As an exercise in Lagrangian and Hamiltonian mechanics, I am looking at a system with the following Lagrangian:

$$L=\dot R \cdot\dot R-\theta\dot R\cdot (SR)+\lambda(R\cdot R-1) $$

$R$ is a vector in 4D Euclidean space, constrained to lie on the unit hypersphere ($\lambda$ is the Lagrange multiplier for this constraint). $S$ is a skew-symmetric matrix that squares to $-1$. The Euler-Lagrange equations result in the following equations of motion:

$$\ddot R-\theta S\dot R-\lambda R=0$$ $$R\cdot R=1$$

My first difficulty is that I cannot figure out how to solve for $\lambda$ to determine the constraint forces. My second difficulty is in finding the general solution; I can find specific solutions of the form: $$R=e^{At} R_0$$ Where $A$ is assumed to be skew-symmetric so that the constraint is guaranteed to be satisfied. This has a unique solution $A=\theta S/2$ when $\lambda=\theta^2/4$ (which is problematic; there does not appear to be enough degrees of freedom to allow for arbitrary initial positions and velocities) and and infinite family of solutions for $\lambda < \theta^2/4$ (which I'm not sure how to combine to get a general solution).

Never mind those problems. Now I try to form the Hamiltonian. The conjugate momenta are:

$$ P=\frac{\partial L}{\partial \dot R}=2\dot R-\theta SR$$

Which is invertible to find the velocity, and:

$$ \pi=\frac{\partial L}{\partial \dot \lambda}=0 $$

Which obviously is not. Not sure what to do about the constraint, I just evaluate it in computing the Hamiltonian and the equations of motion, so that term disappears. I get a Hamiltonian:

$$H=\dot R \cdot\dot R=\frac{1}{4}P\cdot P+\frac{1}{2}\theta P\cdot (SR)$$

Plugging this into the Hamiltonian equations, I get the result:

$$\dot P=\frac{1}{2}\theta SP$$ $$\dot R=\frac{1}{2}P+\frac{1}{2}\theta SR$$

And so:

$$\ddot R=\theta S\dot R+\frac{1}{4}\theta^2 R$$

Which is the same equations of motion that I got before, except that $\lambda=\theta^2/4$ specifically.

So that seemed to work. Except: I wanted to verify that these equations of motion preserve the constraint. Evaluating the Poisson bracket of the constraint with the Hamiltonian shows that it is proportional to $P\cdot R$, so that has to be zero in the initial conditions. But the Poisson bracket of $P\cdot R$ gives the expression:

$$\frac{1}{4}(P\cdot P+\theta P\cdot(SR)+\theta^2)$$

Which should be zero to ensure that the constraint is preserved - but it isn't zero for the unique solution that I found above for $\lambda=\theta^2/4$, even though that solution does preserve the constraint by construction!

So I'm trying to figure out (1) how to solve for the constant $\lambda$ and how to get a general solution to the equations of motion if possible, and (2) how to appropriately deal with the constraint in the Hamiltonian formalism, or (3) if I've just made a mistake somewhere. Hoping there is someone out there who can point me in the right direction. Thanks very much.


1 Answer 1


Okay, I figured out where I went wrong!

$\lambda$ isn't a constant. (I was misunderstanding the role of the Lagrange multiplier.) Then in fact it is easy to solve for, simply by taking the dot product of the equation: $$\ddot R-\theta S\dot R-\lambda R=0$$ with $R$.

For the Hamiltonian formalism, following the generalized Hamiltonian procedure described here, and correcting the error I found that I had made in evaluating the Poisson brackets, I get the same equations of motion as with the Lagrangian formalism.

(Does anyone have a better resource covering the generalized Hamiltonian procedure than the wikipedia link?)


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