HamiltonJacobiEquation - crowlogic/arb4j GitHub Wiki

The Hamilton-Jacobi equation is a partial differential equation that arises in the context of classical mechanics, providing an alternative formulation to the Hamiltonian and Lagrangian approaches. It is named after William Rowan Hamilton and Carl Gustav Jacobi, who made significant contributions to the development of this theory. The Hamilton-Jacobi equation plays a crucial role in the study of integrable systems and provides a powerful method for solving problems in classical mechanics.

The Hamilton-Jacobi equation can be written as follows:

$$ S'(q, t) + H(q, p, t) = 0 $$

where

• $S(q, t)$ is the generating function, also known as the Hamilton's principal function or the action,
• $S'(q, t)$ is the partial derivative of $S$ with respect to time $t$,
• $q$ represents the generalized coordinates of the system,
• $p = \frac{\partial S}{\partial q}$ represents the generalized momenta, expressed as partial derivatives of $S$ with respect to the generalized coordinates $q$,
• $H(q, p, t)$ is the Hamiltonian function of the system, which represents the total energy (the sum of kinetic and potential energy) and governs the system's time evolution.