How are Constants for an n-Dimensional Mechanical System Calculated?

[snoopies622]

According to Landau and Lifgarbagez, an n-dimensional mechanical system has 2n-1 constants, each of which is a function of the generalized coordinates and their derivatives with respect to time.

How does one find these constants?

I know that for a given coordinate $$q$$, if

$$\frac { \partial L } { \partial q} =0$$

then

$$\frac { \partial L}{ \partial \dot q}$$

is a constant, but even if that were the case for every $$q$$, it would produce only n constants, not 2n-1 of them.

 

[snoopies622]

To take a specific example, a particle of mass m moving in orbit around an object with mass M (m<<M) has Lagrangian

$$L = \frac {1}{2} m \dot r ^2 + \frac {1}{2} m r ^2 \dot \theta ^2 + \frac {GMm}{r}$$

Since

$$\frac {\partial L}{\partial \theta} = 0$$

it follows that

$$\frac {\partial L}{\partial \dot \theta} = m r ^2 \dot \theta$$

is a constant. Since time is not one of the generalized coordinates, it also follows that

$$\dot r \frac {\partial L}{\partial \dot r} + \dot \theta \frac {\partial L}{\partial \dot \theta} - L = \frac {1}{2} m \dot r^2 + \frac {1}{2}m r^2 \dot \theta ^2 - \frac {GMm}{r}$$

is a constant.

But this is a two dimensional system, so there should be 2(2)-1=3 constants.

What is the third constant? and is there a method for finding every such constant in any physical system (where the Lagrangian is known) that doesn't rely on intuition or prior knowledge, like knowing in advance that angular momentum and energy are always conserved?

Any clue would be appreciated.

 

[turin]

You have to choose the constants. They are inputs. Have you heard of initial conditions and boundary values? It's the same idea. Physics is differentials. Physics will not tell you what exists, just what will happen to it.

EDIT: I'm sorry, I think I misunderstood your question. After reading your second post again, I'm unaware of a general answer to your question. Noether's theorem may be of interest to you, but it also relies on you recognizing some things (symmetries) for which I am unaware of a general procedure. If you do manage to recognize them, then Noether's theorem will give you conserved quantities. But, I'm afraid Noether's theorem is really nothing more than a formalised (but more general) way of saying that the ignorable coordinates give conserved momenta, like you have already mentioned.

BTW, for the orbital problem, there is another conserved quantity called the Laplace-Runge-Lenz vector, but I can't remember how to derive it. It is basically a conservation of the orientation of the ellipse, i.e. no precession. Of course, there is precession due to GR, so I think this must be a consequence of the inverse square law. I'm sorry, I just don't remember the details.

 

[snoopies622]

Thanks for the Laplace-Runge-Lenz vector reference. I had never heard of it.

After re-reading the relevant passage from Landau and Lifgarbagez, I think you may be right: they might simply be referring to the inital conditions of the system. I'll have to do a little more work on the matter to be certain. I am interested in Noether's theorem, but so far I haven't found a book or any other source of information about it that has satisfied me. Is it really just a special case of the Lagrange equations?

 

[turin]

I am interested in Noether's theorem, but so far I haven't found a book or any other source of information about it that has satisfied me. Is it really just a special case of the Lagrange equations?

No. It is a relationship between symmetry and invariance. I think it is in Goldstein.

 

[atyy]

It does seem to be n invariants, and only for "integrable" Hamiltonians.
http://www.scholarpedia.org/article/Hamiltonian_systems#Integrable_Systems

 

[snoopies622]

Here's the relevant quotation from Landau and Lifgarbagez. It's from chapter 2 of "Mechanics".

"During the motion of a mechanical system, the 2s quantities $$q_i$$ and $$\dot q_i$$ (i=1,2,...,s) which specify the state of the system vary with time. There exist, however, functions of these quantities whose values remain constant during the motion, and depend only on the initial conditions. Such functions are called integrals of the motion.

The number of independent integrals of the motion for a closed mechanical system with s degrees of freedom is 2s-1. This is evident from the following simple arguments. The general solution of the equations of motion contains 2s arbitrary constants (see the discussion following equation (2.6)). Since the equations of motion for a closed system do not involve time explicitly, the choice of the origin of time is entirely arbitrary, and one of the arbitrary constants in the solution of the equations can always be taken as an additive constant $$t_0$$ in the time. Eliminating $$t + t_0$$ from the 2s functions

$$q_i = q_i ( t + t_0 , C_1 , C_2 , ..., C_{2s-1}) , \hspace {3 mm} \dot q_i = \dot q_i (t + t_0 , C_1 , C_2 , ..., C_{2s-1} ),$$

we can express the 2s-1 arbitrary constants $$C_1 , C_2 , ..., C_{2s-1}$$ as functions of $$q$$ and $$\dot q$$, and these functions will be integrals of the motion."

Equation 2.6 is

$$\frac {d}{dt} ( \frac {\partial L}{\partial \dot q_i}) - \frac {\partial L}{\partial q_i}=0 \hspace {10 mm} (i=1,2,...,s)$$

and the following discussion is

"Mathematically, the equations (2.6) constitute a set of s second-order equations for s unknown functions $$q_i (t)$$. The general solution contains 2s arbitrary constants. To determine these constants and thereby to define uniquely the motion of the system, it is necessary to know the initial conditions which specify the state of the system at a given instant, for example the inital values of all the coordinates and velocities."

So I guess finding the conserved quantities in a system involves both finding the solutions to the Lagrange equations, and then rearranging them in such a way that all the initial value constants are functions of $$q$$'s and $$\dot q$$'s.