Solving Euler’s formula for polyhedra, Navier-Stokes equations, and Feynman path integral using Python

Euler’s formula, the Navier-Stokes equations, and the Feynman path integral are three important concepts in the fields of mathematics and physics. Just to clarify, they are not a tall connected to one another directly, however, they are all related to our understanding of the world around us. We will be using the Python programming language to explore and solve these concepts. My assumption is that you understand all these formula’s and hence attempting to solve it using Python.

Euler’s formula is a tool that helps us understand the structure of three-dimensional shapes called polyhedra. It tells us the relationship between the number of vertices, edges, and faces that make up a polyhedron.

The Navier-Stokes equations, on the other hand, are used to study the movement of fluids. These equations are a set of mathematical statements that describe how fluids behave and are used to model the flow of liquids and gases in many different situations.

The Feynman path integral, named after physicist Richard Feynman, is a way of understanding the behavior of particles at the quantum level. It allows physicists to make predictions about the actions of particles by considering all of the possible paths they might take and calculating the chances of each one occurring.

Solving Euler’s formula for polyhedra using Python code

This is a Python function that checks if a three-dimensional shape, called a polyhedron, follows a certain rule called Euler’s formula. To use the function, you need to give it three whole numbers, which represent the number of flat faces, straight edges, and points on the polyhedron. The function will then tell you if the polyhedron follows Euler’s formula by giving you either ‘True’ or ‘False’:

def satisfies_euler(V, E, F):
  return V - E + F == 2

To use this function, you can pass in the values of V, E, and F as arguments. For example, to check whether a cube satisfies Euler’s formula, you could call the function as follows:

V = 8  # A cube has 8 vertices
E = 12 # A cube has 12 edges
F = 6  # A cube has 6 faces

if satisfies_euler(V, E, F):
  print("The polyhedron satisfies Euler's formula.")
  print("The polyhedron does not satisfy Euler's formula.")

This would print “The polyhedron satisfies Euler’s formula.” because a cube has 8 vertices, 12 edges, and 6 faces, and 8 – 12 + 6 = 2, which is the expected result.

Solving Navier-Stokes equations using Python code

To find out how fluids move and behave, we can use a set of equations called the Navier-Stokes equations. These equations describe how the velocity and pressure of a fluid change over time and space. We can’t usually solve these equations using simple math, so we have to use special techniques called numerical methods.

There are some Python programs that have tools for solving these types of equations, such as FEniCS, Firedrake, and PyPDE. Here’s an example of how you can use the FEniCS library to work with the Navier-Stokes equations for a basic fluid flow:

import fenics as fe

# Define the domain and the boundaries
mesh = fe.UnitSquareMesh(8, 8)
boundaries = fe.FacetFunction("size_t", mesh)
fe.CompiledSubDomain("near(x[0], 0)").mark(boundaries, 1)
fe.CompiledSubDomain("near(x[0], 1)").mark(boundaries, 2)
ds = fe.Measure("ds")[boundaries]

# Define the function space for the velocity
V = fe.VectorFunctionSpace(mesh, "CG", 2)

# Define the functions for the velocity and the pressure
u = fe.TrialFunction(V)
v = fe.TestFunction(V)
p = fe.Function(V)

# Define the Navier-Stokes equations
nu = 1.0  # Viscosity coefficient
f = fe.Constant((0, 0))  # External force
a = nu*fe.inner(fe.grad(u), fe.grad(v))*fe.dx + fe.inner(u, v)*fe.dx - p*fe.div(v)*fe.dx + q*fe.div(u)*fe.dx
L = fe.inner(f, v)*fe.dx

# Define the boundary conditions
bcs = [fe.DirichletBC(V, (0, 0), boundaries, 1),
       fe.DirichletBC(V, (1, 0), boundaries, 2)]

# Solve the problem
fe.solve(a == L, u, bcs)

# Extract the pressure field
p = fe.Function(V)
fe.solve(fe.inner(p, q)*fe.dx == -fe.inner(u, fe.grad(q))*fe.dx, p, bcs)

This code defines the domain of the flow (a unit square), the boundary conditions (velocity set to zero at the left boundary and to (1, 0) at the right boundary), and the Navier-Stokes equations in weak form. It then uses the solve function from the FEniCS library to find the solution for the velocity field u, and it extracts the pressure field p by solving an additional equation.

Solving Feynman path integral using Python code

The Feynman path integral is a way of figuring out the chances of a tiny particle moving from one place to another in quantum mechanics. It involves adding up an infinite number of possible paths that the particle could take, each with its own strength. Calculating the path integral involves doing complicated math with all of the possible paths, which can be difficult to do on a computer.

There are some Python programs that have tools for working with path integrals, like QuTiP or PathInt. Here’s an example of how you can use the QuTiP library to find the Feynman path integral for a basic quantum system:

import qutip as qt
import numpy as np

# Define the Hamiltonian and the initial and final states
H = qt.sigmax()  # The Hamiltonian is a Pauli matrix
psi0 = qt.basis(2, 0)  # The initial state is the ground state
psi1 = qt.basis(2, 1)  # The final state is the excited state

# Define the time evolution operator
U = (-1j*H*dt).expm()

# Define the time grid
tlist = np.linspace(0, T, N)

# Calculate the path integral
result = qt.mesolve(U, psi0, tlist, [], psi1)

# Extract the probability of transition from the final state
prob = result.expect[0][-1]

This code defines the Hamiltonian of the system (a Pauli matrix), the initial and final states of the particle, and the time evolution operator. It then defines a time grid and uses the mesolve function from the QuTiP library to calculate the time evolution of the system. Finally, it extracts the probability of transition from the final state from the result of the calculation.

Closing Note: Solving Euler’s formula for polyhedra, Navier-Stokes equations, and Feynman path integral using Python

The use of Python has proven to be an effective tool for solving complex mathematical problems such as Euler’s formula for polyhedra, the Navier-Stokes equations, and the Feynman path integral. By implementing the techniques described in this article, readers can not only gain a deeper understanding of these concepts, but also develop valuable skills in programming and problem-solving. While there may be challenges along the way, the benefits of using Python to tackle these types of problems are numerous and well worth the effort. We hope that this article has provided a helpful introduction to these concepts and that readers will continue to explore and expand upon the ideas presented here.

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