How is This Substitution Made in the Langrangian Derivation? the derivation the following equation is given as a statement of D'Alembert's principle: \vec(F)_(text(net)) * delta vec(r)=m (d^2vec(r))/(dt^2) * delta vec(r)

Makayla Eaton

Makayla Eaton

Answered question

2022-08-10

The derivation the following equation is given as a statement of D'Alambert's principle:
F net δ r = m d 2 r d t 2 δ r
After which the following equation is used: ( x y ) = x y + x y to rewrite the right hand side so that:
m d 2 r d t 2 δ r = m [ d d t ( d r d t δ r ) d r d t d δ r d t ]
What is x, what is y and how is the equation applied?

Answer & Explanation

Dereon Parker

Dereon Parker

Beginner2022-08-11Added 11 answers

x and y are some functions of variable t. The derivative is with respect to this variable, so x′ is equivalent to d x d t . The formula for (xy)′ is the derivative of a product of functions. In this case they use
x = d r d t
and
y = δ r
Then
d d t ( d r d t δ r ) = ( d d t d r d t ) δ r + d r d t d d t δ r
They just move the last term to the other side
erkentrs

erkentrs

Beginner2022-08-12Added 3 answers

( x y ) = x y + x y
is just the product rule in lagrangian notation, and can be used here as x,y are both functions of t,time

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