Particle Trajectory in Curved Space Archives - Time Travel, Quantum Entanglement and Quantum Computing https://stationarystates.com/tag/particle-trajectory-in-curved-space/ Not only is the Universe stranger than we think, it is stranger than we can think...Hiesenberg Mon, 09 Sep 2024 18:16:23 +0000 en-US hourly 1 https://wordpress.org/?v=6.6.2 Particle Trajectory in Curved Space Using Jacobi Equation https://stationarystates.com/general-relativity-and-cosmology/particle-trajectory-in-curved-space-using-jacobi-equation/?utm_source=rss&utm_medium=rss&utm_campaign=particle-trajectory-in-curved-space-using-jacobi-equation Wed, 07 Aug 2024 03:53:48 +0000 https://stationarystates.com/?p=570 To derive the trajectory of a particle in a curved gravitational field using the Hamilton-Jacobi equation, we will follow these steps: 1. **Hamilton-Jacobi Equation**:     Here, is the Hamilton’s […]

The post Particle Trajectory in Curved Space Using Jacobi Equation appeared first on Time Travel, Quantum Entanglement and Quantum Computing.

]]> To derive the trajectory of a particle in a curved gravitational field using the Hamilton-Jacobi equation, we will follow these steps:

1. **Hamilton-Jacobi Equation**:

    \[ H\left( q_i, \frac{\partial S}{\partial q_i}, t \right) + \frac{\partial S}{\partial t} = 0 \]

Here, S is the Hamilton’s principal function, H is the Hamiltonian of the system, and q_i are the generalized coordinates.

2. **Hamiltonian for a Particle in a Curved Gravitational Field**:
The Hamiltonian in a gravitational field described by a metric g_{\mu\nu} is:

    \[ H = \frac{1}{2} g^{\mu\nu} p_\mu p_\nu \]

where p_\mu are the conjugate momenta.

3. **Hamilton’s Principal Function**:
Assume a solution for S of the form:

    \[ S = W(q_i) - E t \]

where W(q_i) is a function of the coordinates and E is the energy of the particle.

4. **Substitute into the Hamilton-Jacobi Equation**:
Substituting S into the Hamilton-Jacobi equation gives:

    \[ H\left( q_i, \frac{\partial W}{\partial q_i} \right) - E = 0 \]

    \[ \frac{1}{2} g^{\mu\nu} \frac{\partial W}{\partial q_\mu} \frac{\partial W}{\partial q_\nu} = E \]

5. **Solve for W**:
This equation is a partial differential equation for W(q_i). Solving this will give us the function W.

6. **Obtain Equations of Motion**:
Once W is known, the trajectory can be found using:

    \[ p_\mu = \frac{\partial W}{\partial q_\mu} \]

The equations of motion are given by Hamilton’s equations:

    \[ \frac{dq_\mu}{dt} = \frac{\partial H}{\partial p_\mu} = g^{\mu\nu} p_\nu \]

    \[ \frac{dp_\mu}{dt} = -\frac{\partial H}{\partial q_\mu} \]

### Detailed Steps:

1. **Start with the Hamilton-Jacobi Equation**:

    \[ \frac{1}{2} g^{\mu\nu} \frac{\partial S}{\partial q_\mu} \frac{\partial S}{\partial q_\nu} + \frac{\partial S}{\partial t} = 0 \]

2. **Assume S = W(q_i) - E t**:

    \[ \frac{1}{2} g^{\mu\nu} \frac{\partial W}{\partial q_\mu} \frac{\partial W}{\partial q_\nu} - E = 0 \]

This simplifies to:

    \[ \frac{1}{2} g^{\mu\nu} \frac{\partial W}{\partial q_\mu} \frac{\partial W}{\partial q_\nu} = E \]

3. **Solve for W(q_i)**:
Solve this PDE for W. In many cases, this requires choosing appropriate coordinates and exploiting symmetries in the metric g_{\mu\nu}.

4. **Calculate p_\mu**:

    \[ p_\mu = \frac{\partial W}{\partial q_\mu} \]

5. **Hamilton’s Equations**:
Use p_\mu to find the equations of motion:

    \[ \frac{dq_\mu}{dt} = \frac{\partial H}{\partial p_\mu} = g^{\mu\nu} p_\nu \]

6. **Integrate the Equations of Motion**:
These differential equations describe the trajectory q_\mu(t). Integrating them provides the trajectory of the particle in the curved gravitational field.

### Example: Schwarzschild Metric
For a particle in a Schwarzschild gravitational field, the metric is:

    \[ ds^2 = -\left(1 - \frac{2GM}{r}\right) dt^2 + \left(1 - \frac{2GM}{r}\right)^{-1} dr^2 + r^2 d\theta^2 + r^2 \sin^2\theta d\phi^2 \]

1. **Hamiltonian**:

    \[ H = \frac{1}{2} \left[ -\left(1 - \frac{2GM}{r}\right)^{-1} p_t^2 + \left(1 - \frac{2GM}{r}\right) p_r^2 + \frac{1}{r^2} p_\theta^2 + \frac{1}{r^2 \sin^2 \theta} p_\phi^2 \right] \]

2. **Hamilton-Jacobi Equation**:
Substitute S = -Et + W(r, \theta, \phi):

    \[ \frac{1}{2} \left[ -\left(1 - \frac{2GM}{r}\right)^{-1} E^2 + \left(1 - \frac{2GM}{r}\right) \left( \frac{\partial W}{\partial r} \right)^2 + \frac{1}{r^2} \left( \frac{\partial W}{\partial \theta} \right)^2 + \frac{1}{r^2 \sin^2 \theta} \left( \frac{\partial W}{\partial \phi} \right)^2 \right] = 0 \]

3. **Separation of Variables**:
Assume W = W_r(r) + W_\theta(\theta) + W_\phi(\phi). Separate variables and solve for each part.

4. **Find p_\mu** and **Integrate**:

    \[ p_t = -E, \quad p_r = \frac{\partial W_r}{\partial r}, \quad p_\theta = \frac{\partial W_\theta}{\partial \theta}, \quad p_\phi = \frac{\partial W_\phi}{\partial \phi} \]

    \[ \frac{dr}{dt} = \left(1 - \frac{2GM}{r}\right) p_r, \quad \frac{d\theta}{dt} = \frac{p_\theta}{r^2}, \quad \frac{d\phi}{dt} = \frac{p_\phi}{r^2 \sin^2 \theta} \]

Integrate these equations to find the trajectory r(t), \theta(t), \phi(t).

This outlines the steps for deriving the trajectory of a particle in a curved gravitational field using the Hamilton-Jacobi equation. Each step involves setting up the problem, solving the Hamilton-Jacobi PDE, and then using the solutions to find the equations of motion.

The post Particle Trajectory in Curved Space Using Jacobi Equation appeared first on Time Travel, Quantum Entanglement and Quantum Computing.

]]>