B. Lagrange equations

Jaime E. Villate. Dynamics and Dynamical Systems,
University of Porto, Portugal, 2019

This appendix shows the derivation of Lagrange equations from Newton's second law. Let us consider a system formed by m rigid bodies whose center of mass position vectors are: r 1 , r 2 , …, r m . Thus, to determine the configuration of the whole system at a given instant, it is necessary to know the values of 3 m coordinates, which could be a combination of distances and angles.

The system is called holonomic, if some of those 3 m are related, making it possible to reduce the number of independent coordinates to n generalized coordinates ( n < 3 m ):

q 1 ( t ) , q 2 ( t ) , . . . , q n ( t )

Each position vector r i can depend on some of the generalized coordinates and on time:

r i ( q 1 , q 2 , . . . , q n , t )

and the velocity vector of the i th body is then

v i = d r i d t = ∂ r i ∂ t + n k = 1 ∂ r i ∂ q k ˙ q k

Therefore, v i depends also on time, the generalized coordinates, and the generalized velocities ˙ q i :

v i ( q 1 , q 2 , . . . , q n , ˙ q 1 , ˙ q 2 , . . . , ˙ q n , t )

and the partial derivatives of v i are then

(B.1)
∂ v i ∂ ˙ q j = ∂ r i ∂ q j ∂ v i ∂ q j = ∂ r i ∂ q j ∂ t + n k = 1 ∂ 2 r i ∂ q j ∂ q k ˙ q k

The acceleration vector of the i th rigid body is:

(B.2)
a i = d v i d t

If at a given instant the value of each coordinate q j is changed to q j + δ q j , where the virtual displacements δ q j are consistent with the constraints, each position vector will undergo a displacement:

(B.3)
δ r i = n j = 1 ∂ r i ∂ q j δ q j

Taking the dot product of each side of this equation times the sides of equation B.2, we obtain

(B.4)
a i · δ r i = n j = 1 d v i d t · ∂ r i ∂ q j δ q j

since the derivative of the product v i · ∂ r i / ∂ q j is,

d d t v i · ∂ r i ∂ q j = d v i d t · ∂ r i ∂ q j + v i · d d t ∂ r i ∂ q j = d v i d t · ∂ r i ∂ q j + v i · ∂ r i ∂ q j ∂ t + n k = 1 ∂ 2 r i ∂ q j ∂ q k ˙ q k

According to equations B.1, the derivative ∂ r i / ∂ q j and the term inside the parenthesis on the right-hand side of the equation are the partial derivatives of v i , with respect to ˙ q j and q j . Therefore, the following result is obtained

d d t v i · ∂ v i ∂ ˙ q j = d v i d t · ∂ r i ∂ q j + v i · ∂ v i ∂ q j

Equation B.4 can then be written as,

(B.5)
a i · δ r i = n j = 1 d d t v i · ∂ v i ∂ ˙ q j − v i · ∂ v i ∂ q j δ q j

Furthermore, notice that the partial derivatives of v 2 i with respect to the generalized coordinates and velocities are:

∂ v 2 i ∂ q j = ∂ ( v i · v i ) ∂ q j = 2 v i · ∂ v i ∂ q j ∂ v 2 i ∂ ˙ q j = ∂ ( v i · v i ) ∂ ˙ q j = 2 v i · ∂ v i ∂ ˙ q j

substituting these two expressions in equation B.5 and multiplying both sides of the equation by the mass m i of the i th body we get,

m i a i · δ r i = n j = 1 d d t m i 2 ∂ v 2 i ∂ ˙ q j − m i 2 ∂ v 2 i ∂ q j δ q j = n j = 1 d d t ∂ E c i ∂ ˙ q j − ∂ E c i ∂ q j δ q j

where E c i is the kinetic energy of the i th body. According to Newton's second law, m i a i is the total force acting on the i th body. Using the expression B.3 and summing over all the i th body, we obtain

m i = 1 n j = 1 F i · ∂ r i ∂ q j δ q j = n j = 1 d d t ∂ E c ∂ ˙ q j − ∂ E c ∂ q j δ q j

which leads to the Lagrange equations:

(B.6)
d d t ∂ E c ∂ ˙ q j − ∂ E c ∂ q j = Q j j = 1 , . . . n

where E c is the total kinetic energy of the system and the generalized force Q j has been defined as

(B.7)
Q j = i F i · ∂ r i ∂ q j