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Lagrangian, Lagrangian Function, Λαγκρανζιανή
- Ένα Φυσικό Μέγεθος.
Έλαβε το όνομά της από τον διάσημο φυσικό Lagrange.
Η εξίσωση που την καθορίζει είναι η εξής:
- L = K - V
The Lagrangian, , of a dynamical system is a function that summarizes the dynamics of the system. It is named after Joseph Louis Lagrange. The concept of a Lagrangian was originally introduced in a reformulation of Κλασσική Μηχανική known as Αναλυτική Μηχανική.
Κάτω από συγκεκριμένες προϋποθέσεις, οι οποίες δίδονται από Lagrangian mechanics, εάν η Λαγρασιανή του συστήματος είναι γνωστή, τότε οι Εξισώσεις Κίνησης (equations of motion) του συστήματος μπορούν να ληφθούν με άμεση αντικατάσταση της συναρτησιακής έκφρασης της Λαγρασιανής στις Εξισώσεις Euler - Lagrange, οι οποίες αποτελούν ειδική οικογένεια Μερικών Διαφορικών Εξισώσεων (differential equations).
The Lagrange formulation of mechanics is important not just for its broad applications, but also for its role in advancing βαθεία κατανόηση της Φυσικής. Although Lagrange only sought to describe Κλασσική Μηχανική, η "Αρχή της Δράσης (action principle)" that is used to derive the Lagrange equation is now recognized to be applicable to Κβαντική Φυσική.
Lagrangian mechanics and Θεώρημα Noether together yield a φυσικό φορμαλισμό for Πρώτη Κβάντωση (first quantization) by including Μεταθέτες (commutators) between certain terms of the Lagrangian equations of motion for a physical system.
- The formulation is not tied to any one coordinate system -- rather, any convenient variables may be used to describe the system; these variables are called "generalized coordinates" and may be any independent variable of the system (for example, strength of the magnetic field at a particular location; angle of a pulley; position of a particle in space; or degree of excitation of a particular eigenmode in a complex system). This makes it easy to incorporate constraints into a theory by defining coordinates which only describe states of the system which satisfy the constraints.
- If the Lagrangian is invariant under a symmetry, then the resulting equations of motion are also invariant under that symmetry. This is very helpful in showing that theories are consistent with either special relativity or general relativity.
- Equations derived from a Lagrangian will almost automatically be unambiguous and consistent, unlike equations just thrown together from various sources.
where the action, S, είναι ένα Μαθηματικό Συναρτησιοειδές (functional)
The equations of motion obtained by means of the [[Συναρτησιακή Παράγωγος (functional derivative) are identical to the usual Εξισώσεις Euler - Lagrange.
Dynamical systems whose equations of motion are obtainable by means of an action principle on a suitably chosen Lagrangian are known as Lagrangian dynamical systems.
Examples of Lagrangian dynamical systems range from the classical version of the Σωματιδιακό Πρότυπο, to Newton's equations, to purely mathematical problems such as geodesic equations and Πρόβλημα Plateau.
Παραδείγμα Κλασσικής ΜηχανικήςEdit
Suppose we have a Τρισδιάστατος Χώρος and the Λαγρασιανή (Lagrangian)
Τότε, η Εξίσωση Euler-Lagrange είναι:
Η παραγώγιση δίδει:
The Euler-Lagrange equations can therefore be written as:
where the time derivative is written conventionally as a dot above the quantity being differentiated, and is the del operator.
Using this result, it can easily be shown that the Lagrangian approach is equivalent to the Newtonian one.
If the force is written in terms of the potential ; the resulting equation is , which is exactly the same equation as in a Newtonian approach for a constant mass object.
A very similar deduction gives us the expression , which is Newton's Second Law in its general form.
Suppose we have a three-dimensional space using spherical coordinates with the Lagrangian
Τότε οι Εξισώσεις Euler-Lagrange είναι:
Here the set of parameters is just the time , and the dynamical variables are the trajectories of the particle.
Despite the use of standard variables such as , the Lagrangian allows the use of any coordinates, which do not need to be orthogonal. These are "generalized coordinates".
