Lagrange Formalism

Recap of Newtonian Mechanics

In Newtonian mechanics, the motion of a particle is described through a few important quantities: for a particle of (inertial) mass , position , we have

and the relations between these quantities, in presence of external forces acting on the particle, are given by Newton’s second law:

The work done by such forces is given by the inner product of the motion and the force:

This is the energy change of the particle through the motion:

meaning that is the energy due to the motion of the particle: the kinetic energy :

Definition: Kinetic Energy

Now, often, an external force acting on the particle is due to a potential :

For example, for a 1D spring, the potential is given by

Example: 1D Spring/ Harmonic Potential

or the electrostatic potential:

Electrostatic potential

D’Alembert’s Principle

Now, for a particle whose the external forces are given by a potential:

This looks as if the forces and are in equilibrium. So, if we move a particle by an infinitisimal distance , the total work done by these forces must be zero:

at any time . We want to apply this for entire path of the motion of the particle, from to . This is called D’Alembert’s Principle.

Action and Lagrangian

Then, the integral of this equation over the time interval gives

now, we can apply integration by parts:

where we used the commutitivity of and . Then the integral becomes

\cite{maeno-leastAction}

\subsection{Lagrangian and Variational Principle} In Lagrangian mechanics, we will use a different approach to describe the motion of a particle than the Newtonian mechanics. Instead of using the usual 3D Euclidean space, we will use a configuration space of a system. This space is spanned by the so-called generalized coordinate , and their time derivatives (or generalized velocity) . Now, let us define quantities called Lagrangian and action :

Lagrangian and Action

Lagrangian is defined as the difference between the kinetic energy and potential energy of a particle:

Now, variation of the action is given by

which is an identical expression to \eqref{eq:newton-variation}.

Variational Principle

So, we postulate that the motion of the particle is such that the action is stationary:

Variational Principle

The motion of a particle is such that the action is stationary, i.e., .

As to show why this maybe useful, let us compare between \cref{eq:eom-variation}:

and \cref{eq:func_deriv_euler_lagrange}:

which immidiately gives that the EoM is the Euler-Lagrange equation, if we set :

The generalization to multiple particles is straightforward:

The Euler-Lagrange equation is given by

Harmonic Oscillator

Let us consider a particle of mass in a 1D harmonic potential:

then the Lagrangian is given by

The Euler-Lagrange equation yields

which is the equation of motion of a harmonic oscillator in Newtonian mechanics. Here, notice that

and

these derivatives of the Lagrangian are the generalized force and conjugate momentum, respectively.