Energy　conserving　and　dissipative　algorithm　designs　in　Hamilton＇s　canonical　equations　via　Petrov–Galerkin　time　finite　element　methodology　are　proposed　in　this　paper　that　provide　new　avenues　with　high-order　convergence　rate,　improved　solution　accuracy,　and　controllable　numerical　dissipation.　Lagrange　quadratic　shape　function　in　time　with　flexible　interpolation　points　are　considered　to　approximate　the　solution　over　a　time　interval.　Instead　of　specifying　the　weight　functions,　two　algorithmic　parameters,　namely,　a　principal　root　(ρq∞)　and　a　spurious　root　(ρp∞),　are　introduced　to　formulate　a　generalized　weight　function,　which　enables　us　to　introduce　controllable　numerical　dissipation　with　respect　to　displacement　and　momenta　while　preserving　high-order　convergence,　and　features　with　improved　solution　accuracy.　A　family　of　third-order　accurate　time　finite　element　algorithms　with　controllable　dissipation　and　improved　solution　accuracy　is　presented　in　both　the　homogeneous　and　non-homogeneous　dynamic　problems;　and　via　setting　(ρq∞,ρp∞)=(1,1),　this　third-order　family　of　algorithms　directly　leads　to　a　new　family　of　fourth-order　accurate　non-dissipative　algorithms　in　general　homogeneous　problems;　and　the　fourth-order　accuracy　is　also　preserved　in　non-homogeneous　problems　when　the　third-order　time　derivative　of　the　external　excitation　has　the　order　of　O(Δtn)(n≤1).　Numerical　examples　are　performed　to　demonstrate　the　pros/cons　for　the　conserving　properties　of　various　schemes　in　the　proposed　Petrov–Galerkin　time　finite　element　of　algorithms.　©　2020　Elsevier　B.V.