Test if 2nd order diff eq. can be derived from a Hamiltonian

In summary, the conversation discusses different approaches to testing whether a given second-order differential equation can be derived from a Hamiltonian without knowing the specific form of the Hamiltonian. One approach involves using a general Lagrangian function and comparing the resulting equation of motion to the given differential equation. However, this method may not work for equations with fractional powers of x and p. Another approach involves numerically integrating the equations of motion and checking for conserved quantities or using canonical transformations and checking for a symplectic criterion. The participants also mention the difficulty of matching terms in complicated cases.
  • #1
Bosh
9
0
Imagine I have a complicated second-order differential equation that I strongly suspect can be derived from a Hamiltonian (with additional momentum dependence beyond p2/2m, so the momentum is not simply mv, but I don't know what it is).

Are there any ways to test whether or not the given second order differential equation can be derived from a Hamiltonian without finding what it is?
 
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  • #2
If the potential energy of the system doesn't have an infinity (like in the potential of two point charges: ##V(r)\propto r^{-1}##), you can try to form a general Lagrangian function

##L =\sum_{k,l=0}^\infty c(k,l)\dot{q}^{k}q^{l}##

where the ##c(k,l)## are constant coefficients, and calculate the corresponding eq. of motion and find by comparison what Lagrangian corresponds to you DE. However, if there's fractional powers of x and p in the equation, this probably isn't so easy.
 
  • #3
This has been my approach. I should have added that the additional terms are a perturbation, so I'm trying to achieve even just the simpler goal of matching terms to the next order in epsilon. Even so, I'm only close in a simple case, and in complicated cases it seems hopeless to get everything to match up (it's not a 1 degree of freedom problem, and there are many terms to simultaneously match).

So I was trying to think if there's a more clever, non-brute-force method. Something like trying to numerically integrate the equations of motion and somehow checking the dimensionality of the manifold on which the trajectories move to check for conserved quantities. I hoped though that there might be something simpler...like connecting the differential equation to some sort of canonical transformation and checking whether a symplectic criterion is satisfied...

Thanks for any thoughts!
 

FAQ: Test if 2nd order diff eq. can be derived from a Hamiltonian

1. What is a Hamiltonian and how is it related to differential equations?

A Hamiltonian is a mathematical function that describes the total energy of a physical system. It is related to differential equations because it can be used to derive the equations of motion for a system, which are often represented as differential equations.

2. How can a Hamiltonian be used to test if a 2nd order differential equation can be derived?

A Hamiltonian can be used to test if a 2nd order differential equation can be derived by checking if the Hamiltonian satisfies the canonical Hamilton's equations, also known as the Hamiltonian equations of motion. If the Hamiltonian satisfies these equations, then it is possible to derive the 2nd order differential equation.

3. What are the canonical Hamilton's equations?

The canonical Hamilton's equations are a set of two equations that describe the evolution of a system over time. They are given by:

dH/dt = -dG/dq
dH/dp = dG/dp


where H is the Hamiltonian, G is the system's total energy, q is the generalized coordinate, and p is the generalized momentum.

4. Can a 2nd order differential equation always be derived from a Hamiltonian?

No, not all 2nd order differential equations can be derived from a Hamiltonian. The Hamiltonian must satisfy certain conditions, such as being a smooth function, in order for the canonical Hamilton's equations to be valid. If these conditions are not met, then a 2nd order differential equation cannot be derived from the Hamiltonian.

5. What are some real-world applications of using Hamiltonians to derive differential equations?

Hamiltonians have many applications in physics and engineering, such as in classical mechanics, quantum mechanics, and control systems. They are used to describe the behavior of physical systems, such as the motion of particles in a magnetic field or the dynamics of a pendulum. They are also used in optimization problems and in the study of chaotic systems.

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