- #1

selim

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In Hamilton's "on a general method in dynamics", he starts with varying the function ##U## and writes the equation:

$$\delta U=\sum m(\ddot x\delta x+\ddot y\delta y+\ddot z\delta z)$$

Then he defines ##T## to be:

$$T=\frac{1}{2}\sum m (\dot x^2+\dot y^2+\dot z^2)$$

Then by ##dT=dU##, he writes:

$$T=U+H$$

Then he varies T and writes:

$$\delta T= \delta U+\delta H$$

note that he is also varying in the initial conditions, that's why he did not omit the term ##\delta H##.

Hamilton then multiplies this expression by dt and integrates and writes it as:

$$\int\sum m(dx \delta \dot x+dy \delta \dot y+dz \delta \dot z)=\int\sum m(d \dot x \delta x+d \dot y \delta y+d \dot z \delta z)+\int\delta H dt$$

Then comes the part where I got confused. He says "that is, by the principles of the calculus of variations" and writes:

$$\delta V=\sum m(\dot x \delta x+\dot y \delta y+\dot z \delta z)-\sum m(\dot a \delta a+\dot b \delta b+\dot c \delta c)+\delta H t$$

where (x,y,z) and (a,b,c) are final and initial conditions then he denotes V by the integral:

$$V=\int\sum m(\dot x \delta x+\dot y \delta y+\dot z \delta z)$$

My questions are as follows:

1-how did he get ##\delta V##, what "principle of the calculus of variations" did he use?

2-then how from that did he get the integral ##V##?

$$\delta U=\sum m(\ddot x\delta x+\ddot y\delta y+\ddot z\delta z)$$

Then he defines ##T## to be:

$$T=\frac{1}{2}\sum m (\dot x^2+\dot y^2+\dot z^2)$$

Then by ##dT=dU##, he writes:

$$T=U+H$$

Then he varies T and writes:

$$\delta T= \delta U+\delta H$$

note that he is also varying in the initial conditions, that's why he did not omit the term ##\delta H##.

Hamilton then multiplies this expression by dt and integrates and writes it as:

$$\int\sum m(dx \delta \dot x+dy \delta \dot y+dz \delta \dot z)=\int\sum m(d \dot x \delta x+d \dot y \delta y+d \dot z \delta z)+\int\delta H dt$$

Then comes the part where I got confused. He says "that is, by the principles of the calculus of variations" and writes:

$$\delta V=\sum m(\dot x \delta x+\dot y \delta y+\dot z \delta z)-\sum m(\dot a \delta a+\dot b \delta b+\dot c \delta c)+\delta H t$$

where (x,y,z) and (a,b,c) are final and initial conditions then he denotes V by the integral:

$$V=\int\sum m(\dot x \delta x+\dot y \delta y+\dot z \delta z)$$

My questions are as follows:

1-how did he get ##\delta V##, what "principle of the calculus of variations" did he use?

2-then how from that did he get the integral ##V##?

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