Solve Hamilton-Jacobi Equation for Hamiltonian w/ Mixed Terms

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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.
  • #1
jamaicanking
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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.
 
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  • #2
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?
 
  • #3
Actually this is quite trivial..thank you anyway.
 

Related to Solve Hamilton-Jacobi Equation for Hamiltonian w/ Mixed Terms

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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