Symplectic Integrator Research: Solver & Reference Suggestions

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In summary, the speaker is working on a physics research project that involves computer modeling and using a numerical ODE solver. They initially wrote their own solver but are now looking for a symplectic integrator specifically for Hamiltonian systems. They are also seeking a good reference on the concept of symplectic quantities in solving ODEs.
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cipher_42
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This summer I am doing a physics research project involving some computer modeling. One of the main aspects of the simulation that I am doing is using a numerical ODE solver. I originally wrote my own solver using the 4th-order Runge-Kutta method with a variable step size just to get a feel for how these programs work, but in the end I'll be using a different program.

Now I just learned about symplectic ODE solvers the other day and, given the nature of the problem that I'm working on, it seems like one of these integrators would be the way to go (my system is Hamiltonian with constant energy). So, after all of that setup, two questions:

1) Does anyone know of any good symplectic integrators out there? I've searched the internet but the only one I can find (named DiffMan) only runs on UNIX and, unfortunately, I don't have access to a UNIX box on which I could install it. Ideally, I'm looking for a MATLAB program, but I could port it if it's not.
2) I still don't understand all of the ideas behind how these programs use symplectic quantities to solve ODEs, so if anyone knows of a good reference, that would be appreciated as well.

Thanks!

- Jason
 
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A good reference for this is E. Hairer et al., Geometric Numerical Integration, 2nd ed. (Springer, Berlin, 2006).
 

1. What is a symplectic integrator?

A symplectic integrator is a numerical method used to solve differential equations in physics and engineering. It is specifically designed to preserve the symplectic structure of the equations, which ensures the conservation of energy and other important physical quantities.

2. How does a symplectic integrator work?

A symplectic integrator works by breaking down a complex differential equation into smaller, simpler equations that can be solved using numerical methods. These smaller equations are then solved iteratively, with each step preserving the symplectic structure of the original equation. This ensures that the solution accurately reflects the behavior of the system over time.

3. What makes symplectic integrators different from other numerical integration methods?

Symplectic integrators are different from other numerical integration methods because they specifically take into account the symplectic structure of the equations being solved. This allows them to accurately preserve important physical quantities, such as energy, momentum, and angular momentum, which may be lost with other numerical methods.

4. What are some applications of symplectic integrators?

Symplectic integrators are commonly used in physics and engineering to solve problems involving systems that exhibit symplectic structure, such as celestial mechanics, molecular dynamics, and Hamiltonian systems. They are also used in computer graphics to simulate the motion of objects in video games and animations.

5. Are there different types of symplectic integrators?

Yes, there are various types of symplectic integrators, each designed for different types of problems and equations. Some common types include the Euler method, Verlet method, and Runge-Kutta method. The choice of which type to use depends on the specific problem and the desired level of accuracy.

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