# Solve Hamilton-Jacobi Equation for Hamiltonian w/ Mixed Terms

• jamaicanking
In summary, the conversation discusses how to solve the Hamilton-Jacobi equation for a Hamiltonian with mixed terms. The individual is unsure of how to separate the variables and solve the differential equation, but later realizes that it is a trivial task.
jamaicanking
how would you solve the hamilton - jacobi equation for something with a hamiltonian with mixed terms like 1/2(p1q2 + 2p1p2 + (q1)^2)

well its quite trivial obtaining the HJ equation since there is no time dependence,

1/2( (ds/dq1)q2 + 2(ds/dq1)(ds/dq2) + (q1)^2 ) = E

I can't see how youw would separate the variables otherwise we could simple set
H(q1,p1) = E1 amd H(q2,p2) = E2 .

However I am stumped on how to do it for the above equation with mixed terms.

I guess I did not phrase the question well.

The issue is I have a given hamiltonian H = 1/2(p1q2 + 2p1p2 + (q1)^2)

I need to solve this and I chose to begin by using the hamilton - jacobi equation and since we have no time dependence . If S is the hamilton action function then

((ds/dq1)*q1 + 2(ds/dq1)(ds/dq2) + (q1)^2) = E where E is now energy.

How would you go about solving this differential equation?

Actually this is quite trivial..thank you anyway.

## What is the Hamilton-Jacobi equation?

The Hamilton-Jacobi equation is a partial differential equation in classical mechanics that describes the evolution of a system over time. It is used to determine the trajectories of particles in a system based on their initial conditions and the system's Hamiltonian, which is a function that represents the total energy of the system.

## Why is the Hamilton-Jacobi equation important?

The Hamilton-Jacobi equation is important because it allows us to solve complex problems in classical mechanics, such as finding the optimal path for a particle to take in a system. It also provides a deeper understanding of the underlying principles of classical mechanics.

## What are mixed terms in the Hamiltonian?

Mixed terms in the Hamiltonian refer to terms that involve both position and momentum variables. These terms are important in systems with non-conservative forces, where the total energy of the system is not conserved.

## How do you solve the Hamilton-Jacobi equation for Hamiltonian with mixed terms?

To solve the Hamilton-Jacobi equation for Hamiltonian with mixed terms, we first separate the Hamiltonian into its position and momentum components. Then, we use a transformation known as the Hamilton-Jacobi transformation to reduce the equation to a simpler form. Finally, we solve the resulting equation using standard techniques such as separation of variables or the method of characteristics.

## What are some applications of the Hamilton-Jacobi equation?

The Hamilton-Jacobi equation has many applications in physics and engineering, including celestial mechanics, quantum mechanics, and control theory. It is also used in the calculation of action-angle variables, which are important in the study of integrable systems. Additionally, the Hamilton-Jacobi equation is used in the development of numerical methods for solving complex problems in classical mechanics.

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