[NOTES/CM-04004] Summary of Lagrangian and Hamiltonian Formalisms

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A summary of Lagrangian and Hamiltonian formalisms is given in tabular form.

	Lagrangian Formalism	Hamiltonian Formalism

1). Basic Variables	\( q_{k}, \dot{q}_{k}\)	\( q_{k}, p_{k}\)

2). States	Rep by a point	Rep by a point
	in configuration space	in phase space

3). EOM	Euler Lagrange	Hamilton's EOM
	EOM	\( \dot{q}_{k} =\frac{\partial H}{\partial p_{k}}\)
	\( \frac{\partial L}{\partial {q}_{k}} - \frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q}_{k}} \right) = 0\)	\( \dot{p}_{k} = - \frac{\partial H}{\partial q_{k}} \)

4). Action	Hamilton's and	Wiess acion
Principle	Wiess action	Principle in phase
	Principle gives	space gives EOM
	Laws of motion

5). System	Contained in	Described \ by
Specific information	Lagrangian	Hamiltonian
(Interactions)

6). Full \ solution	Values of \(q_{k}, \dot{q}_{k}\) \ at	Values of \( q_{k}, p_{k}\)
requires	initial time	at initial time

7). Conservation	Related to	Generators of
\ Laws	Symmetries of \(L\)	transformations have
	in configuration	zero Poisson bracket
	space	with the Hamiltonian

4727:Diamond Point