Search results
Results From The WOW.Com Content Network
The full expanded form of the Standard Model Lagrangian. We can now give some more detail about the aforementioned free and interaction terms appearing in the Standard Model Lagrangian density. Any such term must be both gauge and reference-frame invariant, otherwise the laws of physics would depend on an arbitrary choice or the frame of an ...
In field theory, the independent variable is replaced by an event in spacetime (x, y, z, t), or more generally still by a point s on a Riemannian manifold.The dependent variables are replaced by the value of a field at that point in spacetime (,,,) so that the equations of motion are obtained by means of an action principle, written as: =, where the action, , is a functional of the dependent ...
The Lagrangian and Eulerian specifications are named after Joseph-Louis Lagrange and Leonhard Euler, respectively. These specifications are reflected in computational fluid dynamics , where "Eulerian" simulations employ a fixed mesh while "Lagrangian" ones (such as meshfree simulations ) feature simulation nodes that may move following the ...
The Dirac Lagrangian of the quarks coupled to the gluon fields is given by = ¯, where is a three component column vector of Dirac spinors, each element of which refers to a quark field with a specific color charge (i.e. red, blue, and green) and summation over flavor (i.e. up, down, strange, etc.) is implied.
Being Lagrangian symmetries, gauge symmetries of a Lagrangian satisfy Noether's first theorem, but the corresponding conserved current takes a particular superpotential form = + where the first term vanishes on solutions of the Euler–Lagrange equations and the second one is a boundary term, where is called a superpotential.
A Lagrangian density L (or, simply, a Lagrangian) of order r is defined as an n-form, n = dim X, on the r-order jet manifold J r Y of Y. A Lagrangian L can be introduced as an element of the variational bicomplex of the differential graded algebra O ∗ ∞ ( Y ) of exterior forms on jet manifolds of Y → X .
Lagrangian mechanics describes a mechanical system as a pair (M, L) consisting of a configuration space M and a smooth function within that space called a Lagrangian. For many systems, L = T − V , where T and V are the kinetic and potential energy of the system, respectively.
According to the fundamental lemma of calculus of variations, the part of the integrand in parentheses is zero, i.e. ′ = which is called the Euler–Lagrange equation. The left hand side of this equation is called the functional derivative of J [ f ] {\displaystyle J[f]} and is denoted δ J {\displaystyle \delta J} or δ f ( x ...