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surajshankar
surajshankar
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As the first question has received sufficient exposition, I would like to make a point with regard to the second one. First thing to understand is that integrability and non-linearity of a system are two different concepts. It is true though that all linear systems in classical mechanics (i.e those that are described by systems of linear equations, be them algebraic or differential) are integrable, but integrability does not imply linearity. Typically looking at the case of a system of differential equations describing a system, integrability is then a condition requiring analytical solvability when all the required initial and boundary conditions have been specified (the problem is then well-posed). In classical Hamiltonian systems (or the more general case being a finite system of ODEs), integrability is then specified by the Liouville-Arnold theorem . In all autonomous systems, the hamiltonian automatically becomes a constant of motion and hence in 1 dimension, all hamiltonian systems are integrable. Enlarging the phase space of dynamics, one now requires many more integrals of motion that specify iso-surfaces upon which dynamics occurs and the intersection of all these surfaces will give the required trajectory of the system, once the initial conditions are fixed.

The idea of requiring constants or invariants of an equation is a very broad one and the argument mostly goes through for most other cases too. For infinite dimensional systems (like PDEs, say), we in principle need an infinite set of constants, though it is not sufficient. The existence of such independent constants in involution with each other many times points to a deeper symmetry or degeneracy in the problem. Take for example the Kepler problem, which has three constants of motion
$$ H=\dfrac{\vec{p}^2}{2m}-\dfrac{k}{r}\quad(k>0) $$ The three constants of motion in this problem are $H,\vec{L}^2$ and $L_z$ and they all poisson commute with each other. The dynamical symmetry group of this problem turns out to be $SO(4)$ (this group contains all the canonical transformations that preserve the symmetry of the hamiltonian). An additional constant of motion is the Laplace-Runge-Lenz vector $\vec{A}$, $$ \vec{A}=\vec{p}\times\vec{L}-\dfrac{mk\vec{r}}{r} $$ which also poisson commutes with the other constants $(\{\vec{A},H\}=0,\{\vec{A},\vec{L}\}=0)$. It is these two vectors $(\vec{L}\pm\vec{A})$ modulo constant factors, that generate the group of dynamical symmetries. The fact that $\vec{A}$ is a constant implies that orbits in the Kepler problem do not precess. In the context of the hydrogen atom, this same constancy gives rise to the degeneracy of the energy levels with $\ell$ (the azimuthal quantum number). Degeneracy in $m$ (the magnetic quantum number) is demanded by the spherical symmetry of the potential. From this, we see that the existence of constants of motions that allow a system to be integrable almost always point to a deeper understanding of the problem by exposing hitherto unknown symmetries or degeneracies in the system.

There are a number of non-linear systems like the KdV equation that exhibit soliton and soliton-like solutions and this is a very active area of research in understanding the existence of solutions and integrability of nonlinear systems. One can be pretty sure that if a non-linear equation is found to be integrable, then it is definitely special and has some underlying properties that allow one to solve the equation. There is a nice write up on some of these things given in this document, that I suppose will be helpful in elucidating further details with regard to integrable systems.

As the first question has received sufficient exposition, I would like to make a point with regard to the second one. First thing to understand is that integrability and non-linearity of a system are two different concepts. It is true though that all linear systems in classical mechanics (i.e those that are described by systems of linear equations, be them algebraic or differential) are integrable, but integrability does not imply linearity. Typically looking at the case of a system of differential equations describing a system, integrability is then a condition requiring solvability when all the required initial and boundary conditions have been specified (the problem is then well-posed). In classical Hamiltonian systems (or the more general case being a finite system of ODEs), integrability is then specified by the Liouville-Arnold theorem . In all autonomous systems, the hamiltonian automatically becomes a constant of motion and hence in 1 dimension, all hamiltonian systems are integrable. Enlarging the phase space of dynamics, one now requires many more integrals of motion that specify iso-surfaces upon which dynamics occurs and the intersection of all these surfaces will give the required trajectory of the system, once the initial conditions are fixed.

