Mini Physics

When things go wrong, as they sometimes will,
When the road you’re trudging seems all uphill,
When the funds are low and the debts are high,
And you want to smile, but you have to sigh,
When care is pressing you down a bit,
Rest, if you must, but don’t you quit.

Life is queer with its twists and turns,
As every one of us sometimes learns,
And many a failure turns about,
When he might have won had he stuck it out;
Don’t give up though the pace seems slow–
You may succeed with another blow.

Often the goal is nearer than,
It seems to a faint and faltering man,
Often the struggler has given up,
When he might have captured the victor’s cup,
And he learned too late when the night slipped down,
How close he was to the golden crown.

Success is failure turned inside out–
The silver tint of the clouds of doubt,
And you never can tell how close you are,
It may be near when it seems so far,
So stick to the fight when you’re hardest hit–
It’s when things seem worst that you must not quit.

- Author unknown

 

For most system, the Lagrange function is $L = T – V$, where T is kinetic energy of the system and V is potential energy of the system.

Euler-Lagrange Equation for $L(t,q_1 , … , q_n , \dot{q_1}, … , \dot{q_n})$ is:

$$\frac{\partial L}{\partial q_i} = \frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q_i}} \right), \, i = 1, … , n$$

In cases with constraints, let C be constraint. The E-L equation becomes:

$$\frac{\partial L}{\partial q_i} – \lambda \frac{\partial C}{\partial q_i}= \frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q_i}} \right), \, i = 1, … , n$$

____________

Steps to find Hamilton function:

1. Find $L = T – V$
2. Conjugate momentum: $p_i (q_i, \dot{q_i}, t) = \frac{\partial L}{\partial \dot{q_i}}$
3. $H = \sum_{i}^{} \dot{q_i} p_i – L$
4. Sub. $\dot{q_i}$ into the equation in 3.

Hamilton’s equation of motion:

$$\dot{x_k} = \frac{\partial H}{\partial p_k}$$

$$\dot{p_k} = \, – \frac{\partial H}{\partial x_k}$$

$$\dot{H} = \frac{\partial H}{\partial t}$$

Back To Classical Mechanics