2 d‘Alembert’s Principle and Lagrange’s Equations

Ashok Goyal

1. Introduction

Newtonian mechanics is based on Newton’s laws of motion. In many practical problems it becomes difficult to set up Newton’s equation and solve them particularly in the presence of constraints. In such cases an alternative formalism in needed to simplify the analysis of the system. In this unit we would discuss one such alternative formalism based on D’Alembert’s Principle. In this context we will first introduce the concept of Virtual work.

2. Principle of virtual work

Let a system of N particle is specified by the coordinates . Virtual displacement is defined as an imaginary infinitesimal displacement at a fixed given time t while the displacement satisfies the constraints and the generalized velocities are kept fixed at time t. The virtual displacements when represented in generalized coordinates automatically satisfy the constraints. The relation between the infinitesimal displacement of the generalized coordinates and the virtual displacement is given by= (2.1) ∆

Let be the external applied force and the force of constraints on the ith particle. As the system is in equilibrium the virtual work which is defined as the work done as the system by all the forces that act on the system as the system undergoes virtual displacement is =∙ (2.2) where = + e now restrict ourselves to systems for which the net virtual work done by the constraint forces is zero i.e.∙ = 0 (2.3) Therefore the virtual work done on the system is given by = ∙ (2.4)

Thus the total virtual work done on the system is only by the applied forces. This is the ‘principle of virtual work’.

3. d’Alembert’s Principle 4

The total force acting on a particle is the sum of applied and constraint forces. The virtual work done by the constraint forces is zero and therefore, as we saw, the virtual work done on the system is

3. d’Alembert’s Principle

The total force acting on a particle is the sum of applied and constraint forces. The virtual work done by the constraint forces is zero and therefore, as we saw, the virtual work done on the system is = ∙Where we have dropped the superscript ‘applied’ from the force. From Newton’s second law of motion rate of change of momentum of a particle is equal to the applied force. Thus ==(2.5) We then have = ∙ (2.6) and −   ∙   = 0 (2.7)

This is the d’Alembert’s Principle. It says that the rate of change of momentum is determined only by the non-constraint forces. The principle can be generalized to relate the generalized forces to the rate of change of momenta. In terms of generalized coordinates, we have = = ∙ = (2.8) where the generalized for ℑ = ∙ = = ∙ (2.9)

In equation (2.8) and (2.9) above in runs over from 1 to N and j from 1 to 3N-k.Thus the generalized force on the jth particle can be calculated from the non-constraint applied force alone.Lagrange’s Equation of Motion The kinetic energy T of a dynamical system is defined as 5≡ 1 ∙  =; (2.10) n equation (2.8) and (2.9) above in runs over from 1 to N and j from 1 to 3N-k.Thus the generalized force on the jth particle can be calculated from the non-constraint applied force alone.

4. Lagrange’s Equation of Motion

The kinetic energy T of a dynamical system is defined as ≡ 1 ∙  =; (2.10) 2.This functional relationship of T can be seen by remembering that the position coordinate can be written in terms of generalized coordinates and is obtained by the total derivative of with time and is a function of . We can thus write the partial derivative of T as ==∙(2.11) =∙=∙(2.12).This happens because =(2.13) 0 Which can be easily proved by voting that =∙(2.14) Now =+(2.15).Taking the partial derivative w.r.t.==(2.16).Differentiating equ. (2.12) w.r.t., we get

Differentiating eqn. (2.12) w.r.t., we get =∙+∙2.17The first term can be identified as the generalized forces acting on the jth particle. In the second term, we have =+ =+=(2.18) ∴=+ We thus have the generalized equation of motion –  =(2.19).If we now specialize to conservative forces that is the forces that can be obtained from a potential function U. = −  (2.20)he generalized force =∙=−∙= −,(2.21).The generalized equation of motion obtained by using d’Alembert’s Principle expressed in generalized coordinates is given by −= −(2.22)

Define the Lagrangian

For holonomic constraints we have

=0

This allows us to write the above equation in terms of the Lagrangian as

This is the Euler – Lagrange Equation for each generalized coordinates.

5. Time and velocity dependent Potentials:

In defining the Lagrangian we confined ourselves to potential energy which is time independent.We may have situations where the potential energy even though depends on time but is conservative at each instant of time a for example in the case of planetary motion about the centre of mass.potential is velocity dependent the Euler-Lagrange equations may not hold in generalized force that can be written in terms of velocity dependent potentials called generalized potential energy,through The Euler – Lagrange equations still hold for = − . defined in this way is not the potential energy in the conventional sense of being the work done by a generalized force. That, it cannot be calculated from the line integral of the generalized force but still allows us to use the concept of Lagrangian to obtain the Euler-Lagrange equations of motion. As an example of velocity dependent force, we will take the example of Lorentz force on a charged particle and obtain the corresponding
Lagrange equation.

Lorentz-force Equation:

The equation of motion of a charged particle of mass m and charge q in the presence of electro – magnetic field is given by

where is the velocity of the particle. The electric and magnetic fields can be expressed in terms of scalar potential and the vector potential A through the well known relations from the study of electricity and magnetism and – In terms of X component of the force, equ. (2.26) can be written as .Thus Since vector potential depend on the coordinates and time i.e. =  , , ,We have 1=1+1++(2.31) And substracting substituting for in (2.29) from (2.30), and adding we get 9 And substracting substituting for in (2.29) from (2.30), and adding we get

In deriving the above equation we have used.We can thus write

Where

Is a kind of generalized velocity dependent potential and the Lagrangian of the charged particle in
the presence of electro-magnetic fields is given by

Example : We will illustrate this with the help of a suitable example. A particle of mass m is constrained to move under gravity on an elliptical wire in the x-y plane.

The constraint equation is  choose Which can be parameterized the generalized coordinate to by be 

Which can be parameterized the generalized coordinate to by be

The equation of motion is given by

= 2 2 sin cos − 2 sin cos

6. Summary:

• For conservative systems, generalized force can be obtained from the potential energy function expressed in terms of generalized coordinates as = −
• d’Alembert’s Principle states that the rate of change of momentum is determined only by the non-constraint forces.