The idea of requiring constants or invariants of an equation is a very broad one and the argument mostly goes through for most other cases too. For infinite dimensional systems (like PDEs, say), we in principle need an infinite set of constants, though it is not sufficient. The existence of such independent constants in involution with each other many times points to a deeper symmetry or degeneracy in the problem. Take for example the Kepler problem, which has three constants of motion
$$ H=\dfrac{\vec{p}^2}{2m}-\dfrac{k}{r}\quad(k>0) $$ The three constants of motion in this problem are $H,\vec{L}^2$ and $L_z$ and they all poisson commute with each other. The dynamical symmetry group of this problem turns out to be $SO(4)$ (this group contains all the canonical transformations that preserve the symmetry of the hamiltonian). An additional constant of motion is the Laplace-Runge-Lenz vector $\vec{A}$, $$ \vec{A}=\vec{p}\times\vec{L}-\dfrac{mk\vec{r}}{r} $$ which also poisson commutes with the other constants $(\{\vec{A},H\}=0,\{\vec{A},\vec{L}\}=0)$. It is these two vectors $(\vec{L}\pm\vec{A})$ modulo constant factors, that generate the group of dynamical symmetries. The fact that $\vec{A}$ is a constant implies that orbits in the Kepler problem do not precess. In the context of the hydrogen atom, this same constancy gives rise to the degeneracy of the energy levels with $\ell$ (the azimuthal quantum number). Degeneracy in $m$ (the magnetic quantum number) is demanded by the spherical symmetry of the potential. From this, we see that the existence of constants of motions that allow a system to be integrable almost always point to a deeper understanding of the problem by exposing hitherto unknown symmetries or degeneracies in the system.

There are a number of non-linear systems like the KdV equation that exhibit soliton and soliton-like solutions and this is a very active area of research in understanding the existence of solutions and integrability of nonlinear systems. One can be pretty sure that if a non-linear equation is found to be integrable, then it is definitely special and has some underlying properties that allow one to solve the equation. There is a nice write up on some of these things given in this document, that I suppose will be helpful in elucidating further details with regard to integrable systems.

As the first question has received sufficient exposition, I would like to make a point with regard to the second one. First thing to understand is that integrability and non-linearity of a system are two different concepts. It is true though that all linear systems in classical mechanics (i.e those that are described by systems of linear equations, be them algebraic or differential) are integrable, but integrability does not imply linearity. Typically looking at the case of a system of differential equations describing a system, integrability is then a condition requiring analytical solvability when all the required initial and boundary conditions have been specified (the problem is then well-posed). In classical Hamiltonian systems (or the more general case being a finite system of ODEs), integrability is then specified by the Liouville-Arnold theorem . In all autonomous systems, the hamiltonian automatically becomes a constant of motion and hence in 1 dimension, all hamiltonian systems are integrable. Enlarging the phase space of dynamics, one now requires many more integrals of motion that specify iso-surfaces upon which dynamics occurs and the intersection of all these surfaces will give the required trajectory of the system, once the initial conditions are fixed.

The idea of requiring constants or invariants of an equation is a very broad one and the argument mostly goes through for most other cases too. For infinite dimensional systems (like PDEs, say), we in principle need an infinite set of constants, though it is not sufficient. The existence of such independent constants in involution with each other many times points to a deeper symmetry or degeneracy in the problem. Take for example the Kepler problem, which has three constants of motion
$$ H=\dfrac{\vec{p}^2}{2m}-\dfrac{k}{r}\quad(k>0) $$ The three constants of motion in this problem are $H,\vec{L}^2$ and $L_z$ and they all poisson commute with each other. The dynamical symmetry group of this problem turns out to be $SO(4)$ (this group contains all the canonical transformations that preserve the symmetry of the hamiltonian). An additional constant of motion is the Laplace-Runge-Lenz vector $\vec{A}$, $$ \vec{A}=\vec{p}\times\vec{L}-\dfrac{mk\vec{r}}{r} $$ which also poisson commutes with the other constants $(\{\vec{A},H\}=0,\{\vec{A},\vec{L}\}=0)$. It is these two vectors $(\vec{L}\pm\vec{A})$ modulo constant factors, that generate the group of dynamical symmetries. The fact that $\vec{A}$ is a constant implies that orbits in the Kepler problem do not precess. In the context of the hydrogen atom, this same constancy gives rise to the degeneracy of the energy levels with $\ell$ (the azimuthal quantum number). Degeneracy in $m$ (the magnetic quantum number) is demanded by the spherical symmetry of the potential. From this, we see that the existence of constants of motions that allow a system to be integrable almost always point to a deeper understanding of the problem by exposing hitherto unknown symmetries or degeneracies in the system.

