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1. Classical Dynamics

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   Classical dynamics is the branch of theoretical physics concerned with the motion of macroscopic objects, from infinitesimal celestial bodies to large machines, under the influence of forces. It is the historical predecessor to quantum mechanics and is founded primarily on the principles articulated by Sir Isaac Newton (see Newtonian Mechanics) and later formalized by Lagrange and [Hamilton](/entries…

2. Classical Dynamics

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   $$\dot{q}i = \frac{\partial H}{\partial pi} \quad \text{and} \quad \dot{p}i = -\frac{\partial H}{\partial qi}$$
   A crucial feature of Hamiltonian dynamics is the preservation of phase space volume under time evolution, as mandated by Liouville's Theorem. This theorem implies that the density of representative points in phase space remains constant along trajectories. This conservation is mathematically rigorous but practically complicated by the phenomen…

3. Laplace Transform

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   \mathcal{L}\left\{\frac{d^n f(t)}{dt^n}\right\} = s^n F(s) - \sum_{k=1}^{n} s^{n-k} f^{(k-1)}(0)
   $$
   The initial conditions, $f(0), f'(0), \ldots$, are intrinsically embedded in the transformed algebraic equation. If all initial conditions are zero, the differentiation operation becomes simple multiplication by $s^n$.
   Convolution Theorem