Equations of Motion
The most important equation on modern physics are equations of motions. These equations tell us how a system will evolve as time passes on. We can derive these equations using symmetry considerations from the corresponding Lagrangian using the Euler-Lagrange Equations.
| Important in: | Relationship: | Used For: | ||
| Schrödinger Equation | Quantum Mechanics, Quantum Field Theory | non-relativistic limit of the Klein-Gordon Equation | Describes time evolution | linear |
| Klein-Gordon Equation | Quantum Field Theory | Equation of motion for particles with spin 0 | linear | |
| Pauli Equation | Quantum Mechanics | non-relativistic limit of the Dirac Equation | Equation of motion for particles with spin 1/2 | linear |
| Dirac Equation | Quantum Field Theory | Equation of motion for particles with spin 1/2 | linear | |
| Maxwell Equations | Classical Electrodynamics, Quantum Field Theory | special case of the Yang-Mills equation for a non-abelian gauge theory | Equation of motion for particles with spin 1 in abelian gauge theories | linear |
| Einstein Equation | General Relativity | Describes how spacetime gets curved through energy and matter | non-linear | |
| Yang-Mills Equation | Quantum Field Theory | Equation of motion for particles with spin 1 in non-abelian gauge theories | non-linear | |
| navier_stokes | Hydrodynamics | Describe the flow of fluids | non-linear |
Supplementary equations and boundary conditions
| Equation of Motion | ||||||
| System specific additions, like the interactions/forces acting on the object in question | ||||||
| Boundary conditions | ||||||
| Solutions | ||||||
The equations of motion are usually not enough to describe a system. Especially in the Newtonian framework, we need additional equations that give us, for example, the correct formulas which describe a force that acts on the object in question. For example,
In addition, we always need to specify the boundary_conditions for the system in question.
Unfortunately, knowing how to write down the equations is not the same as being able to solve them. For example, we know very well the equations of motions describing how a river flows. But as soon as it flows quickly over rough grounds, such that it becomes turbulent, we are no longer able to solve the equations. In such cases we are often forced to revert to the simulation methods discussed previously. […] Thus, only for very, very simple theories it is possible to solve these equations exactly. For theories like the standard model, one has to introduce severe approximations (often called truncations) to be able to solve them. If these approximations are made wisely and with insight, these approximations are such that still questions we have to the theory can be answered correctly. But it takes often very long to understand how to do approximations