equations
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| ====== Equations ====== | ====== Equations ====== | ||
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| | | **Important in:** | **Relationship:** | **Used For:** | | | | | **Important in:** | **Relationship:** | **Used For:** | | | ||
| - | | [[equations:schroedinger_equation|Schrödinger Equation]] | [[theories:quantum_mechanics:canonical_quantum_mechanics|Quantum Mechanics]], [[theories:quantum_field_theory|Quantum Field Theory]] | non-relativistic limit of the Klein-Gordon Equation | Describes time evolution | linear | | + | | [[equations:schroedinger_equation|Schrödinger Equation]] | [[theories:quantum_mechanics:canonical|Quantum Mechanics]], [[theories:quantum_field_theory:canonical|Quantum Field Theory]] | non-relativistic limit of the Klein-Gordon Equation | Describes time evolution | linear | |
| - | | [[equations:klein-gordon_equation|Klein-Gordon Equation]] | [[theories:quantum_field_theory|Quantum Field Theory]] | | Equation of motion for particles with [[basic_notions:spin|spin]] 0 | linear | | + | | [[equations:klein-gordon_equation|Klein-Gordon Equation]] | [[theories:quantum_field_theory:canonical|Quantum Field Theory]] | | Equation of motion for particles with [[basic_notions:spin|spin]] 0 | linear | |
| - | | [[equations:pauli_equation|Pauli Equation]] | [[theories:quantum_mechanics:canonical_quantum_mechanics|Quantum Mechanics]] | non-relativistic limit of the Dirac Equation | Equation of motion for particles with spin 1/2 | linear | | + | | [[equations:pauli_equation|Pauli Equation]] | [[theories:quantum_mechanics:canonical|Quantum Mechanics]] | non-relativistic limit of the Dirac Equation | Equation of motion for particles with spin 1/2 | linear | |
| - | | [[equations:dirac_equation|Dirac Equation]] | [[theories:quantum_field_theory|Quantum Field Theory]] | | Equation of motion for particles with spin 1/2 | linear | | + | | [[equations:dirac_equation|Dirac Equation]] | [[theories:quantum_field_theory:canonical|Quantum Field Theory]] | | Equation of motion for particles with spin 1/2 | linear | |
| - | | [[equations:maxwell_equations|Maxwell Equations]] | [[models:classical_electrodynamics|Classical Electrodynamics]], [[theories:quantum_field_theory|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 | | + | | [[equations:maxwell_equations|Maxwell Equations]] | [[models:classical_electrodynamics|Classical Electrodynamics]], [[theories:quantum_field_theory:canonical|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 | |
| - | | [[equations:einstein_equation|Einstein Equation]] | [[theories:general_relativity|General Relativity]] | | Describes how spacetime gets curved through energy and matter | non-linear | | + | | [[equations:einstein_equation|Einstein Equation]] | [[models:general_relativity|General Relativity]] | | Describes how spacetime gets curved through energy and matter | non-linear | |
| - | | [[equations:yang_mills_equations|Yang-Mills Equation]] | [[theories:quantum_field_theory|Quantum Field Theory]] | | Equation of motion for particles with spin 1 in non-abelian gauge theories | non-linear | | + | | [[equations:yang_mills_equations|Yang-Mills Equation]] | [[theories:quantum_field_theory:canonical|Quantum Field Theory]] | | Equation of motion for particles with spin 1 in non-abelian gauge theories | non-linear | |
| | [[equations:navier_stokes]] | [[theories:classical_theories:hydrodynamics|Hydrodynamics]] | | Describe the flow of fluids | non-linear | | | [[equations:navier_stokes]] | [[theories:classical_theories:hydrodynamics|Hydrodynamics]] | | Describe the flow of fluids | non-linear | | ||
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| 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, | 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, | ||
| - | * [[equations:newtons_law|Newton's law of gravity]] | + | * [[formulas:newtons_law|Newton's law of gravity]] |
| - | * [[equations:lorentz_force_law|Lorentz' force law]] | + | * [[formulas:lorentz_force_law|Lorentz' force law]] |
| - | * [[equations:coulombs_law|Coulomb's force law]] | + | * [[formulas:coulombs_law|Coulomb's force law]] |
| In addition, we always need to specify the [[basic_notions:boundary_conditions]] for the system in question. | In addition, we always need to specify the [[basic_notions:boundary_conditions]] for the system in question. | ||
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| **The Equations of Motion yield the Most Important Paths** | **The Equations of Motion yield the Most Important Paths** | ||
| - | In the [[theories:quantum_mechanics:path_integral|path integral formuation]] of [[theories:quantum_mechanics:canonical_quantum_mechanics|quantum mechanics]], particles do not follow one individual path but instead all of them. Hence there can't be one equation whose solution yields the correct particle trajectory. | + | In the [[theories:quantum_mechanics:path_integral|path integral formuation]] of [[theories:quantum_mechanics:canonical|quantum mechanics]], particles do not follow one individual path but instead all of them. Hence there can't be one equation whose solution yields the correct particle trajectory. |
| However, the equations of motion are still important and the path integral formalism tells us why. | However, the equations of motion are still important and the path integral formalism tells us why. | ||
equations.1525513968.txt.gz · Last modified: 2018/05/05 09:52 (external edit)
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