Μηχανική Υλικού ΣωματιδίουEdit
A Υλικό Σωματίδιο (test particle) is a hypothetical simplified point particle with no properties other than mass and charge. Real particles like electrons and up-quarks are more complex and have additional terms in their Lagrangians.
Κλασσική περίπτωση με Νευτώνεια ΒαρύτηταEdit
The Lagrangian is joules. Given a particle with mass kilograms, and position meters in a Newtonian gravitation field with potential joules per kilogram. The particle's world line is parameterized by time seconds. The particle's kinetic energy is:
and the particle's gravitational potential energy is:
Thus the Lagrangian is:
Varying in the integral (equivalent to the Euler Lagrange differential equation), we get
Integrate by parts and discard the total integral. Then divide out the variation to get
is the equation of motion — two different expressions for the force.
Ειδική Σχετιστικιστική περίπτωση με ηλεκτρομαγνητισμόEdit
In Ειδική Σχετικότητα (special relativity), the form of the term which gives rise to the derivative of the momentum must be changed; it is no longer the kinetic energy.
(In special relativity, the energy of a free test particle is )
where meters per second is the speed of light in vacuum, seconds is the proper time (i.e. time measured by a clock moving with the particle) and Notice that the second term in the series is just the classical kinetic energy. Suppose the particle has electrical charge coulombs and is in an electromagnetic field with scalar potential volts (a volt is a joule per coulomb) and vector potential volt seconds per meter. The Lagrangian of a special relativistic test particle in an electromagnetic field is:
Varying this with respect to , we get
which is the equation for the Lorentz force where
Γενική Σχετικιστική περίπτωσηEdit
In Γενική Σχετικότητα (general relativity), the first term generalizes (includes) both the classical kinetic energy and interaction with the Newtonian gravitational potential.
The Lagrangian of a general relativistic test particle in an electromagnetic field is:
If the four space-time coordinates are given in arbitrary units (i.e. unit-less), then meters squared is the rank 2 symmetric metric tensor which is also the gravitational potential. Also, volt seconds is the electromagnetic 4-vector potential. Notice that a factor of c has been absorbed into the square root because it is the equivalent of
Note that this notion has been directly generalized from special relativity
Κλασσική Πεδιακή ΘεωρίαEdit
The time integral of the Lagrangian is called the action denoted by .
In field theory, a distinction is occasionally made between the Lagrangian , of which the action is the time integral:
and the Lagrangian density , which one integrates over all space-time to get the action:
The Lagrangian is then the spatial integral of the Lagrangian density. However, is also frequently simply called the Lagrangian, especially in modern use; it is far more useful in relativistic theories since it is a locally defined, Lorentz scalar field. Both definitions of the Lagrangian can be seen as special cases of the general form, depending on whether the spatial variable is incorporated into the index or the parameters in . Quantum field theories in particle physics, such as quantum electrodynamics, are usually described in terms of , and the terms in this form of the Lagrangian translate quickly to the rules used in evaluating Διαγράμματα Feynman.
To go with the section on test particles above, here are the equations for the fields in which they move. The equations below pertain to the fields in which the test particles described above move and allow the calculation of those fields. The equations below will not give you the equations of motion of a test particle in the field but will instead give you the potential (field) induced by quantities such as mass or charge density at any point .
For example, in the case of Newtonian gravity, the Lagrangian density integrated over space-time gives you an equation which, if solved, would yield .
This , when substituted back in equation (1), the Lagrangian equation for the test particle in a Newtonian gravitational field, provides the information needed to calculate the acceleration of the particle.
Νευτώνιο Βαρυτικό ΠεδίοEdit
The Lagrangian (density) is joules per cubic meter. The interaction term is replaced by a term involving a continuous mass density kilograms per cubic meter. This is necessary because using a point source for a field would result in mathematical difficulties. The resulting Lagrangian for the classical gravitational field is:
where meters cubed per kilogram second squared is the gravitational constant. Variation of the integral with respect to gives:
Integrate by parts and discard the total integral. Then divide out by to get:
(Ειδικό Σχετικιστικό) Ηλεκτρομαγνητικό ΠεδίοEdit
The interaction terms are replaced by terms involving a continuous charge density coulombs per cubic meter and current density amperes per square meter. The resulting Lagrangian for the electromagnetic field is:
Varying this with respect to , we get
which yields Gauss' law.
