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Lagrangian Mechanics
Rating: 3.9 out of 5(20 ratings)
106 students

Lagrangian Mechanics

An Introductory Course in Lagrangian Mechanics
Created bySusan Tayfe
Last updated 11/2025
English

What you'll learn

  • Writting Lagrangian (Lagrange function) of a mechanical system in a generalized coordinate system, and transforming it to a function of independent coordinates.
  • Writting Euler-Lagrange equations (Lagrange's equations of second kind) using Lagrange function of independent coordinates, and finding equations of motion.
  • Writting Lagrange's equations of first kind using Lagrange function of both dependent and independent coordinates, and finding forces of constraint.
  • Finding conserved generalized momentums (corresponing to the cyclic coordinates), and conserved energy (if Lagrangian is not a implicit function of time).

Course content

3 sections • 17 lectures • 1h 50m total length

  • Lagrange Function4:37

    In this lecture Lagrangian of a mechanical system is written as a function of independent generalized coordinates.

  • The Least Action Principle (Hamilton Principle)1:26
  • Euler-Lagrange Equations2:06

    In this lecture Euler-Lagrange equations (or Lagrange equations of second kind) which are derived from the Least Action Principle, are introduced without proof.

  • Lagrangian in Classical Mechanics4:50

    In this lecture you will learn how to derive generalized momentum and force, corresponding to a generalized coordinate in classical systems.

  • Perfectly Elastic Spring7:58

    In this lecture, you will learn how to apply Lagrangian formalism to the perfectly elastic spring and find its equation of motion.

  • Simple Pendulum11:35

    In this lecture, you will learn how to apply Lagrangian formalism to the simple pendulum and find its equation of motion.

  • Double Pendulum11:59

    In this lecture, you will learn how to apply Lagrangian formalism to the double pendulum and find its equations of motion.

  • Moving Plane8:53

    This lecture is the solution of this problem:

    A block of mass m is held motionless on a frictionless plane of mass M and angle of inclination θ. The plane rests on a frictionless horizontal surface. The block is released. Use Lagrangian mechanics to determine the accelerations of both plane and block once the block is released.

  • Pendulum with an Oscillating Support11:56

    In this problem we have a pendulum with an oscillating support and we find the angle of pendulum as a function of time.

Requirements

  • Newtonian Mechanics
  • Calculus

Description

This is an introductory course in Lagrangian mechanics provided for college students and anyone who is familiar with Newtonian mechanics and calculus. 

In this course you will learn how to apply Lagrangian mechanics to the classical systems and find their equations of motion and physical quantities. When applied to the classical systems, Lagrangian mechanics is equivalent to the Newtonian mechanics, but more easier than Newtonian mechanics, especially when you are dealing with more complicated systems.

This course is made of three sections:

  • Lagrangian Dynamics: this section begins with writing Lagrangian (or Lagrange function) of a system in a generalized coordinates in terms of independent coordinates, by finding constraint functions and number of degrees of freedom of the system.  You will learn how to apply Euler-Lagrange equations (Lagrange's equations of the second kind) to the independent coordinates and find the equations of motion of the system. Some example of Lagrangian of classical systems are discussed in this section.

  • Generalized Forces:  This section begins with definition of generalized conservative forces. Then by writing Lagrangian without imposing constraint functions and by applying Lagrange's equations of first kind, you learn how to find constraint forces of a system.

  • Conservation Laws: in this section you will learn how to find conserved generalized momentum (linear and angular momentum) if there is a cyclic coordinate in Lagrangian (i.e. Lagrangian is not implicit function of a coordinate); and how to find conserved energy of the system if Lagrangian is not a implicit function of time.


Register to this course and enjoy learning Lagrangian mechanics!

Who this course is for:

  • Undergraduate Students
  • Anyone who wants to learn Lagrangian Mechanics.

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