The constants of motion such as the energy, linear mlmentum and angular momentum, provide insight in the physical system under study.  it is not necessary to know these constants, but it is very helpful. If you know all constants of motion, the solution if the problem is trivial.

Constants of motion are very helpful for reducing the complexity of the classical equations of motion. Therefore, given a dynamical system in classical mechanics, the first thing to search for are all its constants of motion.

The constants of motion such as the energy, linear mlmentum and angular momentum, provide insight in the physical system under study.  it is not necessary to know these constants, but it is very helpful. If you know all constants of motion, the solution if the problem is trivial.

Hi Anu,

The constants of motion are like the lines in an address for locating someone. You can depend on them (their values) to characterize the system. This is their importance.

For a system without any constants of motion, its scientific treatment becomes extremely difficult if not impossible. If everything about a system is changing (no constants related to it), where do you start? Our aim in science is to study the dynamics of systems, and it requires the existence of something unchanging which you can depend on and then the changing things can be taken care of.

Fortunately, translations in space and time, and rotations in space are symmetries of most systems, and therefore we get conservation laws for the corresponding generators which are constants of motion.

Best

On a more general note, for Hamiltonian (or Lagrangian) systems the constants of motion are closely related to symmetries of the system via the celebrated Noether theorem. Also, as already pointed out by Kåre Olaussen, using the constants of motion allows to reduce the system to a simpler one (roughly speaking, one can study the dynamics on the level surface of the integrals of motion, and in the Hamiltonian (or Lagrangian) case it could be possible to use the associated symmetry to further reduce the system). See the links below for further details :

https://en.wikipedia.org/wiki/Constant_of_motion

https://en.wikipedia.org/wiki/Noether's_theorem

Constants, such as the period of a pendulum, appear naturally in classical physics, although in modern physics they are more appropriately called invariants.  In physics, in general, when we talk about a "constant" we usually mean an "invariant".

Invariants are an expression of what is hidden in a classical system dynamics of motion, they are part of the internal "ropes and pulleys" that make things behave the way they do.

Using invariants allows one to more easily capture how a system changes under a transformation, without concerns about the inner mechanisms.

For example, in periodic motions the theory of mechanics shows that quantities called actions can remain invariant for slow changes in the system. In Maxwell's equations the speed of light is an invariant, although it is not constant at all -- the value depends on the medium, light moves faster in vacuum and slower in water. 

Invariants also play a key role In natural sciences. We multiply the power of our means of observation and reasoning, but our starting and ending points remain sensorial, the phenomena and relationships. In that, as said by Max Planck, "Our task it to find in all these [relative] factors and data, the absolute, the universally valid, the invariant, that is hidden in them."

Cheers,

Ed Gerck

With constants of motion, one can determine the trajectories in the configuration  space by solving a set of first order differential equations instead of the Euler-Lagrange equations which are a set of  second order differential equations. For instance, in a 1-dimensional classical problem, the energy conservation reads,

1/2 m (dr/dt)^2 + V(r) = Energy

This is far easier to solve, compared to the Newton's equation,  if one knows the energy and the potential energy function V(r).

Thank you all.....

But my question is:

If i have a circle of radius R, then the constraint on the system is the fixed radius. But by knowing the value of radius alone, i can find its position in space, where it lies in space. I think it requires something else also to describe the whole system.

The constants of motion are basically a result of universal laws of nature. From the mathematical point of view, the functions that describe physical phenomena fit the observable reality through these constants. In relativistic mechanics and quantum mechanics, we also observe universal amounts for the same reason mentioned above.

Consider one of the simplest initial value problems of classical mechanics

(1) x''(t) + x(t) = 0,

(2) x(0) = x0, x'(0) = y0,

where ' denotes differentiation with respect to time. By multiplying (1) by 2*x'(t), which is not always zero, we find that

(3) [ x'(t)2 + x(t)2 ]' = 0,

We conclude that x'(t)2 + x(t)2 is independent of t. I.e., it is a constant of motion, which we may call R2. For shorthand, rewrite x(t) -> x, y(t) -> y.  Then we have found that the motion takes place in a two-dimensional space of position and velocity (better described as phase space), on a circle of radius R around the  origin, x2 + y2 = R2. We can determine R from the initial conditions, R2 = x02 + y02. Thus we can write x = R sin(φ), y = R cos(φ), where the time-dependence of φ still needs to be determined. We find x' = R cos(φ) φ' = y = R cos(φ). I.e., φ' = 1. Hence it follows that φ = t + φ0, where we must have that tan(φ0) = x0/y0.

