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The time integral of the Lagrangian is another quantity called the action, defined as [33] =, which is a functional; it takes in the Lagrangian function for all times between t 1 and t 2 and returns a scalar value.
The force of friction is negative the velocity gradient of the dissipation function, = (), analogous to a force being equal to the negative position gradient of a potential. This relationship is represented in terms of the set of generalized coordinates q i = { q 1 , q 2 , … q n } {\displaystyle q_{i}=\left\{q_{1},q_{2},\ldots q_{n}\right\}} as
Meshfree methods enable the simulation of some otherwise difficult types of problems, at the cost of extra computing time and programming effort. The absence of a mesh allows Lagrangian simulations, in which the nodes can move according to the velocity field .
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 .
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 ...
In analytical mechanics (particularly Lagrangian mechanics), generalized forces are conjugate to generalized coordinates.They are obtained from the applied forces F i, i = 1, …, n, acting on a system that has its configuration defined in terms of generalized coordinates.
In mathematical optimization, the method of Lagrange multipliers is a strategy for finding the local maxima and minima of a function subject to equation constraints (i.e., subject to the condition that one or more equations have to be satisfied exactly by the chosen values of the variables). [1] It is named after the mathematician Joseph-Louis ...
A Hamiltonian system may be understood as a fiber bundle E over time R, with the fiber E t being the position space at time t ∈ R. The Lagrangian is thus a function on the jet bundle J over E; taking the fiberwise Legendre transform of the Lagrangian produces a function on the dual bundle over time whose fiber at t is the cotangent space T ...