# Why are there 2s -1 independent integrals of motion?

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• yucheng
In summary: Furthermore, isn't the Lagrangian an equation in itself? Wouldn't it be more correct to use an equation like $$\mathbf{L}=\mathbf{A}+\frac{d}{dt}\mathbf{v}$$ instead?The answer provided above seems interesting. However, how correct is it? There are several points that I would like to verify...- Are there other proofs of this?- However, shouldn't it be N position and N velocities? Can it be shown to be equivalent? Plus aren't we working with an autonomous system of equations? Is this why we need t−t0 so that it is independent of time?
yucheng
I was reading Mechanics by Landau and Lifshitz and I am confused when it is stated in chapter 2 section 6 that one of the integrals of motion is not independent and it can be considered an additive constant of time. Hence I tried searching it up online.

https://physics.stackexchange.com/q...f-motion-vs-first-integrals?noredirect=1&lq=1

According to the OP in the link above (first paragraph second sentence), we need to specify 2N initial conditions, one of them is the initial time, the others the initial positions and velocity.

However, shouldn't it be N position and N velocities? Can it be shown to be equivalent? Plus aren't we working with an autonomous system of equations? Is this why we need ##t - t_0## so that it is independent of time?

https://physics.stackexchange.com/questions/13832/integrals-of-motion

The answer provided above seems interesting. However, how correct is it? There are several points that I would like to verify...

Questions:
1. Because the ##\mathcal{L} (q, \dot q)##, the Lagrangian is independent of the acceleration. Hence $$\frac{d}{dt} \frac{\partial \mathcal{L} }{\partial \dot q} - \frac{\partial \mathcal{L}}{\partial q}$$
which only involved one time derivative, only introduces terms linear in ##\ddot q##.
2. According to the author (see a comment below the post as well), $$\ddot q_i =\frac{\text{d}\dot q_i}{\text{d} q_1} \frac{\text{d}q_1}{\text{d}t} =\dot q_1\frac{\text{d}\dot q_i}{\text{d}q_1}$$. How does a total time derivative become a total derivative in ##q_1##? Are we performing a change of variables by inverting ##q_1(t)## to get time as a function of ##q_1## then all coordinates become ##q_i(t(q_1))##?
3. Are there other proofs of this?

Last edited:
yucheng said:
However, shouldn't it be N position and N velocities? Can it be shown to be equivalent? Plus aren't we working with an autonomous system of equations? Is this why we need t−t0 so that it is independent of time?
Hi. Position and velocity change in time in general so they cannot be integral constants e.g. energy.

anuttarasammyak said:
Hi. Position and velocity change in time in general so they cannot be integral constants e.g. energy.
Actually, for that part of the question, I am referring to the initial conditions (which of course determines the ##2s## integrals of motions). I asked it here because it is very relevant! ;)

Ooops by the way, ##N=s## (degrees of freedom)

yucheng said:
Because the L(q,q˙), the Lagrangian is independent of the acceleration. Hence $$\frac{d}{dt} \frac{\partial \mathcal{L} }{\partial \dot q} - \frac{\partial \mathcal{L}}{\partial q}$$
which only involved one time derivative, only introduces terms linear in q¨.
Is this correct?

## 1. Why are there 2s -1 independent integrals of motion?

This is a common question in the field of mechanics and physics. The reason for this is due to the number of degrees of freedom in a system. In a system with s degrees of freedom, there will be s independent integrals of motion. However, in certain cases, there may be constraints or symmetries in the system that reduce the number of independent integrals of motion to 2s - 1.

## 2. What are independent integrals of motion?

Independent integrals of motion are quantities that remain constant throughout the motion of a system. They are derived from the equations of motion and are independent of each other, meaning that they cannot be expressed as a combination of other integrals of motion. These integrals of motion provide valuable information about the dynamics of a system and can be used to solve for other quantities.

## 3. How do independent integrals of motion relate to the conservation laws of physics?

Independent integrals of motion are closely related to the conservation laws of physics, such as the conservation of energy and momentum. In fact, these laws can be derived from the independent integrals of motion. The conservation of energy, for example, can be derived from the integral of motion known as the Hamiltonian.

## 4. Can the number of independent integrals of motion change in a system?

Yes, the number of independent integrals of motion can change in a system. This can occur if the system experiences a phase transition or if there is a change in the number of degrees of freedom. In general, the number of independent integrals of motion will remain constant unless there are significant changes in the system.

## 5. How are independent integrals of motion used in practical applications?

Independent integrals of motion have many practical applications, particularly in the field of mechanics and physics. They can be used to analyze the dynamics of complex systems, such as celestial bodies or fluid flow. They are also used in the development of new technologies, such as spacecraft navigation and control systems. Additionally, independent integrals of motion are crucial in the study of fundamental laws of physics and can help us better understand the behavior of the universe.

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