# Hamilton EOM for Schwarzschild Metric: Problem Solved

• sergiokapone
In summary, the equations of motion in Hamiltonian form for a particle in the Schwarzschild metric are given by equation (5) in Goldstein. The canonical momentum is p_{\mu}, but p^{\mu} is not conserved.
sergiokapone
I have a problem (this is not homework)
Based on covariant Lagrangian ## \mathcal {L} = \frac {m}{2} \frac{dx^{\mu}}{ds} \frac {dx _ {\mu}}{ds} ## record the equations of motion in Hamiltonian form for a particle in the Schwarzschild metric (SM).

Based on Legandre transformations

H =\frac {m}{2} p_{\mu}g^{\mu\nu}p_{\nu}

EOM in Hamilton form (as shown in Goldstein)
\begin{align}
\frac{dx^{\mu}}{ds} = g^{\mu\nu} \frac{\partial H}{\partial p^{\nu}}\\
\frac{dp^{\mu}}{ds} = -g^{\mu\nu} \frac{\partial H}{\partial x^{\nu}}
\end{align}

Canonical momentum is ##p_{\mu}##, but not ##p^{\mu}##. How it is possible to apply this equations for the
Schwarzschild metric?

Again, if I write

\begin{align}
\frac{dx_{\mu}}{ds} = g_{\mu\nu} \frac{\partial H}{\partial p_{\nu}}\\
\frac{dp_{\mu}}{ds} = -g_{\mu\nu} \frac{\partial H}{\partial x_{\nu}} \label{5}
\end{align}

The RHS of equation ##\eqref{5}## in SM for any component allways will give ##0##, because ##H## does not depend on ##x^{\mu}##. But one know the ##p_{r}## does not conserve, ##dp_r/d\tau \neq 0## for SM.

My question, what is the correct view of EOM in Hamilton form for GR in general, or for the SM in specific?

sergiokapone said:
The RHS of equation ##\eqref{5}## in SM for any component allways will give ##0##, because ##H## does not depend on ##x^{\mu}##.

Careful. In the definition of H in (1), the g^{\mu \nu} have explicit coordinate dependence, and thus H does also.

Ok, thanx, I try.

sergiokapone said:
I have a problem (this is not homework)
Based on covariant Lagrangian ## \mathcal {L} = \frac {m}{2} \frac{dx^{\mu}}{ds} \frac {dx _ {\mu}}{ds} ## record the equations of motion in Hamiltonian form for a particle in the Schwarzschild metric (SM).
If you want to see this worked out have a look at http://arxiv.org/abs/1201.5611v1.pdf.

## 1. What is the Hamilton EOM for Schwarzschild Metric?

The Hamilton EOM (Equation of Motion) for Schwarzschild Metric is a mathematical formula that describes the motion of particles in the presence of a massive, non-rotating object, such as a black hole. It is a key equation in the field of General Relativity and is used to understand the behavior of objects in a curved space-time.

## 2. How is the Hamilton EOM derived for the Schwarzschild Metric?

The Hamilton EOM for the Schwarzschild Metric is derived from the Schwarzschild solution, which is a solution to Einstein's field equations in General Relativity. It involves manipulating the equations of motion for a test particle in a curved space-time, using the Hamiltonian formalism, to arrive at a simplified form of the equation.

## 3. What does the Hamilton EOM for Schwarzschild Metric tell us about the behavior of particles near a black hole?

The Hamilton EOM for Schwarzschild Metric tells us that the motion of particles near a black hole is highly affected by the strong gravitational pull of the black hole. It predicts that the particles will follow curved paths and may experience effects such as time dilation and redshift, as they approach the event horizon of the black hole.

## 4. Can the Hamilton EOM for Schwarzschild Metric be applied to other objects besides black holes?

Yes, the Hamilton EOM for Schwarzschild Metric can be applied to any massive, non-rotating object in space. It is commonly used to study the behavior of particles around neutron stars and other massive objects.

## 5. What is the significance of solving the Hamilton EOM for Schwarzschild Metric?

Solving the Hamilton EOM for Schwarzschild Metric is a major achievement in the field of General Relativity and has helped us gain a better understanding of how gravity works in extreme environments, such as near black holes. It has also allowed us to make more accurate predictions about the behavior of particles in curved space-time, which has practical applications in fields such as astrophysics and space travel.

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