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| ====== Equations ====== | ====== Equations ====== | ||
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| **Equations of Motion** | **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 [[basic_tools:symmetry|symmetry]] considerations from the corresponding [[frameworks:lagrangian_formalism|Lagrangian]] using the [[equations:euler_lagrange_equations|Euler-Lagrange Equations]]. | + | 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 [[basic_tools:symmetry|symmetry]] considerations from the corresponding [[formalisms:lagrangian_formalism|Lagrangian]] using the [[equations:euler_lagrange_equations|Euler-Lagrange Equations]]. |
| | | **Important in:** | **Relationship:** | **Used For:** | | | | | **Important in:** | **Relationship:** | **Used For:** | | | ||
| - | | [[equations:schroedinger_equation|Schrödinger Equation]] | [[theories: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|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 | | ||
| ---- | ---- | ||
| - | **Supplementary Equations** | + | **Supplementary equations and boundary conditions** |
| + | <WRAP group> | ||
| + | <WRAP half column> | ||
| + | <diagram> | ||
| + | |AA|AA=Equation of Motion | ||
| + | ||!@4|| | ||
| + | |AA|AA=System specific additions, like the interactions/forces acting on the object in question | ||
| + | ||!@4|| | ||
| + | |AA|AA=Boundary conditions | ||
| + | ||!@4|| | ||
| + | |AA|AA=Solutions | ||
| + | </diagram> | ||
| + | </WRAP> | ||
| + | <WRAP half column> | ||
| + | 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 desribe a force that acts on the object in question. | + | * [[formulas:newtons_law|Newton's law of gravity]] |
| - | + | * [[formulas:lorentz_force_law|Lorentz' force law]] | |
| - | * [[equations:newtons_law|Newton's law of gravity]] | + | * [[formulas:coulombs_law|Coulomb's force law]] |
| - | * [[equations:lorentz_force_law|Lorentz' force law]] | + | |
| - | * [[equations:coulombs_law|Coulomb's force law]] | + | |
| + | In addition, we always need to specify the [[basic_notions:boundary_conditions]] for the system in question. | ||
| + | </WRAP> | ||
| + | </WRAP> | ||
| <tabbox Solutions> | <tabbox Solutions> | ||
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| <tabbox Origin> | <tabbox Origin> | ||
| - | One popular way to derive the fundamental equations of nature is to use the [[frameworks:lagrangian_formalism|Lagrangian formalism]]. | + | One popular way to derive the fundamental equations of nature is to use the [[formalisms:lagrangian_formalism|Lagrangian formalism]]. |
| - The first step is to write down the Lagrangian of the system. We use the Lagrangian to derive the equation of motions. Hence, if we want that our equations of motion are the same in all allowed frames of reference, the Lagrangian must be invariant under all these transformations. This is a powerful constraint that we can use to find the correct Lagrangian. Formulated differently, our Lagrangian must always be invariant under all symmetries of the system. In practice this means we write down all possible terms that respect the symmetries of the system but only include the lowest non-trivial order terms. | - The first step is to write down the Lagrangian of the system. We use the Lagrangian to derive the equation of motions. Hence, if we want that our equations of motion are the same in all allowed frames of reference, the Lagrangian must be invariant under all these transformations. This is a powerful constraint that we can use to find the correct Lagrangian. Formulated differently, our Lagrangian must always be invariant under all symmetries of the system. In practice this means we write down all possible terms that respect the symmetries of the system but only include the lowest non-trivial order terms. | ||
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| **The Equations of Motion yield the Most Important Paths** | **The Equations of Motion yield the Most Important Paths** | ||
| - | In the [[advanced_tools:path_integral|path integral formuation]] of [[theories: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. | ||
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| When in the 19th century people tried to understand how electromagnetism works they also figured this out. However, they made also another intriguing discovery. When writing down the laws which govern electromagnetism, it turns out that electric and magnetic fields are intimately linked, and that they are just two sides of the same coin. That is the reason to call it electromagnetism. | When in the 19th century people tried to understand how electromagnetism works they also figured this out. However, they made also another intriguing discovery. When writing down the laws which govern electromagnetism, it turns out that electric and magnetic fields are intimately linked, and that they are just two sides of the same coin. That is the reason to call it electromagnetism. | ||
| - | //In the early 20th century it then became clear that both phenomena can be associated with a single particle, the photon. But then it was found that to characterize a photon only two numbers at each point in space and time are necessary. This implies that between the six numbers characterizing electric and magnetic fields relations exist. These are known as [[equations:maxwell_equations|Maxwell equations]]// in classical physics, or as quantum Maxwell dynamics in the quantum theory. If you would add, e. g., electrons to this theory, you would end up with [[models:quantum_electrodynamics|quantum electro dynamics - QED]]. | + | //In the early 20th century it then became clear that both phenomena can be associated with a single particle, the photon. But then it was found that to characterize a photon only two numbers at each point in space and time are necessary. This implies that between the six numbers characterizing electric and magnetic fields relations exist. These are known as [[equations:maxwell_equations|Maxwell equations]]// in classical physics, or as quantum Maxwell dynamics in the quantum theory. If you would add, e. g., electrons to this theory, you would end up with [[models:standard_model:qed|quantum electro dynamics - QED]]. |
| <cite>http://axelmaas.blogspot.de/2010/10/electromagnetism-photons-and-symmetry.html</cite></blockquote> | <cite>http://axelmaas.blogspot.de/2010/10/electromagnetism-photons-and-symmetry.html</cite></blockquote> | ||
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