section_path: Relation to other formulations of classical physics/Lagrangian
section_name: Lagrangian
paragraph_text: Lagrangian
  
  Lagrangian mechanics differs from the Newtonian formulation by considering entire trajectories at once rather than predicting a body's motion at a single instant. It is traditional in Lagrangian mechanics to denote position with q {\displaystyle q} and velocity with q ˙ {\displaystyle {\dot {q}}}. The simplest example is a massive point particle, the Lagrangian for which can be written as the difference between its kinetic and potential energies: L (q, q ˙) = T − V, {\displaystyle L(q,{\dot {q}})=T-V,} where the kinetic energy is T = 1 2 m q ˙ 2 {\displaystyle T={\frac {1}{2}}m{\dot {q}}^{2}} and the potential energy is some function of the position, V (q) {\displaystyle V(q)}. The physical path that the particle will take between an initial point q i {\displaystyle q_{i}} and a final point q f {\displaystyle q_{f}} is the path for which the integral of the Lagrangian is "stationary". That is, the physical path has the property that small perturbations of it will, to a first approximation, not change the integral of the Lagrangian. Calculus of variations provides the mathematical tools for finding this path. Applying the calculus of variations to the task of finding the path yields the Euler–Lagrange equation for the particle, d d t (∂ L ∂ q ˙) = ∂ L ∂ q. {\displaystyle {\frac {d}{dt}}\left({\frac {\partial L}{\partial {\dot {q}}}}\right)={\frac {\partial L}{\partial q}}.} Evaluating the partial derivatives of the Lagrangian gives d d t (m q ˙) = − d V d q, {\displaystyle {\frac {d}{dt}}(m{\dot {q}})=-{\frac {dV}{dq}},} which is a restatement of Newton's second law. The left-hand side is the time derivative of the momentum, and the right-hand side is the force, represented in terms of the potential energy.
  
  Landau and Lifshitz argue that the Lagrangian formulation makes the conceptual content of classical mechanics more clear than starting with Newton's laws. Lagrangian mechanics provides a convenient framework in which to prove Noether's theorem, which relates symmetries and conservation laws. The conservation of momentum can be derived by applying Noether's theorem to a Lagrangian for a multi-particle system, and so, Newton's third law is a theorem rather than an assumption.