# What is the advantage of Hamilton's canonical equations?

1. May 8, 2014

### Zoli

Hi!

I would like to know that in what circumstances Hamilton's canonical equations are superior to the Lagrange-equations of the second kind. We know that every second order equation can be rewritten as a system of first order equations.

Thanks,

Zoli

2. May 8, 2014

### UltrafastPED

There are several advantages, but I don't think they are related to the constraints:

1. Hamilton's equations are linear and first order PDEs; Lagrange's equations non-linear and second order.
2. The solutions to Hamilton's equations exist in phase space, and have very nice properties; Lagrange's equations exist in a different space, and the solutions may have some not-so nice properties.
3. The linearity of Hamilton's equations comes at a price: twice as many equations. But they can be easily solved by numerical integration when there are no analytic solutions ... which is most of the time.

See http://en.wikipedia.org/wiki/Symplectic_manifold for some discussion

3. May 8, 2014

### Zoli

1. Why are they linear PDEs? See http://encyclopedia2.thefreedictionary.com/Hamilton's+Canonical+Equations+of+Motion. I do not refer to the Hamilton-Jacobian equation: http://en.wikipedia.org/wiki/Hamilton–Jacobi_equation
2. So you mean that it has nice properties when we use apply stability analysis?
3. It is true since ODE solvers need first order equations, but rewriting Lagrange-equations as first order equations will do the same, doesn't it?

4. May 8, 2014

### UltrafastPED

Hamilton's equations are partial differential equations ... your reference has suffered during translation! You can see the partial derivative. They are also linear in the variables; they are also coupled.

The Lagrange equations are non-linear. Good luck with your hope to convert them to first order ... this is only guaranteed for linear systems. But if you perform a Legendre transform of q'->P then you get the pair of first order Hamilton equations.

For the nice properties see any advanced text on Hamiltonian mechanics ... I'm away from my books.

5. May 8, 2014

### dextercioby

I don't find Hamilton's equations superior in any way. If you can solve a system's Hamilton's equations, you could have solved the (Euler-)Lagrange ones as well.

OTOH, the Hamiltonian formalism as a whole is relevant for quantization which puts in a greater emphasis in teaching it than the one which would be put on the Lagrangian formalism.

6. May 9, 2014

### UltrafastPED

This page summarizes the nicest property of Hamiltonian vs Lagrangian solution space: