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The Lagrangian and Eulerian specifications of the flow field are sometimes loosely denoted as the Lagrangian and Eulerian frame of reference. However, in general both the Lagrangian and Eulerian specification of the flow field can be applied in any observer's frame of reference, and in any coordinate system used within the chosen frame of ...
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.
The central quantity of Lagrangian mechanics is the Lagrangian, a function which summarizes the dynamics of the entire system. Overall, the Lagrangian has units of energy, but no single expression for all physical systems. Any function which generates the correct equations of motion, in agreement with physical laws, can be taken as a Lagrangian.
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 formulation of Lagrange's equations for this system yields six equations in the four Cartesian coordinates x i, y i (i = 1, 2) and the two Lagrange multipliers λ i (i = 1, 2) that arise from the two constraint equations.
Consider a generic (possibly non-Abelian) gauge transformation acting on a component field = =.The main examples in field theory have a compact gauge group and we write the symmetry operator as () = where () is an element of the Lie algebra associated with the Lie group of symmetry transformations, and can be expressed in terms of the hermitian generators of the Lie algebra (i.e. up to a ...
Consequently, gauge symmetries characterize the degeneracy of a Lagrangian system. [5] Note that, in quantum field theory, a generating functional may fail to be invariant under gauge transformations, and gauge symmetries are replaced with the BRST symmetries, depending on ghosts and acting both on fields and ghosts. [6]
where is known as a Lagrangian density; d 4−1 x ≡ dx ⋅ dy ⋅ dz ≡ dx 1 ⋅ dx 2 ⋅ dx 3 for the three spatial coordinates; δ ij is the Kronecker delta function; and ∂ ρ = ∂/∂x ρ for the ρ-th coordinate x ρ. This is an example of a quadratic action, since each of the terms is quadratic in the field, φ.