Since this example is extremely simple to solve directly, it does not give full credit to the power of finding constants of motion (which I think is better wording than invariants, which can mean many other things). But note that a constant of motion (i) gave a restriction on where the motion could take place (depending on initial conditions), here from the two-dimensional plane to a one-dimensional curve in that plane, and (ii) was helpful for finding a first order equation for the motion along that curve.

I do not think knowing the radius alone would give you the position; your constraint still allows the angular degree of freedom. Furthermore, constants of motion or conserved quantities tend to imply aspects and directions of symmetry of a system, which are embedded within them, such as Killing vectors and Killing tensors. Displacing the system along the direction of Killing vectors leaves the system invariant. Also, conserved quantities like the angular momentum are also solutions to some equations of motion, which in turn help simplify our efforts to solve the remaining equations, as in the case of orbital motion under inverse square law forces like gravity.

Dear friends,

Artur pointed to the major reason that generates the conservation laws. But I prefer to say it otherwise: we have constants of motion because our part of the universe has SYMMETRIES. The fact that a motion in the space free of external fields, conserves the linear momentum, is caused by the homogeneity of the space. The conservation of the angular momentum, by the isotropy of the space. Of the energy, by the symmetry I time.

These are important, because, simply, when describing a process we have to respect the conservation laws, otherwise the description is wrong.

An example can be seen in special relativity, where the speed of light is kept constant due to variations of time and external space, i.e., the constancy of speed of light is obtained at the expense of variations in space-time.

This question il linked with the idea of the observables in physical theories. the presence of the symmetry is a constraint on the number of observables. the more you have a symmetries  less you've an obserables. Think about General relativity, because of the huge symmetry (the difeomorphisme) it's very hard to exhibit an observable in this theory (the Hamiltonian is a constraint H =0) . In gauge theory we have the same problem but with a better situation, the symmetry is local and we can still define an observable using the holonomy.

In physics, motion is a change in position of an object with respect to time. So here we need to compare two observables or the same observable in two different times. But to compare you need part of the observabe to be invariants. Indeed, Motion is typically described in terms of displacement, distance (scalar), velocity, acceleration, time and speed. Motion of a body is observed by attaching a frame of reference to an observer and measuring the change in position of the body relative to that frame.

Now the question of observable depend on the measurement methods and instruments. When you do measurement you need some reference, something which does not change; if not you can't do measurement. So you check the symmetry of your system to define the observable of the theory and you use themes as references of mesearument. This is the signification of Poisson brackets when you compare two observables: you compare two results : by changing the first observable following the vector field of the second and vis versa and you sustract the two results to see if the two observables are compatibles or not, i.e. they can be measured in the same time etc..

The conception here is more on conservative laws (i.e.  the symmetries of laws of motions). In contrast Newton can be based on the concept of the force not on energy, so you've a different approach. But still you can connect the two approach.

in case of a system invariant by rotation, the moment p is constant in time and this is natural. etc...

Anu, in case of a circular motion with a radius R in classical mechanics, you will know the position at any moment if you know the position at one single moment, say, at the start.

You need the constant (starting) velocity of the rotating mass, and if the connection with the middle is a rope, you can then find the tension (centripetal force) in the rope. If it is gravity, you can find the value of the central mass.

The main equation is a geometrical one: a = v²/r or if you prefer : r = v²/a. It says: when you have an object at a constant velocity-amplitude of v, and you add a constant acceleration-amplitude a upon it that is always perpendicular upon the instant velocity vector, the object will describe a circle with radius r.

To the quite correct remarks of Kare and Sofia, I would add a rather more mundane point: Newton's equations connect position and acceleration, so that two derivatives are needed. Conservation laws directly allow to evaluate velocities as a function of position, which is technically far simpler, and intuitively straightforward.

An example: take a cylinder rolling without slipping on an inclined plane. Doing things with Newton's equations is quite possible, but energy conservation immediately gives you velocity as a function of position, which solves the problem instantly.

Christian> If they would not exist, physics would not exist.

It is a bit strongly stated, but it illustrates why physics is a more precise theory than economy. Not too many constants of motion in the latter, in particular when "money" can be generated freely.

Thank you so much for providing me valuable information...

Thanks to all.

In a problem of classical motion there are very few quantities which will remain constant even though it is possible to define many many physical quantities which are experimentally measurable. Rareness makes them precious.

Then symmetries are related with conserved quantities (See Mechanics by Landau and Lifshitz). Conservation of momentum is related to translational invariance, conservation of angular momentum is related to rotational invariance, conservation of energy is related with time translational invariance.

In the microscopic world conservation of electric charge is related with the invariance of the Lagrangian density under a local U(1) gauge transformation of the electron field (not electric field). It turns out that to maintain gauge invariance of the Lagrangian density, a simultaneous local transformation of photon field is also required.