Can the Velocity Verlet Algorithm Prove Energy Conservation?

In summary, the conversation discusses the algorithm that conserves energy in a system that also conserves energy. The speaker is looking for a proof of this statement, which is not obvious. The other person mentions Verlet integration as a method that approximately conserves energy and preserves angular momentum. They also mention a paper by Hairer, Lubich, and Wanner on this topic, specifically chapter 5 which focuses on energy conservation.
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whatisreality
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I've read that this algorithm conserves energy if the system it's applied to conserves energy. I can't find a proof, and it's not a particularly obvious statement, so how would you prove it?
 
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  • #2
Actually, Verlet integration is approximately conserving the (total) energy, it does however preserve total angular momentum. This is the reason it is used for orbital mechanics problems. The properties of the propagation matrix of the discretized system can already give you a lot of information, especially if you do it for the hamiltonian. There is a very nice paper by Hairer, Lubich and Wanner on this topic, it is still on my 'have-to-really-read' list,

http://citeseerx.ist.psu.edu/viewdo...CA0E3FCC9?doi=10.1.1.7.7106&rep=rep1&type=pdf

(chapter 5 is on energy conservation)
 
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1. What is the Velocity Verlet algorithm?

The Velocity Verlet algorithm is a numerical integration method used to solve the equations of motion for a physical system. It is commonly used in molecular dynamics simulations to calculate the positions and velocities of particles in a system over time.

2. How does the Velocity Verlet algorithm work?

The Velocity Verlet algorithm works by updating the positions and velocities of particles in a system in discrete timesteps. It uses the positions and velocities at the current timestep, as well as the forces acting on the particles, to calculate the positions and velocities for the next timestep. This process is repeated until the desired simulation time is reached.

3. What are the advantages of using the Velocity Verlet algorithm?

One advantage of the Velocity Verlet algorithm is that it conserves energy in a system, making it a more accurate method for long-term simulations. It is also relatively simple to implement and can handle a wide range of physical systems.

4. Are there any limitations to the Velocity Verlet algorithm?

One limitation of the Velocity Verlet algorithm is that it can become unstable if the timestep used is too large. This can lead to inaccurate results or even cause the simulation to crash. It also assumes that forces acting on particles are continuous, which may not always be the case.

5. How does the Velocity Verlet algorithm compare to other integration methods?

The Velocity Verlet algorithm is considered to be more accurate and stable compared to other integration methods such as the Euler or Verlet algorithms. It also requires fewer evaluations of the forces acting on particles, making it more computationally efficient.

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