Search results
Results From The WOW.Com Content Network
We may write down the Lagrangian in terms of the position coordinates as they are, but it is an established procedure to convert the two-body problem into a one-body problem as follows. Introduce the Jacobi coordinates; the separation of the bodies r = r 2 − r 1 and the location of the center of mass R = (m 1 r 1 + m 2 r 2)/(m 1 + m 2).
Much of the qualitative descriptions of the standard model in terms of "particles" and "forces" comes from the perturbative quantum field theory view of the model. In this, the Lagrangian is decomposed as = + into separate free field and interaction Lagrangians. The free fields care for particles in isolation, whereas processes involving ...
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
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 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 ...
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 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 ∗ E t, which comes equipped with a natural symplectic form, and this latter function is the Hamiltonian.
The solution can be related to the system Lagrangian by an indefinite integral of the form used in the principle of least action: [5]: 431 = + Geometrical surfaces of constant action are perpendicular to system trajectories, creating a wavefront-like view of the system dynamics. This property of the Hamilton–Jacobi equation connects ...