# Questions tagged [integrals-of-motion]

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### How many constants of motion are there in a 2D two body problem?

A system consists of two masses interacting with gravitational force, rotating around their centre of mass. If we only consider the $xy$ plane where the masses rotate, the system has 8 degrees of ...
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### Is 2D rectangular billiard an integrable system? What's the form of explicit solution?

Suppose the free particle moving inside 2D box. $$H=\frac{p_x^2}{2m}+\frac{p_y^2}{2m}+V$$ where the potential is zero inside the box and infinite outside the box. It's clear that $p_x,p_y$ are not ...
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### When is the first integral equal to the total energy of the system?

For a function that is a solution of the Euler-Lagrange equation there is a constant known as the first integral which is given by: $$E=\sum_i(\frac{\partial f}{\partial y'_i}y'_i)-L$$ I am trying to ...
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### ODEs with rational first integrals [closed]

I would like some examples of ODEs (i.e., $\dot{x}=f(x)$, where $x\in\mathbb{R}^n$) that possess one or more rational first-integrals of the form $$H(x)=\frac{a_1^Tx+\alpha_1}{a_2^Tx+\alpha_2},$$ ...
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### Calculate distance based on - Resistance - Acceleration - Time - Initial Velocity

I have been looking at equations that can represent how far something travels given these three variables. Initial Velocity, Resistance, Acceleration, and Time. The main thing I saw which was similar ...
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### Global conserved quantities for point particle coupled to a Schrodinger field

We have a box (represented by a potential $V$) with a classical particle in it. If the box has a finite inertia and it's floating in space, then it shakes as the particle bumps on the walls. The total ...
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### Number of integrals of motion

In Landau-Lifshitz Classical mechanics textbook, it is said that there are generally $2s-1$ integrals of motion where $s$ is the number of degrees of freedom. Why is that? I couldn't find anywhere an ...
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### What term describes the trajectory space splitting behavior when parametrizing a pendulum?

So I was thinking about this post I made earlier: What is the second conserved Quantity of the Pendulum? In which a pendulum appears two have significant properties. It's Kinetic Energy and its Phase. ...
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### 2D harmonic oscillator trajectory

Consider the Hamiltonian for the classic planar harmonic oscillator: $$H = H_x + H_y$$ where $$H_x~=~\frac{1}{2}(p_x^2+x^2), \qquad H_y~=~\frac{1}{2}(p_y^2+y^2).$$ So it is possible to obtain a set of ...
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### Constants of motion [duplicate]

For any system performing any kind of motion with $n$ degrees of freedom, are $2n-1$ integrals of motion and also $2n$ constants of motion always present? If yes, then is there always a symmetry for ...
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### Integral of Motion in the 1D Calogero Model

In section 4 of this article, suppose we have $N$ particles of the same mass $m$ moving in one dimension and interacting with each other via the potential $V_{ij}\equiv V(x_i - x_j)$, where $x_i$ ...
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### Apparent emergence of conserved quantities in non-integrable systems

This question arises from the comments relevant to the post When is the ergodic hypothesis reasonable? Consider a Hamiltonian system having more effective degrees of freedom than conserved quantities....
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Working in 3-dimensions, if we are given a Lagrangian containing $N$ particles. Say, through Noether's theorem, we know that the sum of the linear momentum of all $N$ particles in each direction are ...