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In physics, Lagrangian mechanics is a formulation of classical mechanics founded on the stationary-action principle (also known as the principle of least action). It was introduced by the Italian-French mathematician and astronomer Joseph-Louis Lagrange in his presentation to the Turin Academy of Science in 1760 [ 1 ] culminating in his 1788 ...
This function represents half of the rate of energy dissipation of the system through friction. The force of friction is negative the velocity gradient of the dissipation function, F → f = − ∇ v R ( v ) {\displaystyle {\vec {F}}_{f}=-\nabla _{v}R(v)} , analogous to a force being equal to the negative position gradient of a potential.
The correspondence between Lagrangian and Hamiltonian mechanics is achieved with the tautological one-form. Any smooth real-valued function H on a symplectic manifold can be used to define a Hamiltonian system. The function H is known as "the Hamiltonian" or "the energy function." The symplectic manifold is then called the phase space.
Hamilton's principle states that the true evolution q(t) of a system described by N generalized coordinates q = (q 1, q 2, ..., q N) between two specified states q 1 = q(t 1) and q 2 = q(t 2) at two specified times t 1 and t 2 is a stationary point (a point where the variation is zero) of the action functional [] = ((), ˙ (),) where (, ˙,) is the Lagrangian function for the system.
In classical mechanics, the action, , is defined as the time integral of the Lagrangian, . The Lagrangian is the difference of energies, L = T − U , {\displaystyle L=T-U,} where T {\displaystyle T} is the kinetic energy of a mechanical system and U {\displaystyle U} its potential energy .
In real oscillators, friction, or damping, slows the motion of the system. Due to frictional force, the velocity decreases in proportion to the acting frictional force. While in a simple undriven harmonic oscillator the only force acting on the mass is the restoring force, in a damped harmonic oscillator there is in addition a frictional force ...
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 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 .