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.