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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 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.
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 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 a set of curvilinear coordinates ξ = (ξ 1, ξ 2, ξ 3), the law in tensor index notation is the "Lagrangian form" [18] [19] = (+) = (˙), ˙, where F a is the a-th contravariant component of the resultant force acting on the particle, Γ a bc are the Christoffel symbols of the second kind, = is the kinetic energy of the particle, and g bc ...
The relativistic Lagrangian can be derived in relativistic mechanics to be of the form: = (˙) (, ˙,). Although, unlike non-relativistic mechanics, the relativistic Lagrangian is not expressed as difference of kinetic energy with potential energy, the relativistic Hamiltonian corresponds to total energy in a similar manner but without including rest energy.
File:Lagrangian vs Eulerian [further explanation needed] Eulerian perspective of fluid velocity versus Lagrangian depiction of strain.. In classical field theories, the Lagrangian specification of the flow field is a way of looking at fluid motion where the observer follows an individual fluid parcel as it moves through space and time.
This constant is called the Lagrange multiplier. (In some conventions λ {\displaystyle \lambda } is preceded by a minus sign). Notice that this method also solves the second possibility, that f is level: if f is level, then its gradient is zero, and setting λ = 0 {\displaystyle \lambda =0} is a solution regardless of ∇ x , y g ...