Classical mechanics is often introduced through Newton’s laws, which describe motion in terms of forces acting on particles. While this formulation is intuitive, it can become cumbersome for systems with constraints, generalized coordinates, or many interacting degrees of freedom. Lagrangian mechanics provides an alternative, coordinate-invariant framework that encodes the dynamics of a system through energy rather than force.
The central object in Lagrangian mechanics is the Lagrangian, defined as
where \(q\) denotes a set of generalized coordinates, \(\dot{q}\) their time derivatives, \(T\) the kinetic energy, and \(V\) the potential energy. The dynamics of the system are obtained by requiring that the action functional
be stationary with respect to variations of the path \(q(t)\).
Euler–Lagrange Equations
Applying the principle of stationary action leads to the Euler–Lagrange equations,
which govern the evolution of each generalized coordinate \(q_i\). These equations are equivalent to Newton’s second law for conservative systems, but they naturally incorporate constraints and allow for arbitrary coordinate choices.
Advantages of the Lagrangian Formulation
Lagrangian mechanics shifts the focus from forces to energy and geometry. Constraints can be handled implicitly through coordinate selection, symmetries become transparent, and conserved quantities arise naturally through invariance principles. This formulation is particularly powerful for systems with complex kinematics, coupled degrees of freedom, or underlying symmetries.
Beyond classical mechanics, the Lagrangian framework extends directly to fields, continuum systems, electrodynamics, and modern theoretical physics. It also provides the natural bridge to Hamiltonian mechanics, control formulations, and variational methods in applied mathematics.