|
|
| Line 1: |
Line 1: |
| A '''classical field theory''' is a [[physical theory]] that describes the study of how one or more [[field (physics)|physical fields]] interact with matter. The word 'classical' is used in contrast to those field theories that incorporate quantum mechanics ([[quantum field theory|quantum field theories]]).
| | Hi, I am Nicholas Madruga. After becoming out of my job for many years I grew to become an [http://Www.Officesupervisor.net/ office supervisor] but soon my spouse and I will start our personal business. Hawaii is the only location I've been residing in and my parents live nearby. To play domino is 1 of the issues he loves most. She is operating and sustaining a blog right here: http://www.evolutisweb.com/index.php/Six_Things_You_Didn_t_Know_About_Nya_Internet_Svenska_Casino<br><br> |
|
| |
|
| A physical field can be thought of as the assignment of a [[physical quantity]] at each point of [[space]] and [[time]]. For example, in a weather forecast, the wind velocity during a day over a country is described by assigning a vector to each point in space. Each vector represents the direction of the movement of air at that point. As the day progresses, the directions in which the vectors point change as the directions of the wind change. From the mathematical viewpoint, classical fields are described by sections of [[fiber bundle]]s ([[covariant classical field theory]]). The term 'classical field theory' is commonly reserved for describing those physical theories that describe [[electromagnetism]] and [[gravitation]], two of the [[fundamental force]]s of nature.
| | Check out my page: Nya online casinon 2014 ([http://www.evolutisweb.com/index.php/Six_Things_You_Didn_t_Know_About_Nya_Internet_Svenska_Casino Klikk på neste post]) |
| | |
| Descriptions of physical fields were given before the advent of [[relativity theory]] and then revised in light of this theory. Consequently, classical field theories are usually categorised as ''non-relativistic'' and ''relativistic''.
| |
| | |
| == Non-relativistic field theories ==
| |
| | |
| Some of the simplest physical fields are vector force fields. Historically, the first time fields were taken seriously was with [[Michael Faraday|Faraday's]] [[lines of force]] when describing the [[electric field]]. The [[gravitational field]] was then similarly described.
| |
| | |
| ===Newtonian gravitation===
| |
| | |
| A classical field theory describing gravity is [[gravity|Newtonian gravitation]], which describes the gravitational force as a mutual interaction between two [[mass]]es.
| |
| | |
| Any massive body ''M'' has a [[gravitational field]] '''g''' which describes its influence on other massive bodies. The gravitational field of ''M'' at a point '''r''' in space is found by determining the force '''F''' that ''M'' exerts on a small [[test mass]] ''m'' located at '''r''', and then dividing by ''m'':<ref name="kleppner85">{{cite book|last1=Kleppner|first1=David|last2=Kolenkow|first2=Robert|title=An Introduction to Mechanics|page=85}}</ref>
| |
| :<math> \mathbf{g}(\mathbf{r}) = \frac{\mathbf{F}(\mathbf{r})}{m}.</math>
| |
| Stipulating that ''m'' is much smaller than ''M'' ensures that the presence of ''m'' has a negligible influence on the behavior of ''M''.
| |
| | |
| According to [[Newton's law of gravitation]], '''F'''('''r''') is given by<ref name="kleppner85" />
| |
| :<math>\mathbf{F}(\mathbf{r}) = -\frac{G M m}{r^2}\hat{\mathbf{r}},</math>
| |
| where <math>\hat{\mathbf{r}}</math> is a [[unit vector]] lying along the line joining ''M'' and ''m'' and pointing from ''m'' to ''M''. Therefore, the gravitational field of '''M''' is<ref name="kleppner85" />
| |
| :<math>\mathbf{g}(\mathbf{r}) = \frac{\mathbf{F}(\mathbf{r})}{m} = -\frac{G M}{r^2}\hat{\mathbf{r}}.</math>
| |
| | |
| The experimental observation that inertial mass and gravitational mass are equal to [[equivalence principle#Tests_of_the_weak_equivalence_principle|unprecedented levels of accuracy]] leads to the identification of the gravitational field strength as identical to the acceleration experienced by a particle. This is the starting point of the [[equivalence principle]], which leads to [[general relativity]].
