Transformations of phase space

[Nusc]

Homework Statement

The phase flow is the one-parameter group of transformations of phase space

$$g^t:({\bf{p}(0),{\bf{q}(0))\longmapsto({\bf{p}(t),{\bf{q}(t))$$,

where $${\bf{p}(t)$$ and $${\bf{q}}(t)$$ are solutions of the Hamilton's system of equations corresponding to initial condition $${\bf{p}}(0)$$and $${\bf{q}}(0)$$.

Show that $$\{g^t\}$$ is a group.


Can anyone help me prove the composition?

$$g^t\circ g^s=g^{t+s}$$

Homework Equations

The Attempt at a Solution

 

[lippyka]

The phase flow can be defined as the one-parameter group of transformations of phase space from the initial conditions to the final conditions. The group satisfies the associative property, which states that for any two elements a and b in the group, the product a*b is also an element of the group. This means that for any two elements g^t and g^s, their composition must also be an element of the group. Therefore, we can say that g^t\circ g^s=g^{t+s}