D’ Alembert’s principle

D’ Alembert’s principle, also called  Lagrange–d’Alembert principle. It is a statement of the fundamental classical laws of motion. It is named after its discoverer, the French physicist.

D' Alembert's principle

The principle states that “The sum of the differences between the time derivatives of the momenta differences and force acting on a system of mass particles and the system itself projected onto any virtual displacement consistent with the constraints of the system is zero.

Theory

According to Newton’s second law of the motion, the force acting on the ith particle is given by-

Fi = δPi/ δt = Pi^•    (1)

This equation can be written as-

Fi – Pi^•  = 0     (2)

This equation shows that any particle in the system is in equilibrium under a force. Therefore virtual displacement (δri)-

∑i=1^ N (Fi – Pi^•) δri =0

Where Fi = Fi^a + fi

By equation (3),

∑i=1^ N (Fi^a + fi – Pi^•)δri = 0

∑i=1^ N (Fi^a- Pi^•)δri + ∑i=1^ N fi δri =0  (4)

The virtual work of constraint is zero,

∑i=1^N fi δri =0

By equation 4,

∑i=1^ N (Fi^a- Pi^•)δri =0

And (5) is known as D’Alembert’s principle. Since, the forces of constraints do not appear in the equation. ∴ D’ Alembert’s principle may be written as

∑i=1^ N (Fi- Pi^•)δri =0

Lagrange’s equation from D’ Alembert’s principle

Consider a system of n particles. The transformation equation for the position vector of the particle are-

ri = ri( q1 + q2 +…+ qk…qn,t) ….(1)

Where t is time and qk (k = 1,2,3,…v) is generalised coordinate.

dri/ dt= (δri/δq1)( δq1/δt ) + (δri/δq2) (δq2/δt)+ ……(δri/δqk )(δqk/δt)+…..+(δri/δqn)(δqn/δt) + δri/δt

vi = (δri/δq1 )(q1^• )+ (δri/δq2)  (q2^• ) + ….(δri/δqk )(qk^• )+….+(δri/δqn) (qn^• )+ δri/δt

vi = ∑i=1^ N (δri/δqk)(qk^• )+ δri/δt

vi = ri =  ∑i=1^ N (δri/δqk) (qk^• )+ δri/δt    (2)

Where qk^•  is generalised velocity.

Now, virtual displacement are-

δri = (δri/δqk) (δqk)     (3)

From D’ Alemberts principle

∑i=1^ N (Fi- Pi^•)δri =0

∑i=1^ N(Fi- miri^••) δri =0   (4)

(Note – P = mv = mr^•

P^• = mr^••)

On taking the first term of (4)

Now by (3)

Fi δri = Fi. (δri/δqk) (δqk)

Fi δri = Gk. δqk

Where Gk = Fi. (δri/δqk)

Now taking the second term of equation (4)

∑i=1^ N(miri^••) δri =0

And ,∑i=1^ N [δ/δt(miri^•]δri =0

∑i=1^ N (Pi)δri = ∑i=1^ N(miri^••) δri

∑i=1^ N (Pi)δri = ∑i=1^ N(miri^••) ∑k=1(δri/δqk) (δqk)     … (6)

Then the first part of equation (6),

∑i=1^ N(miri^••)(δri/δqk) = ∑i=1^ N d/dt (miri^• δri/δqk – miri^•• d/dt (δri/δt)

(Note – d/dt (δri/δqk) = δ/δqk (δri/dt)

d/dt (δri/δqk) = δvi / δqk  …(A))

Putting the value from equation (A) we get,

∑i=1^ N(miri^••)(δri/δqk) = [∑i=1^ N d/dt (miri δvi / δqk ) – mivi δvi / δqk  ]      ….(7)

Now, putting these values of equation (7) in equation (6)

∑i=1^ N Piδri = (∑i=1^ N d/dt (mivi δvi / δqk) – mivi δvi / δqk .δ/δqk] dqk

And, ∑i=1^ N Piδri = (∑i=1^ N d/dt {(δ/δqk (∑i=1^ N 1/2 mi(vi.vi))} – δ/δqk{∑i=1^ N 1/2 mi(vi-vi)}dqk

∑i=1^ N Piδri = (∑i=1^ N d/dt {(δT/δqk^•) – δT/δqk] δqk    …(8)

Now put the values from equation (5) and (8) into (4) we get a new equation i.e.

∑i=1^ N Piδri = (∑i=1^ N{ Gk. δqk – ∑i=1^ N[ d/dt (δT/δqk^•) – δT/δqk] δqk] =0

∑i=1^ N Piδri = (∑i=1^ N{ Gk- ∑i=1^ N[ d/dt (δT/δqk^•) – δT/δqk }] =0    …(9)

Gk = { d/dt (δT/δqk^•) – δT/δqk }   ….(10)

This equation represent the general form of Lagrange’s equation.

For a conservative system the force from scalar quantity v,

F = -∇V = -i δv/δx -j δv/δy – k δv/δz

Which can be written as

Gv= -δv/δqk

Now put the value of Gk from above equation in equation (10),

-δv/δqk = { d/dt (δT/δqk) – δT/δqk}

-δv/δqk = { d/dt (δT/δqk^•) – δ/δqk(T- v}=0  …(11)

Since scalar potential v is the function of generalised coordinate qk only not depend on generalised velocity. We can write equation (11) as-

d/dt (δ/δqk(T- v)) – δ/δqk(T- v) =0   …(12)

We define as now function given by-

L= T-v

Then equation (11) can be written as,

[ d/dt (δv/δqk^•) – δv/δqk] =0

Where k = 1,2,3….n

This equation is known as Lagrange’s equation for conservative system.

They are n in number and there is one equation for each generalised coordinate. We must know the Lagrangian function L= T-v in generalised coordinate.

Conclusion

In this article we have discussed D’ Alembert’s principle and Lagrange’s equation from D’ Alembert’s principle along with its derivation. The aim of this article is to clarify the formula, derivation and theory of D’ Alembert’s principle in easiest way possible.

 

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