| |
| | |
| Because the gravitational force '''F''' is [[conservative field|conservative]], the gravitational field '''g''' can be rewritten in terms of the [[gradient]] of a [[gravitational potential]] Φ('''r'''):
| |
| :<math>\mathbf{g}(\mathbf{r}) = -\nabla \Phi(\mathbf{r}).</math>
| |
| | |
| === Electromagnetism ===
| |
| | |
| ==== Electrostatics ====
| |
| {{Main|Electrostatics}}
| |
| | |
| A [[test charge|charged test particle]] with charge ''q'' experiences a force '''F''' based solely on its charge. We can similarly describe the [[electric field]] '''E''' so that {{nowrap|'''F''' {{=}} ''q'''''E'''}}. Using this and [[Coulomb's law]] tells us that the electric field due to a single charged particle as
| |
| | |
| :<math>\mathbf{E} = \frac{1}{4\pi\epsilon_0}\frac{q}{r^2}\hat{\mathbf{r}}.</math>
| |
| | |
| The electric field is [[conservative field|conservative]], and hence can be described by a scalar potential, ''V''('''r'''):
| |
| :<math> \mathbf{E}(\mathbf{r}) = -\nabla V(\mathbf{r}).</math>
| |
| | |
| ==== Magnetostatics ====
| |
| {{Main|Magnetostatics}}
| |
| | |
| A steady current ''I'' flowing along a path ''ℓ'' will exert a force on nearby charged particles that is quantitatively different from the electric field force described above. The force exerted by ''I'' on a nearby charge ''q'' with velocity '''v''' is
| |
| :<math>\mathbf{F}(\mathbf{r}) = q\mathbf{v} \times \mathbf{B}(\mathbf{r}),</math>
| |
| where '''B'''('''r''') is the [[magnetic field]], which is determined from ''I'' by the [[Biot-Savart law]]:
| |
| :<math>\mathbf{B}(\mathbf{r}) = \frac{\mu_0 I}{4\pi} \int \frac{d\boldsymbol{\ell} \times d\hat{\mathbf{r}}}{r^2}.</math>
| |
| | |
| The magnetic field is not conservative in general, and hence cannot usually be written in terms of a scalar potential. However, it can be written in terms of a [[magnetic vector potential|vector potential]], '''A'''('''r'''):
| |
| :<math> \mathbf{B}(\mathbf{r}) = \boldsymbol{\nabla} \times \mathbf{A}(\mathbf{r}) </math>
| |
| | |
| ==== Electrodynamics ====
| |
| {{Main|Electrodynamics}}
| |
| | |
| In general, in the presence of both a charge density ρ('''r''', ''t'') and current density '''J'''('''r''', ''t''), there will be both an electric and a magnetic field, and both will vary in time. They are determined by [[Maxwell's equations]], a set of differential equations which directly relate '''E''' and '''B''' to ρ and '''J'''.<ref name="griffiths326">{{cite book|last=Griffiths|first=David|title=Introduction to Electrodynamics|edition=3rd|page=326}}</ref>
| |
| | |
| Alternatively, one can describe the system in terms of its scalar and vector potentials ''V'' and '''A'''. A set of integral equations known as ''[[retarded potential]]s'' allow one to calculate ''V'' and '''A''' from ρ and '''J''',<ref group="note">This is contingent on the correct choice of [[gauge fixing|gauge]]. ''V'' and '''A''' are not completely determined by ρ and '''J'''; rather, they are only determined up to some scalar function ''f''('''r''', ''t'') known as the gauge. The retarded potential formalism requires one to choose the [[Lorentz gauge]].</ref> and from there the electric and magnetic fields are determined via the relations<ref name="wangsness469">{{cite book|last=Wangsness|first=Roald|title=Electromagnetic Fields|edition=2nd|page=469}}</ref>
| |
| | |
| :<math> \mathbf{E} = -\boldsymbol{\nabla} V - \frac{\partial \mathbf{A}}{\partial t}</math>
| |
| :<math> \mathbf{B} = \boldsymbol{\nabla} \times \mathbf{A}.</math>
| |
| | |
| ===Hydrodynamics===
| |
| {{Main|Hydrodynamics}}
| |
| {{empty section|date=September 2012}}
| |
| | |
| == Relativistic field theory ==
| |
| {{Main|Covariant classical field theory}}
| |
| | |
| Modern formulations of classical field theories generally require [[Lorentz covariance]] as this is now recognised as a fundamental aspect of nature. A field theory tends to be expressed mathematically by using [[Lagrangian]]s. This is a function that, when subjected to an [[action principle]], gives rise to the [[field equations]] and a [[conservation law]] for the theory.
