Remark 0.3.2.
The path of a physical system in classical mechanics is typically attributed to the trajectory \(q(t)\) that minimizes the action.
Subsection 0.3.1 Euler-Lagrange Equation in Classical Mechanics
Definition 0.3.3 Action (Classical Mechanics).
In classical mechanics, the functional for the Euler-Lagrange equation in canonical coordinates is called the action:
\begin{equation*}
S[q] = \int_{t_0}^{t_1}{L\big(t, q(t), \dot{q}(t) \big) \, dt}
\end{equation*}
where \(L=T-V\) denotes the Lagrangian of the system. Here, \(T\) is the kinetic energy of the system and \(V\) is the potential energy.
Remark 0.3.4.
The requirement that the action be stationary gives the physics formulation of the Euler-Lagrange equation and determines the equations of motion for a classical system.
Context 0.3.1.1 Lagrangian Mechanics
Rather than beginning with the forces acting on the system (as in Newtonian mechanics), Lagrangian mechanics begins with a scalar quantity called the action that does not undergo first-order changes due to infinitesimal changes in the system’s physical trajectory (or path).
Connections.
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Built upon calculus of variations.
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Generalizes to field theory, where it produces the Euler-Lagrange field equations.
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Forms the mathematical foundation of variational integrators, which discretize the action rather than the equations of motion.
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Closely parallels other variational principles, such as energy minimization in self-consistent field methods.
