![]() ![]() Calculating a Hamiltonian from a Lagrangian. ![]() The time- derivative of q is the velocity, and so the second Hamilton equation means that the particle’s velocity equals the derivative of its kinetic energy with respect to its momentum. Here q is the space coordinate and p is the momentum mv. The Hamiltonian represents the total energy of the system, which is the sum of kinetic and potential energy, traditionally denoted T and V, respectively. While Hamiltonian mechanics can be used to describe simple systems such as a bouncing ball, a pendulum or an oscillating spring in which energy changes from kinetic to potential and back again over time, its strength is shown in more complex dynamic systems, such as planetary orbits in celestial mechanics. The main motivation to use Hamiltonian mechanics instead of Lagrangian mechanics comes from the symplectic structure of Hamiltonian systems. In fact, as is shown below, the Hamiltonian is the Legendre transform of the Lagrangian when holding q and t fixed and defining p as the dual variable, and thus both approaches give the same equations for the same generalized momentum. This approach is equivalent to the one used in Lagrangian mechanics. In contrast, in Hamiltonian mechanics, the time evolution is obtained by computing the Hamiltonian of the system in the generalized coordinates and inserting it in the Hamilton's equations. Lagrangian mechanics, a previous reformulation of classical mechanics introduced by Joseph Louis Lagrange in 1. Hamiltonian mechanics was first formulated by William Rowan Hamilton in 1. ![]() ![]() Historically, it was an important reformulation of classical mechanics, which later contributed to the formulation of statistical mechanics and quantum mechanics. It uses a different mathematical formalism, providing a more abstract understanding of the theory. Galactic Astronomy Binney Merrifield Djvu To Pdf ![]()
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