Search results
Results From The WOW.Com Content Network
Instead of forces, Lagrangian mechanics uses the energies in the system. 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.
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 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 .
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 ...
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.
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 ...
The time-stepping usually requires a nearest neighbor sorting step to reduce the number of possible contact pairs and decrease the computational requirements; this is often only performed periodically. The following forces may have to be considered in macroscopic simulations: friction, when two particles touch each other;