Establish the equivalence of the Hamiltonian equations (21)...

Establish the equivalence of the Hamiltonian equations (21) and Lagrangian ones (11) in an example of the motion of a single point mass in a stationary potential field, considering it in Cartesian coordinates $x, y, z$.

Key Concepts

Lagrangian Mechanics

Lagrangian mechanics is a reformulation of classical mechanics that expresses the dynamics of a system in terms of the Lagrangian function, defined as the difference between kinetic and potential energies. By applying the principle of least action, it leads to the Euler-Lagrange equations which determine the motion of a system along paths that make the action stationary.

Hamiltonian Mechanics

Hamiltonian mechanics offers an alternative formulation to classical mechanics which uses the Hamiltonian function, typically representing the total energy of the system, derived via a Legendre transformation of the Lagrangian. The evolution of the system is then described by Hamilton's canonical equations, which provide first-order differential equations in terms of coordinates and momenta.

Legendre Transformation

The Legendre transformation is a mathematical procedure used to switch from the Lagrangian to the Hamiltonian formulation. It converts the dependency on generalized velocities into a dependency on generalized momenta, thereby facilitating the derivation of Hamilton's equations from the Euler-Lagrange equations.

Equivalence of Formulations

The equivalence of the Lagrangian and Hamiltonian formulations lies in their ability to produce the same equations of motion for a system. By properly applying the Legendre transformation and recognizing the underlying variational principle, one can show that the Hamiltonian canonical equations are mathematically equivalent to the Euler-Lagrange equations obtained in the Lagrangian framework.

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