Hamilton's equations and Lagrange's equations are two different formulations of classical mechanics that describe the motion of a system. While they are mathematically equivalent, they offer different perspectives and are useful in different scenarios.

Lagrange's equations of motion, formulated by Joseph-Louis Lagrange, are based on the principle of least action and provide a powerful way to derive the equations of motion for a system. They are expressed in terms of generalized coordinates (q) and their time derivatives (q̇), along with the system's Lagrangian (L). The Lagrangian is the difference between the system's kinetic energy (T) and potential energy (V): L = T - V.

The Lagrange's equations of motion are given by:

d/dt (∂L/∂q̇) - (∂L/∂q) = 0

These equations describe the evolution of the generalized coordinates q with respect to time and can be derived for systems with any number of degrees of freedom. They are particularly useful when dealing with complex systems with constraints, as they naturally incorporate the constraints into the formulation.

On the other hand, Hamilton's equations of motion, developed by William Rowan Hamilton, are formulated in terms of generalized coordinates (q) and their conjugate momenta (p). The momenta are defined as the partial derivatives of the Lagrangian with respect to the generalized velocities: p = ∂L/∂q̇.

The Hamilton's equations of motion are given by:

dq/dt = (∂H/∂p)

dp/dt = - (∂H/∂q)

Here, H represents the Hamiltonian, which is the total energy of the system expressed in terms of the generalized coordinates and momenta: H = T + V. Hamilton's equations provide a convenient way to determine the time evolution of the coordinates and momenta of a system.

Hamilton's equations are particularly useful in problems involving transformations and canonical variables, and they have close connections to symplectic geometry. They are often used in classical mechanics, Hamiltonian mechanics, and also find applications in quantum mechanics and other areas of theoretical physics.

In summary, while both Hamilton's equations and Lagrange's equations describe the motion of a system, they use different variables and offer different perspectives on the dynamics. Lagrange's equations are well-suited for problems with constraints, while Hamilton's equations are useful for problems involving transformations and canonical variables. The choice between the two formulations depends on the specific problem at hand and the most convenient mathematical representation.

To add a small comment: The equivalence of both formulations is often used to prove certain characteristics regarding preservation of quantities in the discrete setting, as some proofs are easier to implement in one version or the other. If you see some theory about variational integrators (e.g. in the paper from Marsden & West 2001), you will see that they use the Hamiltonian mechanics to infer some properties on the integrators and add some useful points of view (i.e. matching of momenta, discrete Legendre transforms)

This might not be what you're interested in but Hamiltonian particle tracking is vastly more efficient than Lagrangian, especially when working in finite element spaces. I have lots of examples.