# Summary of Hamilton-Jacobi mechanics

• vanesch
In summary, Hamilton-Jacobi mechanics is a theoretical framework in classical mechanics that uses the Hamilton-Jacobi equation to describe the motion of particles in a conservative system. It differs from Lagrangian mechanics in its approach and has advantages such as the separation of variables and a more intuitive understanding of dynamics. It has many real-world applications, including studying celestial mechanics and optimizing trajectories for vehicles.
vanesch
Staff Emeritus
Gold Member
Some while ago I wrote this short summary, inspired by Goldstein, about Lagrangian and Hamiltonian's formulations of classical mechanics, in a rather informal style. Maybe it can be useful to some people.

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this is good, maybe a derivation of the euler-lagrange equations right from the start (via least action) would be something to add to it

Hamilton-Jacobi mechanics is a powerful mathematical framework for understanding classical mechanics, developed by William Rowan Hamilton and Carl Gustav Jacob Jacobi in the 19th century. It is based on the fundamental principle of least action, which states that a physical system will follow the path that minimizes the action, or the integral of the Lagrangian over time.

In this framework, the dynamics of a system are described by a set of canonical coordinates, which represent the position and momentum of the system's particles. The Hamiltonian, a function of these coordinates, describes the total energy of the system.

One key aspect of Hamilton-Jacobi mechanics is the use of Hamilton's equations, which relate the time derivatives of the canonical coordinates to the partial derivatives of the Hamiltonian. These equations provide a powerful tool for solving complex systems and predicting their behavior over time.

Another important concept in this framework is the Hamilton-Jacobi equation, which is a partial differential equation that relates the Hamiltonian to the action of the system. Solving this equation yields the action as a function of the canonical coordinates, allowing us to determine the path that minimizes the action and thus the behavior of the system.

Overall, Hamilton-Jacobi mechanics provides a rigorous and elegant mathematical framework for understanding classical mechanics and has been instrumental in advancing our understanding of physical systems.

## What is Hamilton-Jacobi mechanics?

Hamilton-Jacobi mechanics is a theoretical framework in classical mechanics that describes the motion of particles in a conservative system. It is based on the Hamilton-Jacobi equation, which is a partial differential equation that relates the position, velocity, and potential energy of a particle in a system.

## What is the Hamiltonian function in Hamilton-Jacobi mechanics?

The Hamiltonian function in Hamilton-Jacobi mechanics is a mathematical function that represents the total energy of a particle in a system. It is derived from the Lagrangian function and contains terms for the kinetic and potential energies of the particle.

## How does Hamilton-Jacobi mechanics differ from Lagrangian mechanics?

While both Hamilton-Jacobi mechanics and Lagrangian mechanics are used to describe the motion of particles in a conservative system, they differ in their approach. Hamilton-Jacobi mechanics uses the Hamilton-Jacobi equation to find the equations of motion, while Lagrangian mechanics uses the Euler-Lagrange equations.

## What are the advantages of using Hamilton-Jacobi mechanics?

One of the main advantages of using Hamilton-Jacobi mechanics is that it allows for the separation of variables, which simplifies the equations of motion and makes them easier to solve. It also provides a more intuitive understanding of the dynamics of a system.

## How is Hamilton-Jacobi mechanics used in real-world applications?

Hamilton-Jacobi mechanics has many applications in physics, engineering, and other fields. It is used to study the motion of particles in celestial mechanics, fluid dynamics, and quantum mechanics, among others. It also has practical applications in the design of control systems and in optimizing trajectories for spacecraft and other vehicles.

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