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I try to understand constructing of Hamiltonian mechanics with constraints. I decided to start with the simple case: free relativistic particle. I've constructed hamiltonian with constraint:

$$S=-m\int d\tau \sqrt{\dot x_{\nu}\dot x^{\nu}}$$

$\phi=p_{\mu}p^{\mu}-m^2=0$ $-$ first class constraint.

Then $$H=H_{0}+\lambda \phi=\lambda \phi.$$

So, I want to show that I can obtain from this Hamiltonian the same equation of motion, as obtained from Lagrangian.

But the problem is that I'm not sure what to do with $\lambda=\lambda(q,p)$. I tried the following thing:

$\dot x_{\mu}=\{x_{\mu},\lambda \phi\}=\{x_{\mu},\lambda p^2\}-m^2\{x_{\mu},\lambda\}=\lambda\{x_{\mu},p^2\}+p^2\{x_{\mu},\lambda\}-m^2\{x_{\mu},\lambda\}=2\lambda \eta_{\mu b} p^b+p^2\{x_{\mu},\lambda\}-m^2\{x_{\mu},\lambda\}=2\lambda \eta_{\mu b} p^b+p^2\frac{\partial \lambda}{\partial p^{\mu}}-m^2\frac{\partial \lambda}{\partial p^{\mu}}$

$\dot \lambda=\{\lambda, \lambda \phi \}=\{\lambda,\lambda p^2\}-m^2\{\lambda,\lambda\}=\lambda\{\lambda,p^2\}+p^2\{\lambda,p^2\}=2\lambda\eta_{ak}p^{a}\frac{\partial \lambda}{\partial x^{k}}$

$\dot p_{\mu}=\{p_{\mu},\lambda p^{2}-m^2\lambda \}=p^{2}\{p_{\mu},\lambda\}-m^2\{p_{\mu},\lambda\}=-p^{2}\frac{\partial \lambda}{\partial x^{\mu}}+m^2\frac{\partial \lambda}{\partial x^{\mu}}$

If we recall that $p^2-m^2=0$, then we get from the third equation: $\dot p=0$, and from the first: $\dot x_{\mu}=2\lambda\eta_{ak}p^{a}$.

So we have

1) $\dot x_{\mu}=2\lambda\eta_{\mu b}p^{b}$

2) $\dot \lambda=2\lambda\eta_{ak}p^{a}\frac{\partial \lambda}{\partial x^{k}}$

3) $\dot p=0$

But I dont know what to do next. Can you help me?

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up vote 4 down vote accepted

Hints to the question (v1):

  1. Let us parametrize the problem wrt. an arbitrary world-line parameter $\tau$ (which does not have to be the proper time).

  2. The Lagrange multiplier $\lambda=\lambda(\tau)$ depends on $\tau$, but it does not depend on the canonical variables $x^{\mu}$ and $p_{\mu}$. Similarly, $x^{\mu}$ and $p_{\mu}$ depend only on $\tau$.

  3. The Lagrange multiplier $\lambda=\frac{e}{2}$ can be identified with an einbein$^1$ field $e$. See below where we outline a simple way to understand the appearance of the on-shell constraint $$\tag{1}p^2+m^2~=~0.$$ Here the Minkowski signature is $(-,+,+,+)$.

  4. Start with the following Lagrangian for a massive relativistic point particle $$\tag{2}L_0~:=~ -m\sqrt{-\dot{x}^2},$$ where dot means differentiation wrt. the world-line parameter $\tau$. Here the action is $S_0=\int \! d\tau~ L_0 $.

  5. Introduce an einbein field $e=e(\tau)$, and Lagrangian $$\tag{3}L~:=~\frac{\dot{x}^2}{2e}-\frac{e m^2}{2}.$$

  6. Show that the Lagrangian momenta$^2$ are $$\tag{4}p_{\mu}~=~\frac{1}{e}\eta_{\mu\nu}~\dot{x}^{\nu}.$$

  7. Show that the Euler-Lagrange equations of the Lagrangian (3) are $$\tag{5} \dot{p}_{\mu}~\approx~0, \qquad \dot{x}^2+(em)^2~\approx~0.$$

  8. Show that the Lagrangian (3) reduces to the original Lagrangian (2) when integrating out the einbein field $e$.

  9. Perform a Legendre transformation$^2$ of the Lagrangian (3), and show that the corresponding Hamiltonian becomes $$\tag{6}H~=~ \frac{e}{2}(p^2+m^2).$$ This Hamiltonian (6) is precisely of the form Lagrange multiplier times constraint (1).

  10. Show that Hamilton's equations are precisely eqs. (4) and (5).

  11. The arbitrariness in the choice of the world-line parameter $\tau$ leads to reparametrization symmetry $$\tau^{\prime}~=~f(\tau), \qquad d\tau^{\prime} ~=~ d\tau\frac{df}{d\tau},\qquad \dot{x}^{\mu}~=~\dot{x}^{\prime\mu}\frac{df}{d\tau},\qquad e~=~e^{\prime}\frac{df}{d\tau},\qquad $$ $$\tag{7} p_{\mu}~=~p_{\mu}^{\prime},\qquad L~=~L^{\prime}\frac{df}{d\tau},\qquad H~=~H^{\prime}\frac{df}{d\tau}\qquad S~=~S^{\prime},$$ where $f=f(\tau)$ is a bijective function.

  12. Thus one may choose various gauges, e.g. $e={\rm const.}$

References:

  1. J. Polchinski, String Theory, Vol. 1, Section 1.2.

--

$^1$ An einbein is a 1D version of a vielbein.

$^2$ Strictly speaking, in the singular Legendre transformation, one should also introduce a momentum $$\tag{8}p_e~:=~\frac{\partial L}{\partial \dot{e}}~=~0$$ for the einbein $e$, which leads to a primary constraint, that immediately kills the momentum $p_e$ again. Note that $\frac{\partial H}{\partial e}\approx 0$ becomes one of Hamilton's equations.

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Excuse me, сan you clarify why $\lambda$ does not depend on the canonical variables, but only depends on time? I obtained hamiltonian which has the same form as your's $(6)$, but I thought that $\lambda=\lambda(x,p)$, while you said $\lambda=\lambda(t)$. Can you explain why? –  xxxxx Dec 18 '13 at 14:16
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@xxxxx : In point mechanics, all fundamental variables, such as, e.g., $x^{\mu}$, $p_{\mu}$, $\lambda$, etc, are just functions of "time", more precisely, the world-line parameter $\tau$. The fundamental variables may get interrelated after we apply eqs. of motion. –  Qmechanic Dec 18 '13 at 14:33
    
Okay, I will try to explain precisely. Thanks for your explanation, I have heard about it a bit. But i wanted to use standart Dirac's approach: I obtained momentum, then identified the type of constraint $\phi$, wrote $H=H_{0}+\lambda \phi$. As I remember, Dirac in his lectures said that $\lambda=\lambda(x,p)$. Then I wrote the Hamilton equations using poisson brackets and hoped to determine from them $\lambda$ and equations of motion, which should have been matched with those that was obtained from the Lagrange's equations. Why in this approach I can use $\lambda=\lambda(t)?$ –  xxxxx Dec 18 '13 at 15:30
    
@xxxxx : Which lectures of Dirac? –  Qmechanic Dec 19 '13 at 9:02
    
Sry, I misunderstood. Of course you are right, that $\lambda=\lambda(t)$. But can you explain, why you did not consider constraint $p_{e}=0$ (by introducing the corresponding lagrange multiplier) when you wrote hamiltonian? –  xxxxx Dec 19 '13 at 15:47
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