| |
| | |
| We use units where c=1 throughout.
| |
| | |
| === Lagrangian dynamics ===
| |
| {{Main|Lagrangian}}
| |
| | |
| Given a field tensor <math>\phi</math>, a scalar called the [[Lagrangian|Lagrangian density]] <math>\mathcal{L}(\phi,\partial\phi,\partial\partial\phi, ...,x)</math> can be constructed from <math>\phi</math> and its derivatives.
| |
| | |
| From this density, the functional action can be constructed by integrating over spacetime
| |
| | |
| :<math>\mathcal{S} = \int{\mathcal{L} \mathrm{d}^4x}.</math>
| |
| | |
| Therefore the Lagrangian itself is equal to the integral of the Lagrangian Density over all space.
| |
| | |
| Then by enforcing the [[Action (physics)|action principle]], the Euler-Lagrange equations are obtained
| |
| | |
| :<math>\frac{\delta \mathcal{S}}{\delta\phi}=\frac{\partial\mathcal{L}}{\partial\phi}-\partial_\mu \left(\frac{\partial\mathcal{L}}{\partial(\partial_\mu\phi)}\right)-\partial_\nu \partial_\mu \left(\frac{\partial\mathcal{L}}{\partial(\partial_\mu \partial_\nu \phi)}\right)-.~.~.-\partial_\sigma \partial_\lambda ...\partial_\nu \partial_\mu \left(\frac{\partial\mathcal{L}}{\partial(\partial_\mu \partial_\nu ...\partial_\lambda \partial_\sigma \phi)}\right)=0.</math>
| |
| | |
| == Relativistic fields ==
| |
| | |
| Two of the most well-known Lorentz-covariant classical field theories are now described.
| |
| | |
| === Electromagnetism ===
| |
| {{Main|Electromagnetic field|Electromagnetism}}
| |
| Historically, the first (classical) field theories were those describing the electric and magnetic fields (separately). After numerous experiments, it was found that these two fields were related, or, in fact, two aspects of the same field: the [[electromagnetic field]]. [[James Clerk Maxwell|Maxwell]]'s theory of [[electromagnetism]] describes the interaction of charged matter with the electromagnetic field. The first formulation of this field theory used vector fields to describe the [[electric]] and [[magnetic]] fields. With the advent of special relativity, a better (and more consistent with mechanics) formulation using [[tensor]] fields was found. Instead of using two vector fields describing the electric and magnetic fields, a tensor field representing these two fields together is used.
| |
| | |
| We have the [[electromagnetic potential]], <math>A_a=\left(-\phi, \vec{A} \right)</math>, and the [[four-current|electromagnetic four-current]] <math>j_a=\left(-\rho, \vec{j}\right)</math>. The electromagnetic field at any point in spacetime is described by the antisymmetric (0,2)-rank [[electromagnetic field tensor]]
| |
| | |
| :<math>F_{ab} = \partial_a A_b - \partial_b A_a.</math>
| |
| | |
| ==== The Lagrangian ====
| |
| | |
| To obtain the dynamics for this field, we try and construct a scalar from the field. In the vacuum, we have <math>\mathcal{L} = \frac{-1}{4\mu_0}F^{ab}F_{ab}.</math> We can use [[gauge field theory]] to get the interaction term, and this gives us
| |
| | |
| :<math>\mathcal{L} = \frac{-1}{4\mu_0}F^{ab}F_{ab} + j^aA_a.</math>
| |
| | |
| ==== The Equations ====
| |
| | |
| This, coupled with the Euler-Lagrange equations, gives us the desired result, since the E-L equations say that
| |
| | |
| :<math>\partial_b\left(\frac{\partial\mathcal{L}}{\partial\left(\partial_b A_a\right)}\right)=\frac{\partial\mathcal{L}}{\partial A_a}.</math>
| |
| | |
| It is easy to see that <math>\partial\mathcal{L}/\partial A_a = \mu_0 j^a</math>. The left hand side is trickier. Being careful with factors of <math>F^{ab}</math>, however, the calculation gives <math>\partial\mathcal{L}/\partial(\partial_b A_a) = F^{ab}</math>. Together, then, the equations of motion are:
| |
| | |
| :<math>\partial_b F^{ab}=\mu_0j^a.</math>
| |
| | |
| This gives us a vector equation, which are [[Maxwell's equations]] in vacuum. The other two are obtained from the fact that F is the 4-curl of A:
| |
| | |
| :<math>6F_{[ab,c]} \, = F_{ab,c} + F_{ca,b} + F_{bc,a} = 0. </math>
| |
| | |
| where the comma indicates a [[partial derivative]].
