Adding DampedOscillator code (#6801)

* Adding DampedOscillator code

* Adding one more test case

* Adding one more test case

* Adding one more test case

* Fixing build issues.

* Fixing build issues.

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

@@ -0,0 +1,109 @@
+package com.thealgorithms.physics;
+
+/**
+ * Models a damped harmonic oscillator, capturing the behavior of a mass-spring-damper system.
+ *
+ * <p>The system is defined by the second-order differential equation:
+ *     x'' + 2 * gamma * x' + omega₀² * x = 0
+ * where:
+ * <ul>
+ *   <li><b>omega₀</b> is the natural (undamped) angular frequency in radians per second.</li>
+ *   <li><b>gamma</b> is the damping coefficient in inverse seconds.</li>
+ * </ul>
+ *
+ * <p>This implementation provides:
+ * <ul>
+ *   <li>An analytical solution for the underdamped case (γ < ω₀).</li>
+ *   <li>A numerical integrator based on the explicit Euler method for simulation purposes.</li>
+ * </ul>
+ *
+ * <p><strong>Usage Example:</strong>
+ * <pre>{@code
+ * DampedOscillator oscillator = new DampedOscillator(10.0, 0.5);
+ * double displacement = oscillator.displacementAnalytical(1.0, 0.0, 0.1);
+ * double[] nextState = oscillator.stepEuler(new double[]{1.0, 0.0}, 0.001);
+ * }</pre>
+ *
+ * @author [Yash Rajput](https://github.com/the-yash-rajput)
+ */
+public final class DampedOscillator {
+
+    /** Natural (undamped) angular frequency (rad/s). */
+    private final double omega0;
+
+    /** Damping coefficient (s⁻¹). */
+    private final double gamma;
+
+    private DampedOscillator() {
+        throw new AssertionError("No instances.");
+    }
+
+    /**
+     * Constructs a damped oscillator model.
+     *
+     * @param omega0 the natural frequency (rad/s), must be positive
+     * @param gamma  the damping coefficient (s⁻¹), must be non-negative
+     * @throws IllegalArgumentException if parameters are invalid
+     */
+    public DampedOscillator(double omega0, double gamma) {
+        if (omega0 <= 0) {
+            throw new IllegalArgumentException("Natural frequency must be positive.");
+        }
+        if (gamma < 0) {
+            throw new IllegalArgumentException("Damping coefficient must be non-negative.");
+        }
+        this.omega0 = omega0;
+        this.gamma = gamma;
+    }
+
+    /**
+     * Computes the analytical displacement of an underdamped oscillator.
+     * Formula: x(t) = A * exp(-γt) * cos(ω_d t + φ)
+     *
+     * @param amplitude the initial amplitude A
+     * @param phase     the initial phase φ (radians)
+     * @param time      the time t (seconds)
+     * @return the displacement x(t)
+     */
+    public double displacementAnalytical(double amplitude, double phase, double time) {
+        double omegaD = Math.sqrt(Math.max(0.0, omega0 * omega0 - gamma * gamma));
+        return amplitude * Math.exp(-gamma * time) * Math.cos(omegaD * time + phase);
+    }
+
+    /**
+     * Performs a single integration step using the explicit Euler method.
+     * State vector format: [x, v], where v = dx/dt.
+     *
+     * @param state the current state [x, v]
+     * @param dt    the time step (seconds)
+     * @return the next state [x_next, v_next]
+     * @throws IllegalArgumentException if the state array is invalid or dt is non-positive
+     */
+    public double[] stepEuler(double[] state, double dt) {
+        if (state == null || state.length != 2) {
+            throw new IllegalArgumentException("State must be a non-null array of length 2.");
+        }
+        if (dt <= 0) {
+            throw new IllegalArgumentException("Time step must be positive.");
+        }
+
+        double x = state[0];
+        double v = state[1];
+        double acceleration = -2.0 * gamma * v - omega0 * omega0 * x;
+
+        double xNext = x + dt * v;
+        double vNext = v + dt * acceleration;
+
+        return new double[] {xNext, vNext};
+    }
+
+    /** @return the natural (undamped) angular frequency (rad/s). */
+    public double getOmega0() {
+        return omega0;
+    }
+
+    /** @return the damping coefficient (s⁻¹). */
+    public double getGamma() {
+        return gamma;
+    }
+}