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I am a Physics undergraduate, so provide references with your responses.

Landau & Lifshitz write in page one of their mechanics textbook:

If all the co-ordinates and velocities are simultaneously specified, it is known from experience that the state of the system is completely determined and that its subsequent motion can, in principle, be calculated. Mathematically this means that, if all the co-ordinates $q$ and $\dot{q}$ are given at some instant, the accelerations $\ddot{q}$ at that instant are uniquely defined.

They justify this as being "known from experience", which is not entirely satisfactory. What is the basis for their assertion?

Similar: Why are there only derivatives to the first order in the Lagrangian?

Is his question equivalent to mine, even though his solely refers to Lagrangian Mechanics?

Furthermore, this might just indicate how mathematically crude my mind is, but why is it not sufficient simply to give the coordinates $q$, and determine $\dot{q}$ from that, i.e. if $q$ is given by some smooth function, can we not determine all further derivatives from that alone?

Community
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Mark Allen
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5 Answers5

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You should think of this by timestepping Newton's laws--- if you know the positions and velocity and one instant, you know the force, and the force determines the acceleration. This allows you to determine the velocity and an infinitesimal time in the future by

$$ v(t+dt) = v(t) + dt F/m $$ $$ x(t+dt) = x(t) + dt v $$

You then find the position and velocity at the next time step, and you find the new force, and continue forever. This is an algorithm to solve Newton's laws, and all that LL are saying is that Newton's laws are known from experience with objects, they are inducted from observations.

Ron Maimon
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"known from experience" in here means "known from experience that first order derivatives in the Lagrangian or second order in the equations of motion are enough". I think their basis for this assertion is very Occamian (but who could know with certainty what L&L were thinking about?) The non-Occamian approach to this answer is given in the post you quoted in your question.

For the last question indeed you can determine $\dot{q},\ddot{q},\ldots,$ from $q$ alone if you already know $q$. But wait! wasn't finding $q$ the problem? and, how are you suppose to determine $q$? Solving a second order equation, for which you need initial conditions ($q_0,\dot{q}_0$).

c.p.
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1

Given a second order differential equation, the solution to it will be uniquely determined by two sets of data.

yca
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1

In order to know about a system, we need to know all the forces acting on that system at some instant. Classically, we always solve Newton's equation of motion i.e conservation of momentum. Again, momentum is a function of velocity and velocity is function of coordinates.

So, if the velocity and coordinates are known, we solve Newton's equation of motion and hence will able to define the system. However, determining all the forces in a general system may not be easy. So, generalized coordinates and velocities are introduced. Using generalized coordinates and velocities, the solution of the equation of motion becomes easier and more efficient.

tpg2114
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sushmita
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i think it's becoz all forces we observe experimentaly are functions of r and v only like the Lorentz force or the gravitational force etc

baz
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