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Section 1.2 Euler-Lagrange Equations in Physics
202607290001 |
Lagrangian mechanics,
Euler-Lagrange,
variational principles
Definition 1.2.1 . Physics Formulation of the Euler-Lagrange Equation.
The
Euler-Lagrange equation in
physics is the necessary condition satisfied by a function that makes a
functional , stationary under admissible
variations . It is written as
\begin{equation*}
\frac{\partial L}{\partial q} - \frac{d}{dt} \Big( \frac{\partial L}{\partial \dot{q}} \Big) = 0 \, .
\end{equation*}
Subsection 1.2.1 Euler-Lagrange Equation in Classical Mechanics
Definition 1.2.3 . Action (Classical Mechanics).
In classical mechanics, the
functional for the
Euler-Lagrange equation 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.
Context 1.2.1.1 . 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.
Generalizes to field theory , where it produces the Euler-Lagrange field equations.
Forms the mathematical foundation of variational integrators , which discretize the action rather than the equations of motion.
Closely parallels other variational principles, such as energy minimization in self-consistent field methods .
