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# Lagrange's Equation

The Cartesian equations of motion of our dynamical system take the form

 (B.9)

for , where are each equal to the mass of the first particle, are each equal to the mass of the second particle, et cetera. Furthermore, the kinetic energy of the system can be written

 (B.10)

Now, because , we can write

 (B.11)

for . Hence, it follows that . According to the previous equation,

 (B.12)

where we are treating the and the as independent variables.

Multiplying Equation (B.12) by , and then differentiating with respect to time, we obtain

 (B.13)

Now,

 (B.14)

Furthermore,

 (B.15)

and

 (B.16)

where use has been made of Equation (B.14). Thus, it follows from Equations (B.13), (B.15), and (B.16) that

 (B.17)

Let us take the previous equation, multiply by , and then sum over all . We obtain

 (B.18)

where use has been made of Equations (B.9) and (B.10). Thus, it follows from Equation (B.6) that

 (B.19)

Finally, making use of Equation (B.8), we get

 (B.20)

It is helpful to introduce a function , called the Lagrangian, which is defined as the difference between the kinetic and potential energies of the dynamical system under investigation:

 (B.21)

Because the potential energy, , is clearly independent of the , it follows from Equation (B.20) that

 (B.22)

for . This equation is known as Lagrange's equation.

According to the previous analysis, if we can express the kinetic and potential energies of our dynamical system solely in terms of our generalized coordinates, and their time derivatives, then we can immediately write down the equations of motion of the system, expressed in terms of the generalized coordinates, using Lagrange's equation, (B.22). Unfortunately, this scheme only works for energy-conserving systems.

Next: Generalized Momenta Up: Classical Mechanics Previous: Generalized Forces
Richard Fitzpatrick 2016-01-25