How do action principles connect to Hamiltonian mechanics?

Analytical mechanics uses scalar quantities and variational principles to describe physical systems, with Hamiltonian mechanics and Lagrangian mechanics forming its two dominant branches. All equations of motion in these formalisms can be derived from the principle of least action. Furthermore, geometric concepts like geodesic equations can be framed either as Euler-Lagrange equations or as sets of coupled first-order Hamiltonian equations.

What we know

• All equations of motion for particles and fields across any analytical formalism can be derived from the principle of least action. [3]
• Lagrangian mechanics and Hamiltonian mechanics are equivalent and connected through a Legendre transformation on generalized coordinates, velocities, and momenta. [3]
• Geodesic equations are commonly presented as Euler-Lagrange equations of motion, but they can also be formulated as a set of coupled first-order equations representing Hamilton's equations. [2]
• Action is a scalar functional that takes a trajectory as its argument and yields a real number, describing how kinetic versus potential energy balance changes along a path. [4]

Where to go next

• the bigger pictureWhat is Lagrangian mechanics?
• how it workedHow does the Legendre transformation link formalisms?
• what caused itWhat is the principle of least action?
• a case in pointHow do Reeb vector flows apply to contact geometry?
• compared withHow does analytical mechanics compare to Newtonian mechanics?
• an unexpected connectionHow does quantum mechanics use the principle of least action?Action and variational principles are not restricted to classical physics, but are also utilized in Feynman's formulation of quantum mechanics.

Sources

• [1]Geodesics as Hamiltonian flows (Wikidata Q5533985) · CC0 1.0
• [2]Geodesics as Hamiltonian flows · CC BY-SA 4.0
• [3]Analytical mechanics · CC BY-SA 4.0
• [4]Action (physics) · CC BY-SA 4.0
• [5]Classical mechanics · CC BY-SA 4.0