Varying instead with respect to , we get
which yields Ampère's law.
(Γενικό Σχετικιστικό) Ηλεκτρομαγνητικό ΠεδίοEdit
For the Lagrangian of gravity in general relativity, see Einstein-Hilbert action. The Lagrangian of the electromagnetic field is:
If the four space-time coordinates are given in arbitrary units, then: joule seconds is the Lagrangian, a scalar density; coulombs is the current, a vector density; and volt seconds is the electromagnetic tensor, a covariant antisymmetric tensor of rank two. Notice that the determinant under the square root sign is applied to the matrix of components of the covariant metric tensor , and is its inverse. Notice that the units of the Lagrangian changed because we are integrating over which are unit-less rather than over which have units of seconds meters cubed. The electromagnetic field tensor is formed by anti-symmetrizing the partial derivative of the electromagnetic vector potential; so it is not an independent variable. The square root is needed to convert that term into a scalar density instead of just a scalar, and also to compensate for the change in the units of the variables of integration. The factor of inside the square root is needed to normalize it so that the square root will reduce to one in special relativity (since the determinant is in special relativity).
The Lagrangian density for a Dirac field is:
The Lagrangian density for Κβαντική Ηλεκτροδυναμική (QED) is:
where is the electromagnetic tensor
- In Κλασσική Μηχανική, in the Hamiltonian formalism, is the one-dimensional manifold , representing time and the target space is the cotangent bundle of space of generalized positions.
- In Πεδιακή Θεωρία, is the spacetime manifold and the target space is the set of values the fields can take at any given point. For example, if there are m real-valued scalar fields, , then the target manifold is . If the field is a real vector field, then the target manifold is isomorphic to . There is actually a much more elegant way using tangent bundles over , but we will just stick to this version.
In order for the action to be local, we need additional restrictions on the action. If , we assume is the integral over of a function of , its derivatives and the position called the Lagrangian, . In other words,
It is assumed below, in addition, that the Lagrangian depends on only the field value and its first derivative but not the higher derivatives.
Given boundary conditions, basically a specification of the value of at the boundary if is compact or some limit on as x approaches (this will help in doing integration by parts), the subspace of consisting of functions, such that all functional derivatives of at are zero and satisfies the given boundary conditions is the subspace of on shell solutions.
Electromagnetism without charges and currentsEdit
- is over space and time.
This means the Lagrangian density is
The far left and far right terms are the same because and are just dummy indices after all. The two middle terms are also the same, so the Lagrangian density is
We can then plug this into the Euler-Lagrange equation of motion for a field:
The second term is zero because the Lagrangian in this case only contains derivatives. So the Euler-Lagrange equation becomes:
The quantity in parentheses above is just the field tensor, so this finally simplifies to
That equation is just another way of writing the two inhomogeneous Maxwell's equations as long as you make the substitutions:
where and take on the values of 1, 2, and 3.
When there are charges or currents, the Lagrangian needs an extra term to account for the coupling between them and the electromagnetic field. In that case is equal to the 4-current instead of zero.
Role in quantum electrodynamics and field theoryEdit
In quantum field theory it is used as the template for the gauge field strength tensor. By being employed in addition to the local interaction Lagrangian it reprises its usual role in QED.
- Φυσική Δράση
- Συναρτησιακή Παράγωγος
- Συναρτησιακό Ολοκλήρωμα
- Αρχή Ελάχιστης Δράσης
- Λογισμός Μεταβολών
- Generalized coordinates
- Χαμιλτονιανός Φορμαλισμός
- Λαγρασιανός Φορμαλισμός
- Σημείο Lagrangian
- Θεώρημα Noether
- Covariant classical field theory
- Βαθμωτή Πεδιακή Θεωρία
- Lagrangian Συντεταγμένη
- Eulerian Συντεταγμένη
- Ευφυής Λαγρασιανή
- Ηλεκτρομαγνητική Λαγρασιανή
- Christoph Schiller (2005), Global descriptions of motion: the simplicity of complexity, Motion Mountain
- David Tong Classical Dynamics (Cambridge lecture notes)
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