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Newtonian vs Hamiltonian Mechanics — bullet gist
Newtonian mechanics
Describes motion using forces and accelerations
Core law: ( F = ma )
Focuses on trajectories in space and time
Intuitive and direct; works well for simple systems
Becomes messy for complex or constrained systems
Hamiltonian mechanics
Describes motion using energy (total energy = Hamiltonian)
Uses positions and momenta instead of forces
Motion follows from elegant mathematical equations
Naturally handles complex, constrained, and multi-body systems
Forms the bridge to quantum mechanics and modern physics
Big difference
Newton: What force causes this motion?
Hamilton: How does energy govern the system’s evolution?
Lagrangian Mechanics — bullet gist
Describes motion using energy, not forces directly
Core idea: Lagrangian = Kinetic Energy − Potential Energy (L = T − V)
System evolves by the principle of least action (nature chooses the optimal path)
Uses generalized coordinates, not just x, y, z
Automatically handles constraints (ropes, surfaces, joints)
Produces equations of motion via Euler–Lagrange equations
More elegant and powerful than Newtonian mechanics for complex systems
Foundation for Hamiltonian mechanics, relativity, and field theory
In one line:
Motion follows the path that makes action stationary, not the path of obvious forces.
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