Solving Hamiltonian equations using differential equations.

Suppose the system is under the influence of a conservative force. The energy is E=T(qi,˙qi)+V(qi)E = T(q_i, \dot q_i) + V(q_i) and TT is a homogeneous function of order 2 in the argument ˙qi\dot q_i. Show that a solution to ddtH=0\frac{d}{dt} H = 0 is:
L=T(qi,˙qi)−V(qi) L = T(q_i,\dot q_i) – V(q_i)

So far I have done the following to work out the problem:

The Lagrangian for a system with NN generalised co-ordinates qq has the form,
L=L(qi,˙qi,t): i∈1,2…..NL = L(q_i,\dot q_i,t):\space i\in {1,2…..N}

Since ddt=0\frac{d}{dt}=0; the Lagrangian does not explicitly depend on time. By chain rule, it follows that,Let p=∂L∂˙qip = \frac{\partial L}{\partial \dot q_i} ddtH=∂H∂p˙p+∂H∂qi˙qi+∂H∂t\frac{d}{dt}H = \frac{\partial H}{\partial p}\dot p + \frac{\partial H}{\partial q_i}\dot q_i + \frac{\partial H}{\partial t}
From this seems like I have to use the fact that T is a homogeneous quadratic function of ˙qi\dot q_i and derive the solution to the equation ddtH=0\frac{d}{dt}H=0 is to be L=T(qi,˙qi)−V(qi) L = T(q_i,\dot q_i) – V(q_i)
I am having trouble deriving the solution, seems like its solving 2n 1st order differential equations,any help or guide will be appreciated.

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