What is the Difference Between Lagrangian and Hamiltonian Mechanics?

Lagrangian and Hamiltonian mechanics are two different formulations of classical mechanics that offer different perspectives and approaches to solving problems. The main differences between them are:

1. Coordinates and momenta: Lagrangian mechanics is expressed in terms of generalized coordinates and their time derivatives, while Hamiltonian mechanics is expressed in terms of generalized coordinates and their conjugate momenta.
2. Equations of motion: In Lagrangian mechanics, the equations of motion are derived from the Euler-Lagrange equations, which determine the coordinates and velocities directly. In Hamiltonian mechanics, the equations of motion are derived from Hamilton's equations, which determine the coordinates and momenta.
3. Numerical solutions: The Lagrangian approach is superior for typical undergraduate problems that require numerical solutions, as it can deal with any type of force (conservative or non-conservative) and directly determines the coordinates and velocities.
4. Philosophical and theoretical questions: Hamiltonian mechanics has a clear advantage for addressing more profound and philosophical questions, as it provides a broader arsenal of powerful techniques, such as the Poisson bracket formulation, canonical transformations, the Hamilton-Jacobi approach, the action-angle variables, and canonical perturbation theory.
5. Relativistic theories: Lagrangian mechanics is more suitable for applications in relativistic theories, while Hamiltonian mechanics naturally allows for a simpler way to deal with constrained systems.

Both formulations are equivalent to Newtonian mechanics and can handle many-body systems and exploit canonical transformations, which are impractical or impossible using vectorial Newtonian mechanics. The choice between Lagrangian and Hamiltonian mechanics depends on the specific problem and the techniques that are more suitable for solving it.

Comparative Table: Lagrangian vs Hamiltonian Mechanics

The main differences between Lagrangian and Hamiltonian mechanics are summarized in the table below:

Both Lagrangian and Hamiltonian mechanics are based on variational principles and can handle many-body systems and exploit canonical transformations. However, they represent physical systems differently and have different advantages in various applications.