Difference between equation of motion and constraint equation
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a system’s classical equations of motionare a set of differential equations describing its time evolution. The question seems to be about the choice of formalism from which those equations are derived: Lagrange’s equations, or Hamilton’s equations, (or Newton’s equations).
Lagrangian mechanics starts by considering a system’s coordinates and their time derivatives as the fundamental variables; constructing a “Lagrangian function” of those variables (basically kinetic minus potential energy); varying that (actually, its integral the “action”) to get Lagrange’s equations which relate symbolic derivatives of the Lagrangian function; and explicitly performing those to get the actual equations of motion.
Hamiltonian mechanics instead considers each coordinate and its associated “canonical momentum”, and constructs a “Hamiltonian function” which is the system’s energy. It is most useful and unambiguous to consider the Lagrangian as fundamental and the Hamiltonian as derived from it; but that isn’t required, and either way the formalism leads to Hamilton’s Equations, another set of symbolic derivatives. Explicitly carrying those out leads to the same final equations of motion.
For simple systems, it’s often easier to just identify the forces acting on each degree of freedom, and directly write down Newton’s equations F = dp/dt (= ma often). For more complicated situations, especially if constraints are involved, this is much harder. But either way, exactly the same final equations of motion result. The formalisms allow different things to be considered easily or with more difficulty, but don’t differ in what equations you end up solving. However identifying conserved quantities, important for many solutions, is much easier at the Lagrangian or Hamiltonian level, usually by considering symmetries.
Those were all classical equations. Quantum theory makes much further (and essential) use of these formalisms; very broadly, the Hamiltonian is most useful for non-relativistic quantum mechanics, and Lagrangian for relativistic field theory. The path integral foundation of quantum theory most naturally uses the action, incorporating a Lagrangian.
Lagrangian mechanics starts by considering a system’s coordinates and their time derivatives as the fundamental variables; constructing a “Lagrangian function” of those variables (basically kinetic minus potential energy); varying that (actually, its integral the “action”) to get Lagrange’s equations which relate symbolic derivatives of the Lagrangian function; and explicitly performing those to get the actual equations of motion.
Hamiltonian mechanics instead considers each coordinate and its associated “canonical momentum”, and constructs a “Hamiltonian function” which is the system’s energy. It is most useful and unambiguous to consider the Lagrangian as fundamental and the Hamiltonian as derived from it; but that isn’t required, and either way the formalism leads to Hamilton’s Equations, another set of symbolic derivatives. Explicitly carrying those out leads to the same final equations of motion.
For simple systems, it’s often easier to just identify the forces acting on each degree of freedom, and directly write down Newton’s equations F = dp/dt (= ma often). For more complicated situations, especially if constraints are involved, this is much harder. But either way, exactly the same final equations of motion result. The formalisms allow different things to be considered easily or with more difficulty, but don’t differ in what equations you end up solving. However identifying conserved quantities, important for many solutions, is much easier at the Lagrangian or Hamiltonian level, usually by considering symmetries.
Those were all classical equations. Quantum theory makes much further (and essential) use of these formalisms; very broadly, the Hamiltonian is most useful for non-relativistic quantum mechanics, and Lagrangian for relativistic field theory. The path integral foundation of quantum theory most naturally uses the action, incorporating a Lagrangian.
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