# Discovering the Origins of Conservation Laws in Particle Simulations

• dyb
In summary, the conversation discusses the concept of conservation laws in a computer simulation of particles. The conservation of linear momentum, angular momentum, and energy is automatically achieved through the use of equations such as F=ma, v=a*t, and p=v*t. The speaker wonders where these conservation laws come from, and it is explained that they are related to symmetries in the equations and their numerical implementation. These symmetries include translation in time, translation in space, and rotation. However, some quantities may only be conserved on average due to time discretization.

#### dyb

Let's assume I simulate a number of particles using a computer program. I teach the particles to move according to F=ma. The F acting on each particle will be the sum of all forces to other particles according to F=m1*m2/distance^2.

I give the particles a set of initial positions and velocities and I automatically get conservation of linear momentum, conservation of angular momentum, and conservation of energy, just like that. I haven't actually programmed those properties into code.

Where does the conservation come from?

Basically I've only used F=m*a, v=a*t, p=v*t, action=reaction, the pythagorean theorem and some sums. Does it come from F=m*a, from action=reaction, is it a property of isometric space, or all of them?

Last edited by a moderator:
To understand the theorem, I'd need to understand what a differentiable symmetry is, what Lagrangian is, and so forth. If I knew all that, I wouldn't be asking.

Can it be pinpointed down to something more simple? Can you give a simplified answer?

The simple answer is that each conservation law is related to a symmetry. Conservation of energy comes from the symmetry of translation in time, conservation of momentum translation symmetry in space, and angular momentum rotation symmetry. If your equations (and importantly their numerical implementation) have these symmetries, then the quantities will be conserved.

Note that depending on the implementation (for instance, because of time discretization), some quantities might be conserved only on average.

Thanks!

## 1. What are conservation laws in particle simulations?

Conservation laws in particle simulations refer to fundamental physical principles that dictate the behavior of particles in a system. These laws state that certain properties, such as energy, momentum, and angular momentum, remain constant throughout a simulation, unless acted upon by an external force.

## 2. Why is it important to study the origins of conservation laws in particle simulations?

Studying the origins of conservation laws in particle simulations allows us to understand the fundamental laws of nature and how they apply to complex systems. This knowledge is crucial for accurately simulating and predicting the behavior of particles in various physical and chemical processes.

## 3. How do scientists discover the origins of conservation laws in particle simulations?

Scientists use a combination of theoretical models, mathematical equations, and computer simulations to investigate the origins of conservation laws in particle simulations. They also conduct experiments to validate their findings and refine their understanding of these laws.

## 4. What are some examples of conservation laws in particle simulations?

Some examples of conservation laws in particle simulations include the conservation of energy, which states that energy cannot be created or destroyed, only transferred or converted into different forms. Another example is the conservation of momentum, which states that the total momentum of a system remains constant unless acted upon by an external force.

## 5. How can our understanding of conservation laws in particle simulations benefit society?

Our understanding of conservation laws in particle simulations can have a wide range of applications in various fields, including physics, chemistry, engineering, and even medicine. It can help us design more efficient and sustainable technologies, improve our understanding of natural phenomena, and advance our knowledge of the universe.