| |
| | |
| ===Gravitation===
| |
| {{Main|Gravitation|General Relativity}}
| |
| After Newtonian gravitation was found to be inconsistent with [[special relativity]], [[Albert Einstein]] formulated a new theory of gravitation called [[general relativity]]. This treats [[gravitation]] as a geometric phenomenon ('curved [[spacetime]]') caused by masses and represents the [[gravitational field]] mathematically by a [[tensor field]] called the [[metric tensor (general relativity)|metric tensor]]. The [[Einstein field equations]] describe how this curvature is produced. The field equations may be derived by using the [[Einstein-Hilbert action]]. Varying the Lagrangian
| |
| | |
| :<math>\mathcal{L} = \, R \sqrt{-g}</math>, | |
| | |
| where <math>R \, =R_{ab}g^{ab}</math> is the [[Ricci scalar]] written in terms of the [[Ricci tensor]] <math>\, R_{ab}</math> and the [[metric tensor (general relativity)|metric tensor]] <math>\, g_{ab}</math>, will yield the vacuum field equations:
| |
| | |
| :<math>G_{ab}\, =0</math>,
| |
| | |
| where <math>G_{ab} \, =R_{ab}-\frac{R}{2}g_{ab}</math> is the [[Einstein tensor]].
| |
| | |
| ==See also==
| |
| *[[Classical unified field theories]]
| |
| *[[Covariant Hamiltonian field theory]]
| |
| *[[Variational methods in general relativity]]
| |
| *[[Higgs field (classical)]]
| |
| | |
| == Notes ==
| |
| {{reflist|group=note}}
| |
| | |
| ==References==
| |
| {{reflist}}
| |
| *{{Cite book
| |
| | first = C.
| |
| | last = Truesdell
| |
| | author-link = Clifford Truesdell
| |
| | first2 = R.A.
| |
| | last2 = Toupin
| |
| | author2-link = Richard Toupin
| |
| | year = 1960
| |
| | contribution = The Classical Field Theories
| |
| | title = Principles of Classical Mechanics and Field Theory/Prinzipien der Klassischen Mechanik und Feldtheorie
| |
| | editor-last = Flügge
| |
| | editor-first = Siegfried
| |
| | editor-link = Siegfried Flügge
| |
| | series = Handbuch der Physik (Encyclopedia of Physics)
| |
| | volume = III/1
| |
| | pages = 226–793
| |
| | place = [[Berlin]]–[[Heidelberg]]–[[New York]]
| |
| | publisher = [[Springer-Verlag]]
| |
| | zbl = 0118.39702
| |
| | postscript = <!-- Bot inserted parameter. Either remove it; or change its value to "." for the cite to end in a ".", as necessary. -->{{inconsistent citations}}
| |
| }}.
| |
| | |
| ==External links==
| |
| *{{cite web | last=Thidé|first= Bo | authorlink=Bo Thidé|title=Electromagnetic Field Theory | url=http://www.plasma.uu.se/CED/Book/EMFT_Book.pdf | accessdate=February 14, 2006 }}
| |
| *{{cite paper | last=Carroll|first= Sean M | title=Lecture Notes on General Relativity | arxiv=gr-qc/9712019|bibcode=1997gr.qc....12019C}}
| |
| *{{cite web| last=Binney|first= James J | title=Lecture Notes on Classical Fields | url=http://www-thphys.physics.ox.ac.uk/user/JamesBinney/classf.pdf| accessdate=April 30, 2007 }}
| |
| * {{cite journal | authorlink = Gennadi Sardanashvily | last = Sardanashvily | first = G. | title = Advanced Classical Field Theory | journal = International Journal of Geometric Methods in Modern Physics | publisher = [[World Scientific]] | volume = 5 | issue = 7 | page = 1163 |date=November 2008 | isbn = 978-981-283-895-7 | arxiv = 0811.0331 | doi = 10.1142/S0219887808003247 | bibcode = 2008IJGMM..05.1163S }}
| |
| | |
| {{DEFAULTSORT:Classical Field Theory}}
| |
| [[Category:Theoretical physics]]
| |
| [[Category:Lagrangian mechanics]]
| |