User:Maschen/Lagrangian mechanics (damped and dissipative forces)

All the damped/dissipation stuff in Lagrangian mechanics articles to be collected into one article?

In physics, Lagrangian mechanics is a formulation of classical mechanics often used to study the configuration and its time evolution of a mechanical system. The formulation is particularly suitable to undamped conservative forces, for one or many particles, because the Lagrangian is straightforwards to write down and the equations of motion can be derived readily, even if the actual equations may be difficult or impossible to solve exactly.

In the Newtonian formulation of mechanics, Newton's laws use forces and the forces acting on a particular constituent of a system (e.g. a particle, a point inside a massive continuum) are additive, making it easy to include any forces into the equations. By contrast, it is not as easy to incorporate every type of force into the Lagrangian. A number of physicists or mathematicians have developed ways of including dampening effects, and accounting for non-conservative forces, in the Lagrangian formulation.

Conservative and non-conservative forces

A conservative force can be expressed as the gradient of a potential, and it is straightforwards to include the potential into the Lagrangian and recover the gradient in the Euler-Lagrange equations after evaluating the derivatives of the Lagrangian. Examples include the non-relativistic gravitational potential, and electrostatic field.

Other forces are non-conservative and are not expressible as a gradient of a potential function. Sometimes, it is possible to find a "potential" with dependence on velocities as well as positions, and use the general Lagrange equations (for generalized forces), the prototypical example is the electromagnetic field.

More sophisticated forces associated with friction, drag, viscosity, and so on cannot be handled the same way, and require special treatment.

Rayleigh dissipative function

Dissipation (i.e. non-conservative systems) can also be treated with an effective Lagrangian formulated by a certain doubling of the degrees of freedom; see.

In a more general formulation, the forces could be both conservative and viscous. If an appropriate transformation can be found from the F_i, Rayleigh suggests using a dissipation function, D, of the following form:

$D={\frac {1}{2}}\sum _{j=1}^{m}\sum _{k=1}^{m}C_{jk}{\dot {q}}_{j}{\dot {q}}_{k}\,.$

where C_jk are constants that are related to the damping coefficients in the physical system, though not necessarily equal to them. If D is defined this way, then

$Q_{j}=-{\frac {\partial V}{\partial q_{j}}}-{\frac {\partial D}{\partial {\dot {q}}_{j}}}\,$

and

${\frac {\mathrm {d} }{\mathrm {d} t}}\left({\frac {\partial L}{\partial {\dot {q}}_{j}}}\right)-{\frac {\partial L}{\partial q_{j}}} {\frac {\partial D}{\partial {\dot {q}}_{j}}}=0\,.$

Others?

See Jose and Salatan

Damped forces

Many realistic oscillatory systems have dampening effects which reduce the vibration eventually. Accounting for this in Lagrangian mechanics is possible in certain cases.

(See Landau Lifshitz Mechanics)

References