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Lagrange vs Hamilton: Clarifying the Distinction
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[QUOTE="vanhees71, post: 5510837, member: 260864"] The Hamilton principle of least action is different for the Lagrangian and the Hamilton version. In the Lagrange formulation the action functional is a functional of trajectories in [b]configuration space[/b], $$S[q]=\int_{t_1}^{t_2} \mathrm{d} t L(q,\dot{q},t).$$ The equations of motion are given by the stationary point of this functional for trajectories in configuration space with fixed boundaries, i.e., for ##\delta q(t_1)=\delta q(t_2)=0##, i.e., the Euler-Lagrange equations, $$\frac{\delta S}{\delta q}=0 \; \Rightarrow \; \frac{\partial L}{\partial q}-\frac{\mathrm{d}}{\mathrm{d} t} \frac{\partial L}{\partial \dot{q}}.$$ The formalism is form-invariant under arbitrary diffeomorphisms in configuration space, i.e., the EL equations are of the same form in any generalized configuration-space variables. In the Hamiltonian formalism of the least-action principle, you consider trajectories in [b]phase space[/b] and the variational principle is for variations of phase-space trajectories. The action functional reads $$A[q,p]=\int_{t_1}^{t_2} \mathrm{d} t [\dot{q} \cdot p-H(q,p,t)].$$ The equations of motion are the stationary points of this functional under variations of the phase-space trajectories ##(q,p)##, with the ##\delta p## arbitrary and ##\delta q(t_1)=\delta q(t_2)=0##. The equations of motion are the Hamilton canonical equations, $$\dot{q}=\frac{\partial H}{\partial p}, \quad \dot{p}=-\frac{\partial H}{\partial q}.$$ The transformations that leave these equations form-invariant are the much larger set of canonical transformations (symplectomorphisms on phase space), i.e., those transformations, which leave the canonical Poisson-bracket relations invariant, $$\{q^j,q^k \}=\{p_j,p_k \}=0, \quad \{q^j,p_k\}={\delta^j}_k,$$ where the Poisson bracket of any pair of phase-space functions ##A,B## is defined as (Einstein summation convention implied) $$\{A,B \}=\frac{\partial A}{\partial q^k} \frac{\partial B}{\partial p_k} - \frac{\partial B}{\partial q^k} \frac{\partial A}{\partial p_k}.$$ The great thing with the Hamilton formalism in phase space is that together with the Poisson brackets the function space of phase-space functions becomes a Lie algebra, and from the point of view of modern physics it's the most fundamental way to describe classical mechanics. With a little "deformation" (in the mathematical sense) you get quantum theory for (almost) free! [/QUOTE]
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Lagrange vs Hamilton: Clarifying the Distinction
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