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12
votes
4answers
2k views

If all conserved quantities of a system are known, can they be explained by symmetries?

If a system has $N$ degrees of freedom (DOF) and therefore $N$ independent1 conserved quantities integrals of motion, can continuous symmetries with a total of $N$ parameters be found that deliver ...
11
votes
3answers
399 views

What could cause an asymmetric orbit in a symmetric potential?

My question can be summarized as: Given a potential with a symmetry (e.g. $z\rightarrow-z$), should I expect orbits in that potential to exhibit the same symmetry? Below is the full motivation for ...
8
votes
1answer
3k views

Constants of motion vs. integrals of motion

Since the equation of mechanics are of second order in time, we know that for $N$ degrees of freedom we have to specify $2N$ initial conditions. One of them is the initial time $t_0$ and the rest of ...
6
votes
2answers
214 views

Why are there only 3 Additive Integrals of Motion?

1. I was reading Landau & Lifschitz's book on Mechanics, and came across this sentence on p.19: "There are no other additive integrals of the motion. Thus every closed system has seven such ...
5
votes
1answer
649 views

Non-integrability of the 2D double pendulum

Context: For a system with $n$ degrees of freedom (DOF), one has to deal with $2n$ independent coordinates ($2n$ dimensional phase space), of position $q$ and $\dot{q}$ in Lagrangian formulation, ...
3
votes
2answers
2k views

Integrals of Motion

Landau & Lifshitz write on the first page of chapter 2 of their Mechanics book (p.13) The number of independent integrals of motion for a closed mechanical system with $s$ degrees of freedom ...
3
votes
3answers
1k views

Complete vs General Integral of first order PDE

The following is an excerpt from Landau's Course on Theoretical Physics Vol.1 Mechanics: ... we should recall the fact that every first-order partial differential equation has a solution depending ...
3
votes
0answers
50 views

Integrals of motion for a free particle

I'm struggling to understand the argument on p. 13 in Landau and Lifshitz that for a system with $N$ degrees of freedom there must be $2N-1$ integrals of motion. In particular, I can't understand ...
3
votes
0answers
154 views

What variable is the conjugate momentum for angular momentum?

From the definition of conjugate momentum for a generalized coordinate we get that the conjugate for angular momentum should be proportonal to its integral with respect to time. According to my ...
2
votes
2answers
1k views

Applying velocity Verlet algorithm

I want to implement a simple particules system using the velocity form of the Verlet algorithm as integrator. Initial conditions at $t=0$ for a given particule $p$: mass: $ m $ position: $\...
2
votes
0answers
20 views

Constancy of Coefficients of Additive Integrals Throughout Subsystems of a Closed System

I'm studying Landau and Lifshitz's Statistical Physics, Part 1, 3rd edition and am looking for clarification on the following statement, which appears on page 11 in the section on The Significance of ...
2
votes
1answer
581 views

Angular momentum components as independent integrals of motion

I was told that in order to solve the Kepler problem (6 degrees of freedom in total) you have to proceed, step by step, to reduce those degrees of freedom using the integrals of motion. You do so ...
1
vote
1answer
137 views

Integrals of Motion for s Degrees of Freedom

From Landau & Lifshitz, Classical Mechanics, the number of integrals of independent integrals of motion for a system of $s$ degrees of freedom is $2s-1$. I am considering a spherical pendulum in ...
1
vote
0answers
93 views

Finding conserved quantities from Hamiltonian when Symmetry is not evident [closed]

A particle is moving in 3D space, under a potential $$V = -\frac{\alpha}{r}-\frac{\vec{r} \cdot \vec{\mu}}{r^3 } $$ where $\vec{\mu}$ is some constant vector. I need to show there are three ...
1
vote
0answers
143 views

Single particle trajectory in a quadrupole potential

I am wondering if there are any studies of a single (classical) particle trajectory in quadrupole potential: $$ V(x,y,z)=A\sqrt[]{\frac{x^2 + y^2}{a} + \frac{z^2}{b}} $$
1
vote
0answers
747 views

Do symmetries increase the number of conserved quantities? [closed]

Let us consider a classical mechanical system of N particles in a constant external field. We have 3N coordinates and 3N velocities, so totally 6N unknown variables. We have 6N ordinary differential ...
1
vote
2answers
78 views

Action variables in canonical transformations

Let's suppose we have a Hamiltonian $H(p_k, q_k)$ and we want to transform it via a canonical transformation to one Hamiltonian which doesn't depend on the new coordinates $w_k$, but only in the ...
0
votes
1answer
64 views

Determining constants of motion for this Hamiltonian? [on hold]

Let $S$ be a Hamiltonian system with three degrees of freedom, let $q_1, q_2, q_3$ be the generalized coordinates. Suppose the Hamiltonian of the system is $$ H = \alpha \sum_{i=1}^3 p_i + V(q_1^2 + ...
0
votes
0answers
22 views

What is the Relationship Between Poisson Brackets and Additive Integrals of Motion?

Question in the title: what is the relationship, if any, between Poisson brackets and additive integrals of motion? Context: Is there anything we can say about additive integrals of motion in ...
0
votes
1answer
36 views

Regarding $f$ degrees of freedom & $f\!-\!1$ constants & inclusion of these constants

In the classic & famous book "Electromagnetic fields & Interactions" by Richard Becker (Dover publishing), on page 55 (of volume 2) , author says: If the system possesses f degrees of ...
0
votes
1answer
56 views

Procedure for finding integrals of motion

In some examples I have read that if you want to find the integral of motion for some equation of motion, say on the form $\ddot{x}+ax=0$ for some constant $a$, you multiply the EOM by $$\dot{x}=q(x) \...
0
votes
0answers
48 views

First integrals for a particle in a central-force field

Consider an arbitrary dimension $n>3$. What are the independent first integrals for a particle? The Hamiltonian is $$ H = \frac{p^2}{2m} +V (|r|) . $$
0
votes
0answers
79 views

Motion Integrals of a Particle in a Force Field

I am trying to wrap my head around the following problem: A point particle is moving in a field, where its potential energy is U=-α/r. Find first motion integrals. In our university we have no ...