- #1

JonnyMaddox

- 74

- 1

'Lagrange's equations can be put in the form [itex] \frac{d}{dt}(\frac{\partial{L}}{\partial{\dot{q}}})-\frac{\partial{L}}{\partial{q}}=0[/itex](eq. 1.57) even if there is no potential function [itex]V[/itex] in the usual sense, providing the generalized forces are obtained from a function [itex]U(q, \dot{q})[/itex] by the prescription: [itex]Q=- \frac{\partial{U}}{\partial{q}}+\frac{d}{dt}(\frac{\partial{U}}{ \partial{ \dot{q}}} )[/itex](1.58). In such case eqs. 1.57(or plus another term [itex] \frac{dF}{dt}[/itex]) still follow from [itex] \frac{d}{dt}(\frac{\partial{T}}{\partial{\dot{q}}})-\frac{\partial{T}}{\partial{q}}=Q[/itex] with the Lagrangian given by [itex]L = T - U [/itex]. Here U may be called a "generalized potential, or "velocity-dependent potential"..."

(not an exact quotation)

Then he gives an example of an electromagnetic force potential(which is velocity depedent), which he then calls U and writes it like [itex]L=T-U[/itex] and later he writes: "Note that if not all the forces acting on the system are derivable from a potential, then Lagrange's equations can always be written in the form [itex] \frac{d}{dt}(\frac{\partial{L}}{\partial{\dot{q}}})-\frac{\partial{L}}{\partial{q}}=Q[/itex] where L contains the potential of the conservative force as before, and Q represents the forces not arising from a potential..."

I don't really understand what the difference between U, V and Q is, and why U should be derivable from 1.58, what does 1.58 even mean? It looks like the normal Langrangian equation but only with the potential U and Q. Should V also be obtained from 1.58 ? But I think V should be obtained when I integrate the force. And when I can do the same with U, then why he writes it as U and not as V. And should a force not derivable from a potential still be obtained from 1.58?? Then in the next page he adds a dissipation function to the Lagrangian. And should this dissipation function also be derivable from 1.58?? I'm not sure but the whole derivation of the Lagrangian seems a bit like magic to me :((

Greets