Added SimplePendulumRK4 (#6800)

* Added SimplePendulumRK4

* Fixed build issue.

* Fixed build issue.

* Fixed build issue.

src/main/java/com/thealgorithms/physics/SimplePendulumRK4.java

@@ -0,0 +1,126 @@
+package com.thealgorithms.physics;
+
+/**
+ * Simulates a simple pendulum using the Runge-Kutta 4th order method.
+ * The pendulum is modeled with the nonlinear differential equation.
+ *
+ * @author [Yash Rajput](https://github.com/the-yash-rajput)
+ */
+public final class SimplePendulumRK4 {
+
+    private SimplePendulumRK4() {
+        throw new AssertionError("No instances.");
+    }
+
+    private final double length; // meters
+    private final double g; // acceleration due to gravity (m/s^2)
+
+    /**
+     * Constructs a simple pendulum simulator.
+     *
+     * @param length the length of the pendulum in meters
+     * @param g the acceleration due to gravity in m/s^2
+     */
+    public SimplePendulumRK4(double length, double g) {
+        if (length <= 0) {
+            throw new IllegalArgumentException("Length must be positive");
+        }
+        if (g <= 0) {
+            throw new IllegalArgumentException("Gravity must be positive");
+        }
+        this.length = length;
+        this.g = g;
+    }
+
+    /**
+     * Computes the derivatives of the state vector.
+     * State: [theta, omega] where theta is angle and omega is angular velocity.
+     *
+     * @param state the current state [theta, omega]
+     * @return the derivatives [dtheta/dt, domega/dt]
+     */
+    private double[] derivatives(double[] state) {
+        double theta = state[0];
+        double omega = state[1];
+        double dtheta = omega;
+        double domega = -(g / length) * Math.sin(theta);
+        return new double[] {dtheta, domega};
+    }
+
+    /**
+     * Performs one time step using the RK4 method.
+     *
+     * @param state the current state [theta, omega]
+     * @param dt the time step size
+     * @return the new state after time dt
+     */
+    public double[] stepRK4(double[] state, double dt) {
+        if (state == null || state.length != 2) {
+            throw new IllegalArgumentException("State must be array of length 2");
+        }
+        if (dt <= 0) {
+            throw new IllegalArgumentException("Time step must be positive");
+        }
+
+        double[] k1 = derivatives(state);
+        double[] s2 = new double[] {state[0] + 0.5 * dt * k1[0], state[1] + 0.5 * dt * k1[1]};
+
+        double[] k2 = derivatives(s2);
+        double[] s3 = new double[] {state[0] + 0.5 * dt * k2[0], state[1] + 0.5 * dt * k2[1]};
+
+        double[] k3 = derivatives(s3);
+        double[] s4 = new double[] {state[0] + dt * k3[0], state[1] + dt * k3[1]};
+
+        double[] k4 = derivatives(s4);
+
+        double thetaNext = state[0] + dt / 6.0 * (k1[0] + 2 * k2[0] + 2 * k3[0] + k4[0]);
+        double omegaNext = state[1] + dt / 6.0 * (k1[1] + 2 * k2[1] + 2 * k3[1] + k4[1]);
+
+        return new double[] {thetaNext, omegaNext};
+    }
+
+    /**
+     * Simulates the pendulum for a given duration.
+     *
+     * @param initialState the initial state [theta, omega]
+     * @param dt the time step size
+     * @param steps the number of steps to simulate
+     * @return array of states at each step
+     */
+    public double[][] simulate(double[] initialState, double dt, int steps) {
+        double[][] trajectory = new double[steps + 1][2];
+        trajectory[0] = initialState.clone();
+
+        double[] currentState = initialState.clone();
+        for (int i = 1; i <= steps; i++) {
+            currentState = stepRK4(currentState, dt);
+            trajectory[i] = currentState.clone();
+        }
+
+        return trajectory;
+    }
+
+    /**
+     * Calculates the total energy of the pendulum.
+     * E = (1/2) * m * L^2 * omega^2 + m * g * L * (1 - cos(theta))
+     * We use m = 1 for simplicity.
+     *
+     * @param state the current state [theta, omega]
+     * @return the total energy
+     */
+    public double calculateEnergy(double[] state) {
+        double theta = state[0];
+        double omega = state[1];
+        double kineticEnergy = 0.5 * length * length * omega * omega;
+        double potentialEnergy = g * length * (1 - Math.cos(theta));
+        return kineticEnergy + potentialEnergy;
+    }
+
+    public double getLength() {
+        return length;
+    }
+
+    public double getGravity() {
+        return g;
+    }
+}