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By
Wikipedia, The Lagrangian, L, 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 classical mechanics known as Lagrangian mechanics. In classical mechanics, the Lagrangian is defined as the kinetic energy, T, of the system minus its potential energy, V. In symbols, Under conditions that are given in Lagrangian mechanics, if the Lagrangian of a system is known, then the equations of motion of the system may be obtained by a direct substitution of the expression for the Lagrangian into the Euler–Lagrange equation, a particular family of partial differential equations. The Lagrange formulationImportanceThe Lagrange formulation of mechanics is important not just for its broad applications, but also for its role in advancing deep understanding of physics. Although Lagrange only sought to describe classical mechanics, the action principle that is used to derive the Lagrange equation is now recognized to be applicable to quantum mechanics. Physical action and quantum-mechanical phase are related via Planck's constant, and the principle of stationary action can be understood in terms of constructive interference of wave functions. The same principle, and the Lagrange formalism, are tied closely to Noether's theorem, which relates physical conserved quantities to continuous symmetries of a physical system. Lagrangian mechanics and Noether's theorem together yield a natural formalism for first quantization by including commutators between certain terms of the Lagrangian equations of motion for a physical system. Advantages over other methods
"Cyclic coordinates" and conservation lawsAn important property of the Lagrangian is that conservation laws can easily be read-off from it. E.g., if the Lagrangian To repeat: the conservation of the generalized momentum As an aside: If the time, t, does not appear in ExplanationThe equations of motion are obtained by means of an action principle, written as: where the action, and where The equations of motion obtained from this functional derivative are the Euler–Lagrange equations of this action. For example, in the classical mechanics of particles, the only independent variable is time, t. So the Euler-Lagrange equations are 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 Standard Model, to Newton's equations, to purely mathematical problems such as geodesic equations and Plateau's problem. An example from classical mechanicsIn the rectangular coordinate systemSuppose we have a three-dimensional space and the Lagrangian
Then, the Euler–Lagrange equation is: where i = 1,2,3. The derivation yields: 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 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 A very similar deduction gives us the expression In the spherical coordinate systemSuppose we have a three-dimensional space using spherical coordinates r,θ,φ with the Lagrangian Then the Euler–Lagrange equations are: Here the set of parameters si is just the time t, and the dynamical variables φi(s) are the trajectories Despite the use of standard variables such as x, the Lagrangian allows the use of any coordinates, which do not need to be orthogonal. These are "generalized coordinates". Lagrangian of a test particleA test particle is a particle whose mass and charge are assumed to be so small that its effect on external system is insignificant. It is often 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. Classical test particle with Newtonian gravitySuppose we are given a particle with mass and the particle's gravitational potential energy is: Then its Lagrangian is Varying Integrate the first term by parts and discard the total integral. Then divide out the variation to get and thus is the equation of motion — two different expressions for the force. Special relativistic test particle with electromagnetismIn 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. It becomes: (In special relativity, the energy of a free test particle is where Varying this with respect to which is which is the equation for the Lorentz force where General relativistic test particleIn general relativity, the first term generalizes (includes) both the classical kinetic energy and interaction with the Newtonian gravitational potential. It becomes: The Lagrangian of a general relativistic test particle in an electromagnetic field is: If the four space-time coordinates Note that this notion has been directly generalized from special relativity Lagrangians and Lagrangian densities in field theoryThe time integral of the Lagrangian is called the action denoted by S. and the Lagrangian density The Lagrangian is then the spatial integral of the Lagrangian density. However, Selected fieldsTo 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 Newtonian gravityThe Lagrangian (density) is where Integrate by parts and discard the total integral. Then divide out by and thus which yields Gauss's law for gravity. Electromagnetism in special relativityThe interaction terms Varying this with respect to which yields Gauss' law. Varying instead with respect to which yields Ampère's law. Electromagnetism in general relativityFor the Lagrangian of gravity in general relativity, see Einstein-Hilbert action. The Lagrangian of the electromagnetic field is: If the four space-time coordinates Lagrangians in quantum field theoryDirac LagrangianThe Lagrangian density for a Dirac field is: where Quantum electrodynamic LagrangianThe Lagrangian density for QED is: where Quantum chromodynamic LagrangianThe Lagrangian density for quantum chromodynamics is [1] [2] [3]: where Mathematical formalismSuppose we have an n-dimensional manifold, M, and a target manifold, T. Let Examples
Mathematical developmentConsider a functional, In order for the action to be local, we need additional restrictions on the action. If 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 φ at the boundary if M is compact or some limit on φ as x approaches The solution is given by the Euler–Lagrange equations (thanks to the boundary conditions), The left hand side is the functional derivative of the action with respect to φ. See also
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Published - July 2009
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