hamiltonian systems?

Kendall wrote: "I've heard of Lagrangian systems but what are Hamiltonian systems?"

They are essentially two different ways to skin the same cat: the Hamiltonian is the Legendre transform of the Lagrangian, so they can be converted into each other.

Superficially speaking, the Lagrangians are functions of positions and velocities, while the Hamiltonians are functions of positions and momenta. Moreover, one approach results into one second order differential equation, while the other results into two first order differential equations.

Conceptually, though, they are quite different in term of overall approach: the Lagrangian, for example, is related to the concept of action.
This is a good summary of the key differences between the two approaches: https://profoundphysics.com/lagrangia....