There are a number of non-linear systems like the KdV equation that exhibit soliton and soliton-like solutions and this is a very active area of research in understanding the existence of solutions and integrability of nonlinear systems. One can be pretty sure that if a non-linear equation is found to be integrable, then it is definitely special and has some underlying properties that allow one to solve the equation. There is a nice write up on some of these things given in this document, that I suppose will be helpful in elucidating further details with regard to integrable systems.

Source Link
surajshankar
surajshankar
  • 1.9k
  • 15
  • 17

As the first question has received sufficient exposition, I would like to make a point with regard to the second one. First thing to understand is that integrability and non-linearity of a system are two different concepts. It is true though that all linear systems in classical mechanics (i.e those that are described by systems of linear equations, be them algebraic or differential) are integrable, but integrability does not imply linearity. Typically looking at the case of a system of differential equations describing a system, integrability is then a condition requiring solvability when all the required initial and boundary conditions have been specified (the problem is then well-posed). In classical Hamiltonian systems (or the more general case being a finite system of ODEs), integrability is then specified by the Liouville-Arnold theorem . In all autonomous systems, the hamiltonian automatically becomes a constant of motion and hence in 1 dimension, all hamiltonian systems are integrable. Enlarging the phase space of dynamics, one now requires many more integrals of motion that specify iso-surfaces upon which dynamics occurs and the intersection of all these surfaces will give the required trajectory of the system, once the initial conditions are fixed.

The idea of requiring constants or invariants of an equation is a very broad one and the argument mostly goes through for most other cases too. For infinite dimensional systems (like PDEs, say), we in principle need an infinite set of constants, though it is not sufficient. The existence of such independent constants in involution with each other many times points to a deeper symmetry or degeneracy in the problem. Take for example the Kepler problem, which has three constants of motion
$$ H=\dfrac{\vec{p}^2}{2m}-\dfrac{k}{r}\quad(k>0) $$ The three constants of motion in this problem are $H,\vec{L}^2$ and $L_z$ and they all poisson commute with each other. The dynamical symmetry group of this problem turns out to be $SO(4)$ (this group contains all the canonical transformations that preserve the symmetry of the hamiltonian). An additional constant of motion is the Laplace-Runge-Lenz vector $\vec{A}$, $$ \vec{A}=\vec{p}\times\vec{L}-\dfrac{mk\vec{r}}{r} $$ which also poisson commutes with the other constants $(\{\vec{A},H\}=0,\{\vec{A},\vec{L}\}=0)$. It is these two vectors $(\vec{L}\pm\vec{A})$ modulo constant factors, that generate the group of dynamical symmetries. The fact that $\vec{A}$ is a constant implies that orbits in the Kepler problem do not precess. In the context of the hydrogen atom, this same constancy gives rise to the degeneracy of the energy levels with $\ell$ (the azimuthal quantum number). Degeneracy in $m$ (the magnetic quantum number) is demanded by the spherical symmetry of the potential. From this, we see that the existence of constants of motions that allow a system to be integrable almost always point to a deeper understanding of the problem by exposing hitherto unknown symmetries or degeneracies in the system.

There are a number of non-linear systems like the KdV equation that exhibit soliton and soliton-like solutions and this is a very active area of research in understanding the existence of solutions and integrability of nonlinear systems. One can be pretty sure that if a non-linear equation is found to be integrable, then it is definitely special and has some underlying properties that allow one to solve the equation. There is a nice write up on some of these things given in this document, that I suppose will be helpful in elucidating further details with regard to integrable systems.