Generalized Coordinates - Landau & Lifshitz

[sisyphusRocks]

If suppose only if the velocities are determined for all N particles can the system be completely determined, can we not extend and say that only if acceleration for all particles are provided can the system be completely determined? For instance can there not be two systems of N particles with same velocities and same positions but different accelerations. I am not sure I am getting this part.

 

[sisyphusRocks]

To add to that:

Again in the second para it is said that the mechanical system is completely defined when the coordinates and velocities arent given. How so? I am missing a fundamental point here. Does that mean Lagrangian action principle only applies to certain kind of mechanical systems and there could systems which are non lagrangian? Please help.

 

[sisyphusRocks]

To add to that:

Here as we started with a Lagrangian which depended on only q and q dot, we are left with 2s constants and s second order equations and hence we need initial conditions of s coordinates and s velocities and we are done. Suppose we had a lagrangian which depends on q dot dot also then we would be left with a complicated problem. I am not sure what type of lagrange equations will come up if we try to minimize the action which is based on such a lagrangian? Am i making sense here? Please read all my three posts in order to understand my question. Thanks for any help. I am not able to proceed with the next chapter as this is nagging me.

 

[DrClaude]

If suppose only if the velocities are determined for all N particles can the system be completely determined, can we not extend and say that only if acceleration for all particles are provided can the system be completely determined?

You have to read carefully (which is always the case with L&L): "it is known from experience[...]" It is the study of the natural world that tells us that $q$ and $\dot{q}$ are sufficient.

One way to understand why that is the case is to consider that if you can calculate all internal and external forces, then you can calculate the acceleration of all particles. If the forces only depend on $q$ and $\dot{q}$ (which they usually do), then the system is fully specified.

For instance can there not be two systems of N particles with same velocities and same positions but different accelerations. I am not sure I am getting this part.

If they have different accelerations, it means that the forces acting on the particles are not the same, which means that it is not the same physical system.

 

[DrClaude]

T
Again in the second para it is said that the mechanical system is completely defined when the coordinates and velocities arent given. How so? I am missing a fundamental point here. Does that mean Lagrangian action principle only applies to certain kind of mechanical systems and there could systems which are non lagrangian? Please help.

To quote from Wikipedia:

No new physics is introduced by Lagrangian mechanics; it is actually less general than Newtonian mechanics.[7] Newton's laws can include non-conservative forces like friction, however they must include constraint forces explicitly and are best suited to Cartesian coordinates. Lagrangian mechanics is ideal for systems with conservative forces and for bypassing constraint forces, and some (not all) non-conservative forces, in any coordinate system. Generalized coordinates can be chosen by convenience, to exploit symmetries in the system or the geometry of the constraints, which considerably simplifies describing the dynamics of the system.

 

[sisyphusRocks]

If the forces only depend on $q$ and $\dot{q}$ (which they usually do), then the system is fully specified.

This clarifies my doubt. Now I am more peaceful. Thank you. The wikipedia also threw light to my problem.

If they have different accelerations, it means that the forces acting on the particles are not the same, which means that it is not the same physical system.

This makes sense. Thank you.

 

[Dale]

For instance can there not be two systems of N particles with same velocities and same positions but different accelerations.

Yes. A system is characterized by its Lagrangian. Different systems will have different Lagrangians. In order to specify the future behavior of a single system you need:
1) The Lagrangian which characterizes the system
2) The initial coordinates
3) The initial velocities

You do not need the initial accelerations because those are determined by the combination of the initial coordinates and the Lagrangian.

 

[vanhees71]

In other words, as long as your Lagrangian is only a function of the generalized coordinates $q$ and the generalized velocities $\dot{q}$ (and maybe also explicitly on time), your Euler-Lagrange equations, solving the variational principle of least (stationary) action,
$$\frac{\mathrm{d}}{\mathrm{d} t} \frac{\partial L}{\partial \dot{q}}=\frac{\partial L}{\partial q},$$
are a system of 2nd-order ordinary differential equations, determining the trajectory $q(t)$ of the system, if you have given $q(t_0)=q_0$ and $\dot{q}(t_0)=\dot{q}_0$ (initial-value problem).

That the Lagrangian is of this specific form is plausible from Newton's 2nd Law, which is a 2nd order differential equation for the position of the particle(s). That the equations of motion are of this form, cannot be derived but is, as Landau-Lifshitz states, just based "on experience", as any fundamental law of nature are just based on empirical evidence. In this sense all our scientific knowledge is always tentative since, if new empirical facts contradicting a so far "valid" natural law become known, we have to adapt our theories to this new facts. To determine whether an observation is really a fact or based on mistakes in the experimental setup and/or observation, is of course another difficult story, but now I drift too much into the philosophy of science...