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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 ...
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.
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 Lagrangian is then [45] [46] [nb 4] = ˙ ⏟ + ˙ (| |) ⏟ where M = m 1 + m 2 is the total mass, μ = m 1 m 2 /(m 1 + m 2) is the reduced mass, and V the potential of the radial force, which depends only on the magnitude of the separation | r | = | r 2 − r 1 |. The Lagrangian splits into a center-of-mass term L cm and a relative motion ...
Here, A stands for the electromagnetic potential 1-form, J is the current 1-form, F is the field strength 2-form and the star denotes the Hodge star operator. This is exactly the same Lagrangian as in the section above, except that the treatment here is coordinate-free; expanding the integrand into a basis yields the identical, lengthy expression.
Lagrangian analysis is the use of Lagrangian coordinates to analyze various problems in continuum mechanics. Lagrangian analysis may be used to analyze currents and flows of various materials by analyzing data collected from gauges/sensors embedded in the material which freely move with the motion of the material. [ 1 ]
Lagrangian (field theory), a formalism in classical field theory; Lagrangian point, a position in an orbital configuration of two large bodies; Lagrangian coordinates, a way of describing the motions of particles of a solid or fluid in continuum mechanics; Lagrangian coherent structure, distinguished surfaces of trajectories in a dynamical system
In continuum mechanics, the generalized Lagrangian mean (GLM) is a formalism – developed by D.G. Andrews and M.E. McIntyre (1978a, 1978b) – to unambiguously split a motion into a mean part and an oscillatory part. The method gives a mixed Eulerian–Lagrangian description for the flow field, but appointed to fixed Eulerian coordinates. [1]